searchindex.js

Search.setIndex({envversion:42,terms:{finite_r:[5,34],interchang:[33,50,40],four:[14,45,17,18,40,25,53],secondli:39,greater_tuple_degrevlex:45,whose:[10,12,8,2,14,20,5,33,34,52,27,40,25,13],typeerror:[8,2,14,45,20,5,49,18,19,33,34,6,40,53,39,52,13],sorri:52,interval_root:9,swap:[7,51,13],under:[23,33,40,14,22],some_el:52,worth:25,digit:21,everi:[2,14,17,34,25,13],risk:52,polynomial_element_gener:[41,0,27,42,48,21,52],map_coeffici:[5,34],jacm:26,sage_object:[23,45],upstream:[51,13],affect:[13,53,2,25,9],"_lmul_":[35,0,21,39,34],fractal:14,kbase_libsingular:24,mpolynomialideal_singular_base_repr:14,disturb:21,min_max_delta_intvec:25,polynomial_singular_repr:[37,29,53,42],upload:21,kbar:21,factori:[23,12,13,25],vector:[10,13,14,45,17,21,52,15,24,25],greater_tuple_block:45,matric:[12,8,25],zdd:13,though:[38,40,36,19],initialis:[18,19],c_bitsiz:25,realintervalfield:[34,52,25,6],naiv:34,ngen:[12,8,2,18,19,23,52,13],direct:[],nski:25,consequ:[33,14],second:[36,2,14,45,40,17,53,7,51,50,25,13],neg_err:25,chen:[5,8],ill:34,is_singular:14,absprec:[1,27],volker:[7,2,14],even:[2,14,36,19,32,33,34,23,40,25,13],euclidean:[42,34,39,27],neg:[49,0,45,42,20,15,5,34,33,21,35,52,6,39,25],steiner:14,"new":[41,0,8,2,14,42,20,5,17,19,34,27,21,52,7,53,39,50,25,13],symmetr:[],linear_map:25,told:25,var_arrai:6,elimin:[17,13,14],behavior:[48,2,14,53],never:[23,52,25],here:[10,2,14,20,5,17,19,33,34,6,23,27,40,25,13],groebner_fan:14,accur:25,path:25,interpret:[23,33,39,14],gens_valu:19,anymor:52,characterst:34,subsitut:2,precis:[41,0,43,9,14,36,19,47,34,52,8,27,53,39,40,25,26],credit:20,vishkautsan:[5,8],fraction_field:[34,17,18,52,14],slimgb:[40,13,14,24],portabl:[9,25],carl:[9,25],compare_tuples_matrix:45,gpol:51,strat:[51,13],interreduc:[40,17,14],linearli:[10,17,25],residue_field:[3,52],stdhib:[40,14],infinite_polynomial_r:[23,33,40],ord:34,unit:[41,0,43,2,42,20,5,19,33,34,35,27,53,39,13],plot:[34,14],describ:[10,13,14,34,33,21,52,40,25],would:[12,2,5,17,33,21,23,53,15,25,34],taylor_shift1_intvec:25,is_simpl:5,has_coerce_map_from:52,hensel:39,c112:13,vol:[10,8],call:[0,2,5,6,7,8,12,13,14,15,17,18,19,21,23,25,27,28,45,20,32,33,34,35,37,38,39,41,42,43,49,40,51,52,53],k003:[17,13],type:[0,2,5,9,13,14,39,18,19,21,22,23,27,33,34,35,15,42,43,49,40,52,53],until:[23,40,34,52,25],matlab:52,relat:[],mod_var_set:13,notic:[34,25,14],libsingularopt:14,warn:[39,14,6],exce:32,w201:17,"_is_category_initi":12,hold:[14,33,34,40,39,25],unpack:[5,8],gf2x:[],must:[0,26,5,6,2,12,13,14,17,18,19,21,23,25,20,33,34,35,39,41,8,47,40,52,53],beta_1:23,beta_0:23,beta_3:23,beta_2:23,beta_5:23,beta_4:23,err:25,unpickle_mpolynomialring_gener:8,generalis:8,flori:52,magma_str:45,norm:[37,34],max_index:33,hansen:[23,33],root:[],pierr:52,overrid:[41,0,5,34],i_n:33,divided_differ:52,i_j:33,give:[8,2,14,45,33,34,35,52,40,51,15,25,13],has_root:25,want:[13,14,5,34,50,52,27],boothbi:[20,38],david:[41,0,14,28,42,3,45,36,18,20,21,15,34,53],unsign:39,groebnernew2:7,krull_dimens:[23,8,18,19,52],vanish:[5,17,8,13,14],end:[15,34,39,25],is_block_ord:45,thing:[0,12,28,2,25],ordinari:34,hom:[5,2,28,19],how:[12,27,14,40,51,39,25,2],schonhag:36,answer:[12,27,14,33,34,40,25],verifi:[41,0,9,19,6,15,53],ancestor:25,conwaysconst:25,wolfram:25,x500:2,new_nr:12,recogn:14,chines:21,after:[17,18,34,6,40,51,25],diagram:13,befor:[41,0,27,14,21,6,39,25],wrong:[12,2,28,32,34,40,39,25],adic:[],all_don:25,parallel:13,i_1:33,attempt:[10,0,9,14,20,34,41,21,23,7,51,25],third:[51,39,25],interpol:[13,52],add_up_polynomi:13,prec_degre:27,clean_top_by_chain_criterion:13,polynomialbaseringinject:34,think:[12,2,36,34,25,53],maintain:25,zz_px:21,enter:34,fan:14,unramifi:19,order:[],oper:[41,13,14,48,34,49,15,25],composit:[21,35,34],is_squar:34,atiyah:[42,34],over:[],failur:25,becaus:[8,13,45,15,34,18,33,21,6,37,39,25,27],nonzero_posit:15,monomial_coeffici:[12,2,20,13,15,53],phi:[34,19],flexibl:[12,17,2,53],vari:25,logging_not:25,"_cache_kei":34,clo:[5,8],fit:[34,39,13,25],zmodn:21,fix:[0,1,9,6,2,12,13,14,17,19,23,25,27,32,34,35,38,43,49,51,52,53],monomial_pairwise_prim:[12,2,28],"__class__":19,better:[52,12,2,25,53],trager:14,mf99:52,persist:19,hidden:21,easier:25,them:[13,45,34,19,33,48,21,6,40,39,25,9],var1:6,thei:[41,12,9,5,17,19,34,52,6,23,40,25],var2:6,proce:[40,34],promin:17,database_gap:39,safe:15,"break":[17,52,6],promis:17,interrupt:[34,49],suggest_plugin_vari:13,choic:[10,18,34,52,6,50,39,25],alex:17,unpickl:[12,13],bonu:21,cyclotomic_coeff:[32,52],each:[12,8,9,14,45,53,50,5,17,18,19,34,21,6,23,27,51,13,25,2],debug:[34,25],side:[23,12,2,14],mean:[8,13,14,20,40,34,33,21,52,23,50,24,25],prohibit:53,slen:25,bradshaw:[29,32,26,28,42,20,47,21,35,38,34,53],buchberg:[40,7,51,13,14],coerceabl:39,laurentpolynomial_univari:20,lavasani:29,logo:14,fast_when_tru:47,is_singleton_or_pair:13,emul:15,goe:[20,39],newli:[50,19],gf2x_buildsparseirred_list:26,crucial:[50,25],content:[41,0,27,45,5,34,39],rewrit:[41,0,21,15],newton_slop:[34,27],eprint:17,zentral:13,got:[35,0,34,39],sqr_pd:38,forth:19,compare_tuples_block:45,laurent_polynomi:20,log2:25,linear:[10,13,14,17,32,34,25],navig:13,situat:[42,34,52],infin:[42,8,13,14,34,17,19,21,52,23,25,27],free:[41,0,12,2,14,34,35,23,40,24,25],ineffici:52,nth:52,"_groebner_basis_libsingular":12,ntl:[],"65211563729e":34,infix:34,a122:13,silent:[8,13,6],lsign:25,gwalk:14,angl:[13,14],traceback:[0,2,5,6,12,13,14,39,18,19,23,27,28,45,20,33,34,35,37,8,15,41,42,43,49,40,52,53],good_input:50,filter:[51,34],ish:13,isn:[52,53],inverse_mod:[5,34],mult_fact_sim_c:13,subtl:34,confus:[23,2,22],sageobject:[23,45],polyhedron:5,kemper:14,rang:[26,0,36,2,14,33,9,34,41,47,27,21,35,52,49,7,25,13],get_msb_bit:25,pyrex:[38,28],grade:[12,45,2,14,13],independ:[10,5,17],wast:6,rank:[17,34,45],restrict:[12,2,14,5,17,33,53,15,13],unlik:2,alreadi:[12,2,14,5,19,34,50,40,25,13],messag:[21,6],robust:[34,14],principalidealdomain:52,agre:[20,52],primari:14,hood:22,ans_pari:34,cartesian:13,rewritten:53,de_casteljau_doublevec:25,top:[32,39,13,25],sometim:[34,2,25],else_branch:13,toi:[40,14],too:[13,14,21,35,39,25,53],tol:[34,25],tom:[20,38],john:[10,14],f_1:[5,53,2,13],precisionerror:[39,34,43,27,25],toon:[12,2,28],tool:25,is_nilpot:34,inexact:[34,14],somewhat:[41,0,2,34,22,25,13],technic:[34,6],monomial_lcm:[12,2,28],hillar:[50,40],ebar:13,target:25,keyword:[12,17,2,14,34],provid:[2,5,6,7,12,13,14,17,19,21,23,25,45,20,32,34,8,15,50,43,40,52,53],nzz:2,project:[5,17,8,14],matter:34,solomon:[5,8],is_laurentpolynomialr:18,uniniti:49,boolemonomi:13,w210:17,pretend:34,raw:[12,13],seed:[13,25],further:[50,39,25],seen:[23,13,19],seem:[13,25],incompat:[23,29,45],incompar:33,no_list:27,boolesetiter:13,ellipticcurv:[39,34,43,14],strength:13,burcin:[0,12,35,13],inter_reduct:[7,17,14],unhash:34,galoi:39,latter:[21,34],mpolynomialideal_singular_repr:14,thorough:39,iterindex:13,conwai:[52,25],factor_mod:[41,0,39,27],monomialfactori:13,fuzzi:34,thoma:[7,14],thoroughli:23,polynomialring_dense_padic_ring_capped_absolut:52,simplifi:[23,50],shall:[20,50,6],object:[2,14,40,5,17,18,19,34,32,33,21,52,6,23,38,7,15,50,25,13],palindrom:32,compare_tuples_degneglex:45,unsophist:20,monomi:[2,5,7,10,12,13,14,17,21,23,24,28,45,20,33,34,15,40,42,8,50,51,53],regular:[51,13,14],s001:[17,13],specifi:[10,0,12,8,9,14,45,20,5,18,32,34,35,52,6,50,53,39,13,25,2],get_var_map:13,s002:[17,13],sparsiti:6,jean:[52,14],tradit:[5,8],random_set:13,doi:34,don:[23,17,15,21,49],split_for_target:25,doc:[23,13],add_gener:[50,13],polynomial_generic_dense_field:[42,48,52],doe:[2,6,7,10,12,13,14,39,17,19,21,25,27,28,20,33,34,35,15,40,42,49,50,52],declar:19,yann:47,left:[12,14,15,34,21,35,6,37,40,39,25],sum:[26,8,2,14,20,15,5,36,47,21,35,53,39,34,13],came:34,quadraticfield:[34,19,14],opposit:[23,13,25],chapui:47,random:[41,0,8,2,14,9,5,17,19,20,21,52,51,25,13],sage:[0,1,2,3,5,6,7,8,9,10,29,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,45,30,32,33,34,35,36,37,38,39,40,41,42,43,47,48,49,50,51,52,53],a_5:23,syntax:[14,45],"_mul_trunc_":[35,0,21,39,34],nuigalwai:[23,33,50,40],protocol:[50,7,13,14],polynomial_singular_interfac:[29,28,42,37,52,53],involv:[2,14,45,20,5,19,37,34,23,40,53,25,13],absolut:[21,48,34,52,27,25,2],new_cr:12,pychkin:17,greater_tuple_deglex:45,greater_tuple_matrix:45,"__call__":[34,8,13],is_commut:[12,8],quotient_by_principal_id:[19,52],other_r:14,ncpolynomial_plur:12,"8x8":21,stop:[25,14],infti:40,congruent:27,cryptographi:13,report:[17,33,50,40,25,53],net:13,bar:2,"_add_":[35,0,21,39,34],"public":21,twice:[2,25],bad:[5,8],septemb:52,fair:52,absolute_diamet:[9,25],result:[0,2,5,6,7,9,12,13,14,21,23,24,25,27,28,45,32,34,35,39,40,41,8,47,50,52,53],greater_tuple_lex:45,respons:[23,25],ybar:12,hash:[12,15,13],weispfen:[7,14],best:[0,50,25,14],subject:2,symmetris:40,awar:[48,12,2,53],tensor:23,hopefulli:34,databas:52,irt:9,figur:[5,53],greater_etupl:15,unpickle_booleanpolynomial0:13,mul:[5,17,2,14,53],awai:25,approach:[7,34,25,14],is_mon:34,attribut:[19,45],accord:[23,50,25],extend:[12,13,14,34,47,33,21,35,27,25,26],pointer:[12,18,13,6],xrang:[34,13,52],extens:[29,14,42,17,19,37,48,34,22,23,15,52,53],lazi:[13,52],add:[13,14,34,52,50,51,15,25],newr:12,lex_lead:13,first_term:13,reduced_normal_form:13,dictref:15,cox:[8,14],howev:[10,12,2,14,53,40,5,19,33,34,6,23,7,51,50,25,13],"97936298566e":34,against:[23,2,13],shimoyama:14,dictionari:[12,2,14,20,16,17,32,33,34,28,23,53,15,5,52,13],hamming_weight:34,unpickle_booleanpolynomialr:13,maxlength:17,rshift_coeff:27,uni:[29,21,13,53],com:[23,33,13,25],splitting_field:34,suggest:[38,40],widen:25,kwd:[12,8,2,14,20,17,18,19,21,28,52,23,35,34,13],deglex:[12,8,2,45,17,18,6,23,53,13],height:32,linear_represent:10,permut:[23,33,50,39,40],theoret:13,universal_discrimin:34,assum:[10,12,2,28,20,19,34,23,50,39,40,25,13],speak:34,bp1:25,degener:[5,8],termorder_from_pb_ord:13,union:[17,13,14],numpi:[34,9,49],three:[14,5,18,19,6,50,24,25],been:[41,0,12,1,2,45,17,19,34,35,52,6,23,15,25,27],much:[2,14,17,34,23,40,15,25],interest:[13,25],basic:[19,48,34,23,50,25],tini:9,quickli:[32,25],macaulay2:[14,28,45,40,52,53],suppress:13,base_ring_categori:6,ani:[17,2,6,7,10,12,13,14,39,36,19,23,25,27,45,20,32,33,34,15,50,40,51,52,53],rational_reconstruct:35,lift:[2,14,5,17,19,37,53,39,27],"catch":2,mpolynomialid:[7,17,13,14],ident:[10,0,14,18,32,34,52,6,40,39,25,53],properti:[25,18,13,14,6],sourceforg:13,lesser:25,freealgebra:[12,14],calcul:[],pairwis:[12,14,2,28,17,9],finder:34,bnd:25,davenport:34,laurentpolynomialring_mpair:18,need:[41,0,8,13,14,45,5,34,19,21,52,53,23,51,15,25,27],zmod:[21,14,34,35,52,2],kwarg:[17,8,34],cone:14,exterior:12,gianni:14,precompute_degree_reduction_cach:25,sever:[12,14,8,2,28,16,17,33,34,50],class_group:19,ezgcd:2,incorrectli:14,perform:[36,8,26,14,33,40,17,27,48,34,52,37,38,7,50,25,13],lex_lead_deg:13,make:[41,0,13,28,45,20,17,18,19,33,34,37,52,6,23,53,15,25,2],nonzero_valu:15,complex:[],split:[41,0,43,42,34,17,21,23,25],big:[23,0,34,41],complet:[41,0,12,8,36,18,15,52],"92238652153286e225":34,xbar:[12,19,52],fairli:[25,6],rais:[10,41,2,14,42,20,15,33,34,35,52,23,27,53,39,40,37,13],refin:[9,25,19],studi:[17,14],tune:35,kept:[8,13,25],thu:[13,14,17,19,33,22,23,51,53],inherit:[29,8,13,18,19,47,33,35,22,23,26],fraunhof:13,cheap_reduct:13,greatest:[26,0,2,41,47,33,21,35,39,34,13],thi:[0,36,2,3,5,6,7,8,26,10,29,12,13,14,9,15,17,18,19,20,21,23,25,27,28,45,30,32,33,34,35,37,38,39,40,41,42,43,47,49,50,51,52,53],lagrang:[52,53],everyth:[23,17],quotient:[],unchang:14,identifi:14,just:[29,13,14,20,5,36,18,32,34,6,25],irreducible_el:52,interval_bernstein_polynomial_float:25,minimal_weight:52,newton:[5,34,27,52],constant_coeffici:[12,2,20,13,34,53],root_bound:25,extend_vari:52,yet:[34,13,25],previous:[50,52],vars_set:13,add_as_you_wish:13,easi:[10,13,21,6,7,51,39,34],hao:[5,8],bw93:[7,51,14],conjug:34,polynomial_relative_number_field_dens:[48,52],jefferson:17,s003:[17,13],els:[20,34,13,14],expon:[42,12,2,14,45,20,15,16,33,21,28,39,34,53],compare_tuples_wdeglex:45,change_r:[12,8,2,14,20,5,18,53,21,52,49,51,39,34,27],applic:[23,21],esub:15,bd89:34,preserv:[10,33,50,40,6],unpickle_mpolynomialring_libsingular:2,poliscycloprod:34,raphson:34,background:50,lcm:[41,0,12,2,28,5,19,34,7,39],is_polynomi:34,specif:[],arbitrari:[36,14,30,17,34,6,25],lead_divisor:13,bpf:25,manual:[39,14],pdiv:34,noth:[2,13],mpolynomialring_polydict:[28,53],minpoli:37,sebastian:39,polynomial_number_field:[48,52],unnecessari:13,underli:[41,0,12,14,42,20,33,48,21,23,50,15,34,53],fcp:37,www:[21,13,25],right:[26,0,12,32,2,14,20,15,5,34,41,47,21,35,37,27,53,39,40,25,13],nilpot:[42,34],zca:13,deal:[34,13,14,25],scalar_lmult:15,interv:[41,0,9,15,34,52,6,39,25],linear_algebra_in_last_block:13,intern:[21,14,34,39,25,13],g_i:13,g_j:13,polynomialring_dense_padic_ring_gener:52,indirect:[12,2,14,45,17,34,6,23,53,39,13],prime_divisor:[21,34],witti:[9,25],total:[12,2,14,45,5,34,40,53,25,13],"_right":21,termorder_from_singular:45,subclass:[5,34,25],convert_map_from:34,canni:8,ideal_1poly_field:3,condit:[17,2,34,13],polynomialquotientring_field_with_categori:19,quo_rem:[41,0,2,42,20,19,47,21,35,49,27,53,39,34,26],plu:[20,0,14],hyperellipticcurv:14,get_base_order_cod:13,g_1:[51,13],confer:14,g_2:51,quaternion:52,descart:25,post:9,"super":12,mpolynomialring_libsingular:[12,2,52],chapter:14,alexand:[12,21,13],slightli:[34,39,9,25,19],precision_rel:27,surround:15,unfortun:9,algo:34,span:[17,13,14],pow_pd:38,produc:[23,8,27,14],ppc:34,soc:40,subr:[8,13],xyz:[28,14],"float":[34,13,14,25],encod:13,bound:[32,9,17,47,21,52,49,25,13],newton_raphson:34,down:[23,34,21,13,25],jena:13,old:19,symmgb_f2:13,manfr:[5,8],accordingli:19,moeller:[7,51,14],suffici:[39,27,25],support:[29,12,8,9,14,45,5,17,34,21,28,52,6,35,53,15,25,2],polynomialring_dense_padic_ring_lazi:52,gens_dict:[23,52],why:34,avail:[8,13,14,34,17,18,47,21,52,6,23,40,25],stuck:2,reli:[10,34],formul:10,fraction:[14,18,19,34,15,52],overhead:[40,14],stopiter:[15,13],requirefield:14,lowest:[10,20,21],head:[51,7,12,2,28],dr_cach:25,form:[10,0,12,2,14,20,34,17,19,32,33,21,53,23,40,51,39,25,13],offer:15,forc:[48,50,17,13,45],epsilon:[21,34],xgcd:[41,0,26,42,19,47,21,35,39,34,27],kroneck:[],maxim:[12,8,2,17,33,34,23,40,52,13],setgen:50,"true":[0,1,2,3,5,6,7,26,10,29,12,13,14,9,15,17,18,19,21,23,25,27,28,45,20,32,33,34,35,36,37,8,39,40,41,42,43,47,48,49,50,51,52,53],polynomialsequence_gener:[10,17,14],reset:50,base_field:[2,19],curcnt:12,emin:15,bugfix:2,coeff:[12,2,28,34,25,53],maximum:[12,8,2,14,5,21,52,27,53,15,25,13],"_mul_karatsuba":52,exp_err_shift:25,erdo:32,"37a1":43,symmetric_id:40,is_unique_factorization_domain:52,featur:[28,14,53],classic:[23,33,40,36],skip_squarefre:[9,25],hardcod:48,"abstract":[19,25],polynomial_default_categori:6,proven:25,exist:[41,8,2,14,45,5,34,6,25,13],localvar:6,cover_r:[13,19],check:[0,1,2,3,5,9,10,12,13,14,17,25,27,32,33,34,35,37,8,15,42,43,48,49,40,52,53],reductionstrategi:13,encrypt:21,groebnerstrategi:13,rationalfield:[51,2,19],basis_is_groebn:[7,17,14],when:[26,5,6,2,13,14,39,17,19,21,25,45,33,34,35,37,15,40,8,47,50,51],refactor:[12,2,14,45,17,53],role:13,cube_root:34,test:[0,1,2,5,6,7,9,29,12,13,14,15,17,18,19,21,23,24,25,27,28,45,20,32,33,34,35,37,8,39,40,41,42,43,48,49,50,51,52,53],lui:48,presum:15,shrink:25,degreerank:45,symmetric_basi:40,intend:[32,38,12],some_spolys_in_next_degre:13,ringhomomorphism_from_bas:30,warp:25,consid:[10,41,43,13,14,45,15,34,17,33,21,52,53,7,51,39,40,25,27],in_dict:[20,13,14],uniformli:[8,13,14],is_polynomialsequ:17,mathew:52,faster:[41,8,2,15,5,17,34,23,38,40,39,25,13],furthermor:13,anywher:17,max_exp:15,pseudo:23,flag:25,ignor:[12,8,2,14,5,34,28,6,25,53],freemonoid:15,time:[2,14,5,34,33,21,52,6,23,27,53,50,25,13],push:25,integermod_int:[5,34],backward:[8,45],mpfr:[],concept:[12,14],chain:[23,14],skip:[25,17,9,14,19],global:[13,45,20,5,18,34,6,25],roe:[3,18],signific:[38,25],polynomial_quotient_r:19,ring_el:[34,52],lll:21,polynomial_generic_spars:42,strzebo:25,row:[10,52,2,25],"__rshift__":27,decid:[10,13,25],bernstein_down:25,depend:[10,12,9,17,34,38,27,53,50,25,13],insatisfi:17,random_matrix:8,iacr:17,syzygi:14,intermedi:[13,14,40,50,51,7,25],rueth:34,opt_red_tail:13,quasi:33,decis:[41,0,13],userwarn:[34,14],booleanpolynomi:13,stdfglm:[40,14],isinst:[5,34,19],sourc:[13,14,25],string:[12,8,2,14,45,20,5,18,47,37,21,52,6,23,40,15,34,13],asymptot:[36,25],zxflint:6,isle_num:25,springer:[7,17,8,21,14],more_bit:25,booleanid:13,complex_embed:19,is_groebn:[17,14],word:[10,34,7,51,40,25],exact:[41,9,18,34,39,25,27],foo:[13,2,52,6],make_el:[47,21,35,26],level:[],did:[25,14],frobeniu:[2,28],m00:6,iter:[12,2,28,20,5,17,19,47,34,52,15,25,13],mhansen:[23,33],item:33,unsupport:45,booleanpolynomialid:13,quick:34,round:[33,34,21,25],mpolynomialring_polydict_domain:[28,53],prevent:[14,6],work:[10,8,2,14,40,5,17,19,32,33,34,52,49,27,15,50,25,13],slower:[8,13,14,17,21,15,25],courtoi:17,degneglex:[45,6],magali:14,sign:[41,0,8,5,15,25],krull:[23,52],cost:50,complex_interv:9,dbp:25,greater_tuple_wdeglex:45,bold:40,booleanpolynomialvector:13,appear:[10,42,12,8,2,14,20,17,34,27,53,15,13],unsort:15,current:[8,13,30,15,19,33,48,34,52,6,23,50,39,25],cyclic:[17,13,14,24],monomialconstruct:13,boost:13,padic:[1,43,27,52],deriv:[2,20,5,34,51,39,25,9],coerce_map_from:[30,34,2],"1275702593849246e":25,gener:[],coeffici:[0,2,5,6,7,26,10,12,13,14,9,15,16,17,19,21,25,27,28,20,32,33,34,35,38,39,41,42,8,47,49,40,51,52,53],satisfi:[13,52],linalgerror:34,explicitli:[2,18,33,35,6,50,40,25],tangent:34,address:[34,15],coincid:[40,5,13,14],pari_polynomi:34,make_polydict:15,wait:51,booleanmonomialiter:13,shift:[42,26,20,34,47,21,35,49,25,27],trial:34,behav:[12,15,2,19],compare_tuples_lex:45,weird:[48,2],deg:13,coeff_pd:38,semant:34,regardless:25,ipd:9,extra:[8,45,20,5,18,34,52,25],immediatli:13,modul:[11,12,2,14,29,30,45,32,21,35,6,23,40,15,24],kedlaya:[14,28,45,34,52,53],alpha_4:23,alpha_5:23,alpha_2:[23,33],alpha_3:23,alpha_0:[23,33],alpha_1:[23,33],instal:[7,51,14],"50746621104340e451":34,red:13,g_algfactori:12,prove:[9,25],"7755575615628914e":34,univers:[5,17,8,21,34],baser:45,prec:[41,0,43,9,18,19,34,8,39,52,27],polynomial_absolute_number_field_dens:[48,52],laurentpolynomial_gener:20,live:[8,2,5,17,19,34,23,15,13],criteria:[13,14,25],is_cyclotom:34,bigc:34,tightli:25,get_order_cod:13,plug:[47,35,26],finit:[2,3,5,6,7,29,12,13,14,17,18,19,23,24,27,28,33,34,37,8,40,52],b_100:33,nav_els:13,templat:[47,35,26],examin:[51,25],de_casteljau_intvec:25,citro:[37,19],graphic:[34,14],ulp:[34,25],cap:[],set_karatsuba_threshold:52,uniqu:[10,12,9,14,18,37,34,6,23,40,52,13],groebner_basi:[10,13,14,40,17,23,7,51,24],whatev:14,grbner:17,purpos:[10,23,34,37,45],fld:[34,9],print_po:43,then_branch:13,beta_10:23,stream:25,bernstein_expand:25,predict:9,curri:17,scrimshaw:20,yy_3:23,yy_2:23,yy_1:23,yy_0:23,topic:17,critic:[51,7,34,13],subregion:25,"3ss4s6":14,dummy_pd:38,grenet:42,occur:[2,14,20,17,33,34,23,50,53,40,13],verlag:17,alwai:[0,2,5,17,19,32,33,34,52,23,27,53,25,13],differenti:[20,5,34,8,52],multipl:[],weil_restrict:[17,14],first_lexicograph:52,write:[0,27,6],criterion:[7,51,14],pure:[53,34,36,22],unpickle_booleanpolynomi:13,nffactor:34,map:[10,36,2,14,9,5,17,19,33,34,52,53,25,13],product:[12,2,14,45,34,19,33,21,35,23,40,53,39,25,13],mat:34,max:[32,25,14],clone:13,is_equ:13,paper_35_ful:13,membership:[40,14],spol:[7,51],repeatedli:13,lenstra:52,polynomialring_dense_mod_n:52,t_0:10,goal:25,practic:[13,22],hensel_lift:[39,27],tbar:42,explicit:[25,6],marshal:7,algebraicr:[9,25],inform:[27,14,45,17,33,34,50,51,15,40,25],"switch":36,preced:[33,50,15,40],combin:[10,34,14,45],gamma:19,callabl:[33,5,17,34,38],zbar:12,talk:[41,0,25],poor_xgcd:34,pdict:15,variable_has_valu:13,thomson:52,x1_:23,equip:[23,33,40],singular_funct:12,still:[13,14,19,34,23,50,25],refine_al:25,tord:2,group:[19,34,23,40,39,52],thank:[34,19],polynomialring_dense_mod_p:52,sylvest:[5,34],kettner:25,std_grade:2,xx_1:23,xx_0:23,xx_3:23,main:[10,36,13,52],countabl:[23,33,40],small_spolys_in_next_degre:13,non:[0,9,5,6,2,29,12,14,39,36,21,25,45,32,33,34,35,15,40,8,50,51,52,53],recal:25,halv:25,get_verbos:7,initi:[10,32,12,43,2,45,3,17,47,30,48,34,35,26,38,7,51,39,24,25,9],sol:[17,13],is_polynomialquotientr:19,flint:[],half:[35,25],poly_modn_dens:21,refine_root:9,now:[2,14,34,19,21,52,40,51,39,25,13],hall:52,nor:20,introduct:14,laurent_series_ring_el:20,compare_tuples_negwdegrevlex:45,term:[],name:[2,3,5,6,29,12,13,14,17,18,19,23,25,28,45,20,33,34,37,8,52,53],booleanpolynomialentri:13,polynomialring_integral_domain:52,revers:[49,0,12,2,14,45,15,34,17,18,32,33,21,35,6,23,40,39,25,27],revert:6,crypto:17,"_singular_":45,separ:[45,17,18,25,6,52],x19:[14,6],eadd_p:15,x10:14,x11:[14,6],x12:14,updat:[7,51,39,25],x14:14,x15:14,x16:14,x17:[14,6],compil:[],domain:[],w200:17,arg1:[17,18,6],arg2:[17,18,6],atomic_expon:15,gaussian:[34,9,13],low_degre:18,w100:[17,13],w101:[17,13],w102:[17,13],w103:[17,13],coprim:[41,0,34,39],ahmad:29,happen:[12,2,28,34,25,13],dispos:17,accomplish:36,"3rd":52,space:[10,13,14,24],y_10:23,decreasingli:[20,5,12,2,53],formula:32,correct:[12,27,14,40,19,33,34,7,39,50,25,53],earlier:40,greater_tuple_invlex:45,hull:5,runtimeerror:52,a112:13,theori:[8,13,14,33,34,23,50,40],toy_varieti:10,org:[34,17,13,25],s_unit:19,care:[8,13,14,17,33,23,25],wai:[8,2,14,18,19,33,34,6,7,51,39,50,25,13],polynomialring_dense_padic_ring_capped_rel:52,full_tabl:52,make_padic_poli:27,recov:21,turn:[8,13,5,33,34,50,40,25],place:19,unwis:6,principl:23,imposs:[33,8,13],frequent:34,first:[17,9,5,6,7,2,10,12,13,14,36,18,21,23,25,45,34,39,40,41,42,8,50,51,52,53],origin:[],reimplement:13,directli:[41,0,13,14,34,32,33,21,25],methoddecor:14,carri:23,onc:[50,34],arrai:[34,15,6],x1_5:23,can_rewrit:13,transform:[13,25],reverse_intvec:25,allow_zero_invers:17,symmetri:14,"95090418262362e":34,open:14,lexicograph:[10,12,8,2,14,45,18,33,26,6,23,40,52,13],size:[12,8,2,14,3,5,17,18,19,34,33,21,28,52,6,23,7,25,13],given:[0,9,5,6,2,10,12,13,14,15,18,19,20,21,23,25,45,30,33,34,35,37,38,39,41,8,40,52,53],rightmost:25,breviti:34,sylvester_matrix:[5,34],convent:[17,39,14],width:25,"574630599127759e":34,adiclli:27,euclideandomainel:42,find_root:25,defer:29,numberfield:[42,12,2,14,5,19,32,48,34,37,53,52,9],necessarili:[40,25,14,27],demonstr:[19,33,34,50,40,52],circl:14,handbook:[51,21,14],proposit:[5,8],conveni:[5,13],cl_maximum_root_first_lambda:25,knowledg:21,negdegrevlex:[17,14,45],copi:[41,0,13,42,20,5,21,50,15,34],braun:2,minimair:[5,8],mathworld:25,g_algebra:[12,14],newton_polygon:27,polynomial_generic_domain:[42,27],domin:34,prec_cap:39,than:[0,17,2,6,13,14,39,36,19,21,22,23,25,27,33,34,35,37,15,41,8,40,52],"_macaulay2_":45,wide:[25,14],ciphertext:17,is_glob:45,f_n:13,bateman_bound:32,booleanmonomialmonoid:13,force_int_expon:15,f_r:[5,2,53],were:[10,8,2,17,6,50,52],posit:[14,45,15,5,18,19,34,52,23,40,39,25],lmap:25,"_macaulay_resultant_universal_polynomi":[5,8],seri:[8,26,14,20,18,34,52,53],pre:[17,13],wdegrevlex:[5,45],sai:25,is_on:[13,19,47,34,35,39,26],nicer:14,algebraicnumb:9,multiples_of:13,argument:[0,12,2,14,5,17,18,19,33,34,35,6,23,38,53,39,52,13],min_exp:15,slimgb_libsingular:24,wolpert:25,notimplementederror:[2,18,34,35,37,40,52,53],sat:17,is_mpolynomialr:[8,52],engin:[],squar:[41,0,2,45,34,35,6,53],use_full_group:40,moreov:[23,40],"gr\u00f6bner":14,note:[],other:[0,26,5,6,7,2,13,14,9,15,17,18,21,22,23,25,27,45,33,34,35,39,40,41,42,43,47,48,49,50,51,52,53],ideal:[],denomin:[8,2,5,34,39,25],take:[41,0,36,2,17,19,32,33,34,23,40,25,13],advis:[33,50,40,13],functori:19,contains_zero:9,surf:14,algebraic_depend:17,begin:6,sure:[0,36,2,17,34,23,15,52,53],infinitegendict:23,trace:[37,5],normal:[12,2,7,51,15,40,25,13],multipli:[41,0,36,27,42,20,5,34,21,49,7,51,40,25],refcount:2,sagemath:[34,25],random_vector:5,beta:[23,33,21,19],stdhilb:[40,14],pair:[0,12,2,14,9,17,19,34,52,53,23,7,51,15,25,13],univar_pd:38,latex:[23,33,40,15],krandick:25,subinterv:25,quaternionalgebra:52,adopt:13,typeset:23,"0235750533041806e":34,sigma:33,booleanmonomialvariableiter:13,mathfrak:40,slope:[34,27,25],cbj07:17,show:[12,2,14,40,19,33,34,35,53,50,51,24,13],cheat:10,cheap:13,demey:[43,52],nefari:34,threshold:52,corner:[13,5,48,34,6,39],codomain:[5,34],galois_group:39,down_degre:25,ground:[20,12,34,14],onli:[9,5,6,7,2,10,12,13,14,39,17,18,22,23,25,27,28,45,20,33,34,37,15,40,42,8,50,51,52],slow:[34,25,14],transformed_basi:14,bbar:13,"457101633618084e":34,is_singleton:13,ideal_nc:14,ll_red_nf_noredsb_single_recurs:13,defecit:34,term_ord:[12,8,2,14,45,18,6,13],dict:[42,12,27,20,13,34,38,53,15,2],fld_in:34,offici:13,noncommut:[],startswith:19,variou:[12,17,2,34],get:[0,8,13,33,9,34,20,21,52,53,39,25,2],twalk:14,monomial_reduc:[12,2,28],secondari:10,kxy:2,cannot:[13,14,20,49,18,34,6,39,25,2],progress:[33,50,17,40,25],gen:[0,2,3,5,6,10,12,13,14,15,17,18,19,21,23,24,28,20,32,33,34,37,39,40,41,42,8,48,50,52,53],requir:[26,36,2,14,15,34,47,21,35,23,53,39,25,13],truli:34,prime:[41,0,12,8,2,14,29,5,19,32,21,28,6,51,39,34,53],delay_f:13,reports_on_ca:13,held:25,polcyclo:52,yield:[5,39,40,6],complete_primary_decomposit:[2,14],xmax:34,where:[0,17,2,5,6,10,12,13,14,15,36,19,21,23,25,27,28,45,20,32,33,34,35,39,41,42,8,49,40,51,52,53],interpolation_polynomi:13,loeffler:18,abc_pd:38,polynomial_integer_dense_ntl:[41,42,52],assumpt:[33,40,14,6],dbar:13,parentwithgen:18,scale_intvec_var:25,nonstandard:[17,14],sub_m_mul_q:2,infinit:[],sage_ev:23,detect:[34,17,40,25],number_field:[34,19],monoid:13,enumer:[34,39],inaccur:[9,14,25],label:[18,34,39,6],enough:[21,25,14,27],x29:[14,6],x28:14,simplest:39,between:[13,14,34,17,19,21,52,25],x25:14,"import":[9,5,6,7,26,10,29,12,13,14,17,18,21,23,25,28,45,32,33,34,15,40,49,50,51,52,53],x27:14,x26:14,x21:14,is_term:[34,53],x23:[14,6],x22:14,spars:[42,8,17,18,32,33,34,6,23,38,40,15,52],parent:[0,1,26,5,6,2,12,13,15,19,21,23,25,27,20,32,33,34,35,37,8,39,41,42,43,47,48,49,50,52,53],screen:23,flist:[5,8],w113:17,w112:17,w111:17,w110:17,integermodr:[51,42,34,14],to_ocean:25,symbolic_express:[5,34],come:[17,13,19],region_width:25,attributeerror:19,make_polynomialrealdens:49,complex_root:[34,9,49],buchmann:17,region:25,polynomialring_dense_padic_field_lazi:52,tutori:13,substitute_vari:13,mani:[12,2,14,16,17,19,34,23,50,40,25,13],rstrat:40,interreduced_basi:[40,13,14,24],etupleit:15,is_monomi:[20,12,34,2,53],rescal:27,eadd:15,inspir:[32,2],uniquefactorizationdomain:6,gf2x_buildrandomirred_list:26,pol:[34,43,13],colon:14,confirm:0,cancel:[33,53],curr:12,nicola:17,poli:[10,2,34,25,52,13],boolepolynomi:13,didier:[5,34,2],zimmermann:26,coupl:34,stretch:33,invert:[42,13,5,32,34,35,37,52],hardli:33,invers:[2,45,20,5,19,34,37],mark:25,hilbert_seri:14,greater_tuple_degneglex:45,u11:[5,8],u10:[5,8],valueerror:[41,0,8,2,14,20,5,18,34,35,52,6,23,53,39,25,13],polynomialquotientring_field:19,hyperellipt:14,polynomial_dense_mod_p:[21,52],cyclotom:[],resolut:14,andrew:32,truncate_ab:49,polynomial_dense_mod_n:21,singular_default:14,matrix_integer_dens:21,var_pd:38,lectur:21,unary_pd:38,former:[40,21,8,25,14],those:[12,13,14,17,18,34,6,23,7,51],"case":[9,5,6,2,10,13,14,17,23,24,25,27,20,32,33,34,35,8,39,43,48,40,52,53],infinitepolynomialring_dens:23,max_bitsize_intvec_doctest:25,new_r:[12,13],nevil:52,polynomialquotientring_domain:19,arnold:32,cast:38,booleanpolynomialvectoriter:13,hilbert_numer:14,a_15:23,"6812407475e":34,exterioralgebra_plur:12,advantag:[23,39],canon:[10,2,30,5,19,34,53,13],"2_22":34,worri:22,ar_rt:25,create_key_and_extra_arg:12,booleanmonomi:13,polresult:34,"__init__":8,var3:6,add_bigoh:34,pbori:[16,13,6],media:13,get_be_log:25,same:[0,2,5,6,12,13,14,17,18,19,21,25,27,45,20,33,34,35,37,8,39,40,41,43,47,48,50,52,53],mpolynomial_el:53,binari:13,inconsist:[17,14],pad:[35,0,21,39,34],document:[10,12,14,20,5,34,19,21,35,52,40,39,24,25],exhaust:17,finish:[33,50,21],closest:25,laurentpolynomialr:[20,34,18],currring_wrapp:12,small_root:[21,35],assist:38,dreyer:[12,13],change_var:52,capabl:13,two:[17,9,5,6,2,10,12,13,14,39,36,19,21,23,24,25,28,45,33,34,15,41,48,52],improv:[12,8,26,13,38,7,53,25,2],extern:[26,14,47,34,35,40],mk_ibpf:25,"_data":15,defn:[30,5,34,19],polynomialquotientr:19,appropri:[10,13,34,6,25,53],polynomialsequence_gf2:17,paderborn:21,weinmann:17,s0s0:13,s0s1:13,polroot:34,without:[12,2,14,17,21,50,40,34],model:[34,13],syzygy_modul:14,negwdegrevlex:45,polygen:[0,2,19,48,34,52,49,53,39,25,9],polr:13,execut:13,vars_as_monomi:13,maximum_root_local_max:25,resp:[33,24,14],rest:[40,25,14],top_sugar:13,"1e100":34,aspect:13,harvei:[41,0,42,36,21,34],speed:[34,15,13,25],get_realfield_rndu:25,concentr:13,versu:23,polynomialringhomomorphism_from_bas:30,mcdonald:[42,34],except:[10,41,2,14,20,34,37,53,39,25,13],littl:[38,34,39,8,14],exercis:[42,34],laurentpolynomialring_univariate_with_categori:18,of_degre:52,vulner:21,dp_asc:[13,45],real:[],size_doubl:13,prepend_str:18,around:[23,7,51,13],recreat:[5,8],handl:[13,45,36,32,34,49,23,53],s3s1:13,s3s0:13,insist:53,recursively_insert:13,x_i:13,mod:[32,26,42,47,21,35,52,34],m11:6,commutativealgebrael:[20,34],patrick:[5,8],multi_polynomial_sequ:[10,17,14],triangular_factor:10,integ:[0,9,3,5,6,2,12,13,14,15,36,18,19,20,21,23,25,27,28,30,32,33,34,35,37,39,41,42,8,49,40,51,52,53],kbit:21,real_root:[34,25],either:[26,12,2,14,9,15,5,17,34,21,52,23,7,39,25,13],output:[0,2,5,6,7,10,12,13,14,17,18,19,21,23,24,25,27,28,20,32,33,34,35,37,39,40,41,8,47,48,50,51,52,53],tower:15,inter:[17,24,14],manag:14,node:[51,13],division_polynomial_0:43,elimpolmin:10,laurentpolynomialring_univari:18,varname_cmp:23,magma:[13,14,45,21,40,51,39],has_degree_ord:13,total_degre:[12,2,5,15,13,53],use_polybori:17,"_factor_univariate_polynomi":34,nonzero:[27,20,5,19,34,25,2],is_maxim:40,s_class_group:19,is_triangular:10,definit:[10,0,45,41,19,34,37,39,25],iff:[10,25],inject:[34,36,2],"1102230246251565e":34,base_r:[12,8,2,28,17,18,19,21,52,6,34],complic:[33,34,25,14,42],permutationgroup:39,polynomial_compil:38,refer:[10,8,13,14,5,17,21,52,23,7,15,40,34],power:[42,12,43,2,14,20,5,34,19,21,52,49,37,8,27,53,39,40,25,26],ration:[0,9,5,6,2,29,12,14,17,18,19,20,23,25,27,45,30,33,34,35,39,40,41,8,50,51,52,53],b212:13,b211:13,tabera:48,karatsuba_threshold:52,awalk1:14,immut:[38,17,18,34,6],"throw":14,comparison:[23,33,13,37],x30:14,x31:6,ffprimroot:52,monomial_quoti:[12,2,28],symmetricid:[23,33,40],gregori:17,joyner:[14,28,45,20,34,15,53],fauger:13,degre:[0,2,3,5,6,26,10,12,13,14,15,17,18,19,21,23,25,27,45,20,32,33,34,35,37,38,39,41,42,8,47,48,49,40,52,53],stand:52,cbar:13,"0212861992297117e":34,refine_recurs:25,morphism:[2,28,30,5,19,34],routin:34,effici:[13,17,21,35,52,7,50,25],eval:[38,45],infinitepolynomialring_spars:23,elementari:40,stable_hash:13,terminolog:25,method_decor:14,quotient_r:33,your:[41,0,34,52,25],weighted_degre:5,log:[36,25],area:13,bernstein_polynomial_factory_ratlist:25,start:[23,25,14],interfac:[],low:[],lot:[41,0,51],"728374398092689e":34,y_30:33,machin:25,lambda:[9,14,5,17,34,51,25],hard:10,polytop:5,multi_polynomial_r:[2,28,45,33,52,53],verbatim:20,tupl:[2,5,6,7,10,12,13,14,39,16,17,18,19,21,25,28,45,20,33,34,37,15,42,8,52,53],regard:[8,14],padded_list:34,"\u00e9tale":19,cryptanalysi:17,satur:14,shamelessli:10,cryptograph:17,buggi:0,inject_vari:[12,18,13,14,6],longer:[40,13],notat:[53,18,6],tripl:[0,15],laurentpolynomialring_mpair_with_categori:18,possibl:[0,8,13,14,45,20,5,17,19,33,21,52,23,50,39,40,25,34],"default":[0,9,5,6,26,10,12,13,14,15,17,18,21,23,25,27,28,20,33,34,35,8,39,40,41,2,47,50,51,52,53],x79:6,polynomial_generic_sparse_field:[42,52],ntl_set_directli:21,tqdegre:14,cl_maximum_root_local_max:25,embed:[48,34,13,25],extrem:38,cfld:34,taylor:25,creat:[10,12,8,2,14,17,18,19,33,37,28,52,6,23,27,50,25,13],certain:[12,2,14,21,51,25],check_irreduc:34,connected_compon:17,mainli:[13,14],salem:[41,0,39],decreas:14,file:[10,41,43,26,47,35,15,25],var_set:13,proport:25,bp2:25,incorrect:[34,27],again:[52,25],integral_point:5,mathematica:[24,14],bremen:[29,13,53],qqbar:[53,34,9,14,45],field:[],binomi:14,valid:[5,34,2,14],polynomialquotientring_gener:19,weyl_algebra:[8,52],is_unit:[42,2,20,19,33,34,37,53,13],"12757025938e":25,ring_or_el:52,architectur:22,poor:34,create_kei:23,genuin:34,integraldomain:[52,19,28],sequenc:[],symbol:[10,8,2,14,5,32,34,49,13],vertex:5,docstr:[34,18,36],echelon_form:13,remove_zero:15,reduc:[41,0,12,8,2,14,40,5,17,32,33,53,7,51,39,50,25,13],compare_tuples_negdegrevlex:45,redud:13,descript:[23,33,40,34,25],data:[34,15,13,52,49],tricki:[17,15],gradient:[5,2,14],tailreduct:50,reduced_groebner_bas:14,potenti:[14,34,17,21,23,39,25],k100:[17,13],is_gener:[5,53],cpu:9,x_3:[23,33,50,40],represent:[8,13,14,45,34,21,52,6,23,38,15,25],all:[26,5,6,7,2,9,29,12,13,14,39,17,18,19,23,25,28,45,20,33,34,8,15,40,43,49,50,52],"_order":33,consider:[40,13],illustr:[26,14,47,34,35,39,53],k103:[17,13],sca:12,scalar:[0,14,17,33,34,15],disc:[0,39,27],deprecationwarn:34,maxima:34,follow:[2,5,6,7,29,12,13,14,39,36,18,21,23,25,45,20,33,34,35,8,15,40,43,50,52],rewrot:[41,0],ugap:25,init:13,daniel:10,ncpolynomialid:14,gauss_on_poli:13,gendict:23,introduc:[34,13,52,6],adapt:52,normalis:[40,39],straightforward:[10,25],evaul:32,candid:13,is_homogen:[12,2,14,5,34,53,15,13],reset_root_width:25,fall:[34,35,52,14],veri:[41,0,12,9,14,20,15,13,17,34,32,21,6,23,38,53,39,25,2],fac:13,ticket:[0,1,2,5,6,12,13,14,17,18,19,23,27,45,32,34,37,8,39,41,43,49,52,53],strang:15,cartesian_product:[13,19],retur:47,list:[0,36,2,5,6,7,26,10,12,13,14,9,15,17,18,19,21,23,25,27,28,45,20,32,33,34,35,37,39,40,41,42,8,47,49,50,51,52,53],arithmeticerror:[41,42,2,20,5,34,52,53],small:[9,34,17,21,35,23,50,25,13],a121:13,becker:[7,14],dimens:[23,52,13,14],extra_arg:12,map_every_x_to_x_plus_on:13,bp_done:25,pyobject:32,past:2,zero:[0,1,2,5,6,7,9,10,12,13,14,15,17,19,21,23,25,27,20,32,33,34,35,8,39,40,41,42,43,48,49,50,52],design:[38,13],pass:[8,14,45,34,17,19,32,33,21,52,23,39,25],val2:32,val1:32,what:[12,2,17,53,21,6,7,51,40,25,13],abc:[18,2,52,6],sub:[2,14,20,17,34,53,25,13],itwm:13,section:[34,14],ast:40,set_random_se:[17,13],euler_phi:32,delet:5,version:[],dir:19,is_irreduc:[47,35,34,39,26],intersect:[13,14],polynomial_real_mpfr_dens:[52,49],method:[2,5,6,7,29,13,14,39,17,18,19,21,23,25,45,20,32,33,34,38,15,40,48,50,52,53],contrast:[20,7,12,2,53],monomialmonoid:13,full:[12,2,34,35,52,53,15,25,13],themselv:2,revert_seri:[35,0,39],variat:[8,18,25,6,15,52],behaviour:[5,34,39],modular:[21,8,2],shouldn:13,middl:25,inher:25,n_var:13,standard:[2,14,5,39,24,25,27],modifi:[50,25,14],cmpfn:15,valu:[9,6,7,2,13,14,39,16,17,19,21,25,27,28,32,33,34,8,15,43,49,52,53],search:[11,17],interval_bernstein_polynomial_integ:25,divisor:[26,0,2,32,20,41,19,47,33,21,35,39,34,13],indetermin:[17,8,52],primary_decomposition_complet:14,xflint:6,prior:[15,39],amount:[25,14],doctest:[12,9,14,45,13,17,34,6,23,7,53,39,25,2],variable_names_recurs:[8,52],y15:53,minsb:13,magnitud:25,y13:53,y12:53,diamet:25,booleanmonomialmonoid__call__:13,polynomial_generic_dens:[42,1,52,34],primit:[34,17,39,52,14],transit:39,c_0:[23,33],c_3:33,c_2:23,c_4:33,famili:[23,25,14],heurist:[50,13,25],spoli:13,zacharia:14,select:[13,34,6,7,51,25],x47:6,x43:6,infinitepolynomial_dens:[23,33],x41:6,distinct:[10,8,9,14,17,32,34,39],selmer:19,get_cpar:[47,35,26],change_variable_nam:34,buchberger2:[40,14],massag:17,eigenwillig:25,irreduc:[26,47,34,35,39,52,53],taken:[10,2,45,33,52,40,25],minor:13,more:[0,5,6,13,14,39,36,23,25,27,20,33,34,35,37,8,15,41,42,43,48,40,52],flat:[],diamond:[18,6],desir:[10,34,14],henc:[23,50,40,14,6],yokoyama:14,an_el:19,convinc:40,objgen:[34,28],right_r:20,fmpq_poly_mullow:39,particular:[12,2,14,5,17,34,6,25],known:[12,27,6,23,40,25,2],lazard:[10,14],first_class:19,cach:[12,13,14,19,34,52,6,23,25],none:[0,1,26,3,5,6,2,12,13,14,9,15,17,18,19,21,23,25,27,20,32,33,34,35,8,39,40,41,42,43,48,49,50,52,53],endpoint:25,polynomial_rational_flint:[34,39,52],polynomialquotientring_generic_with_categori:19,minimalize_and_tail_reduc:13,polynomialring_dense_finite_field_with_categori:52,outlin:25,det:34,histori:25,remain:14,del:34,max_err:25,polynomial_construct:20,dec:34,malb:[29,13,14,53],def:[34,8,9],pick:[8,25,14],pari_group:39,bogu:39,"_sub_":[35,0,21,39,34],ambient:19,share:[12,2,14,5,17,34,16,25],accept:[13,5,17,33,34,23,52],ncpolynomialring_plur:12,is_noetherian:[23,8,18,52],pointwis:36,can_convert_to_singular:29,to_bernstein_warp:25,action:[23,33,40,13],whichev:25,huge:[33,34],cours:[2,14,5,33,34,52,23,51,25,53],mai:[0,17,2,5,6,7,10,12,13,14,39,36,21,23,24,25,27,28,45,20,33,34,15,41,50,52,53],newton_polytop:5,awkward:37,divid:[41,0,12,2,14,20,17,27,34,28,52,35,7,51,25,13],rather:[8,13,6,33,22,38,40],anoth:[],gebauer:[7,51],mapl:[5,8],divis:[41,0,42,20,15,19,32,34,39],nf_lex_point:13,deshomm:[5,34,2],strong:[15,2],simpl:[26,8,9,14,5,17,47,35,51,25,13],unabl:[39,49],regener:14,augment:10,referenc:21,algebra:[10,12,8,9,14,53,5,17,19,34,52,6,23,7,51,40,25,13],alt_var:12,minimum:[52,15,25,14],invok:[32,33,34],x112:17,x113:17,x110:17,x111:17,associ:[2,14,20,13,18,53],diagon:52,"short":25,onto:25,polynomialring_dense_finite_field:52,author:[0,36,2,3,5,7,8,26,10,29,12,13,14,9,15,17,18,19,20,21,23,24,25,27,28,45,30,32,33,34,35,37,38,39,40,41,42,43,47,48,50,51,52,53],libsingular:[],ambigu:34,caus:14,zerodivisionerror:[0,12,2,28,42,34,39],compare_tuples_deglex:45,multivari:[],bitsiz:25,functionfield:34,constitu:23,rotat:36,infinitepolynomialr:[23,33,50,40],faugere_step_dens:13,fft:36,common_nonzero_posit:15,logar:14,eroc:[0,12,35,13],degree_lowest_rational_funct:53,paper:[10,25],through:[28,45,16,21,25,52],hierarchi:[53,28,22],hell:52,paramet:[10,8,9,14,40,5,17,34,47,33,21,52,6,23,38,50,53,15,13,25,2],style:[13,25],variablefactori:13,rr_gap:25,try_rand_split:25,bernstein_polynomial_factori:25,binary_pd:38,resort:14,html:[13,25],atomic_coeffici:15,all_spolys_in_next_degre:13,might:[27,5,17,19,25,13],alter:2,unique_represent:13,z_0:40,good:[50,34,25],"return":[0,36,2,3,5,6,7,26,10,29,12,13,14,9,15,17,18,19,21,23,25,27,28,45,20,32,33,34,35,37,8,39,40,41,42,43,47,49,50,51,52,53],coeffic:53,parallel_reduc:13,termord:[12,8,2,45,5,18,6,13],var_nam:23,do_log:25,wordsiz:25,framework:[50,13,19],xntl:6,ll_reduce_al:13,mpolynomi:[12,2,14,5,17,28,53,13],bigger:[17,8,13],kbase:[24,14],booleanpolynomialr:[17,13,6],elimination_id:[17,14],bernstein_polynomial_factory_ar:25,solver:17,easili:[34,17,13,19,6],achiev:[0,45],compris:[23,40,8],found:[12,14,2,28,34,52,25,9],awalk2:14,difficult:[41,0,34,39],is_pair:13,z16x:37,truncat:[0,26,20,5,47,21,35,49,39,34],valuat:[42,27,20,21,15,34],subsystem:17,weight:[5,45,13,14,26],mul_pd:38,idea:[12,2,28,47,34,35,26],functor:[23,8,14],realli:[41,0,17,33,34,23,15,53],linkag:[47,35,26],expect:[27,34,19,14,25],polynomial_padic_capped_relative_dens:[27,52],todo:[],reduct:[],robert:[29,32,26,28,42,20,47,21,35,38,34,53],dprod_imatrow_vec:25,x_4000:33,footnot:34,multi_polynomial_ideal_libsingular:24,probabilist:47,is_field:[12,8,28,18,19,23,52],print:[43,9,14,50,5,17,40,19,32,33,34,52,6,23,8,7,15,13,25,2],variable_nam:[20,34],divisors_of:13,zeros_in:13,ring_refcount_dict:12,fiber:51,polynomialconstruct:13,advanc:[21,15],gf2:[17,13],normal_basi:[24,14],guess:34,reason:[13,36,33,34,38,50,15],base:[],believ:14,put:[50,17,15],phd:21,basi:[],add_generator_delai:13,k101:[17,13],omit:[25,6],perhap:[34,25,14],previous_prim:53,kung:26,selection_s:13,base_extend:[34,43,52],expected_err:25,contained_var:13,major:[21,2],upper:[17,25],articl:12,misc:[12,2,14],number:[],placehold:34,is_univari:[20,53,2,13],pop:25,done:[8,2,14,40,5,34,32,33,21,50,24,25,13],construct:[1,2,6,12,13,14,16,17,18,19,23,27,28,45,20,33,35,8,42,43,48,52],stabl:13,miss:[13,14],differ:[12,8,2,14,5,34,18,19,21,52,6,40,39,25,13],exponenti:38,msign:25,polynomialfactori:13,multiplic:[12,2,28],unrestrict:[12,15,2,53],fld_out:34,least:[29,2,14,5,17,34,23,39,25],expand:[34,2,25],statement:14,store:[21,15,19,27],assign:[50,8,27,14],imperfect:34,is_zero:[26,0,12,2,14,20,47,21,35,49,39,34,13],autoselect:[40,14],groebner:[],similarli:[34,39,52],is_mpolynomi:[5,53],pari:[41,0,9,48,34,39,52],part:[26,12,43,2,9,17,19,47,20,34,35,23,39,25,13],pars:14,ll_red_nf_redsb:13,eventu:50,std:[37,40,12,24,14],king:[12,2,45,19,33,34,23,50,40,52,13],kind:17,x50:13,doubli:25,min_prec:9,remov:[8,13,34,52,50,15,40,25],symmetric_cancellation_ord:33,compiledpolynomialfunct:38,x59:6,horizont:34,firstli:39,blanklin:[7,34,12,25,13],a42:52,str:[23,17,25],lead_deg:13,infinitepolynomi:[33,50],toward:[12,25],field_extens:[37,19],naqi:34,randomli:[8,13],jacobian_id:5,comput:[0,9,5,7,8,26,10,13,14,17,19,21,23,24,25,27,33,34,35,38,39,40,41,2,47,48,50,51,53],polynomial_r:[23,8,18,19,52],interreduct:[40,13],identifiert:13,macaulai:[5,8],compare_tuples_negdeglex:45,polynomial_padic_flat:[1,52],packag:[17,25],perri:[10,14],univariate_polynomi:[20,53,2,13],brickenstein:[12,13,14],dedic:17,fwalk:14,option:[0,9,5,6,2,12,13,14,39,17,18,19,23,25,45,33,34,35,15,50,8,40,52,53],negwdeglex:45,lie:[41,0,21,47,34,35,25,13],built:13,equival:[13,14,45,19,34,35,49,37],randint:[34,52],full_prot:13,self:[0,26,5,2,12,13,14,15,17,18,19,21,23,25,27,28,45,20,33,34,35,37,39,40,41,42,8,47,48,49,50,52,53],violat:[12,2,28],npart:17,alonso:48,build:[10,38,17,25],target_width:25,jacobian:5,analogu:[28,14,53],coefficient_matrix:[10,17,13,14],distribut:[15,14],tailreduc:[33,50,40],previou:[50,17,19,52],reach:[8,13,14],most:[0,2,5,6,12,13,14,15,16,17,18,19,22,23,25,27,28,45,20,33,34,35,37,8,39,41,42,43,49,40,52,53],plai:[13,25],charl:[17,13,14],rho:[12,2],alpha:[23,33,34,19,37],prentic:52,polynomial_zmod_flint:[21,35,52],clear:14,cover:[13,14,25],dimension:[10,12,14,5,40,24],clean:[10,20,17,18,34,49,7,51],uniquerepresent:13,greater_tuple_negwdegrevlex:45,subgroup:19,cdf:[34,52],x101:[17,13],x100:[17,13],x103:[17,13],x102:[17,13],fink:52,particularli:[42,12,2,14,25,53],algebraic_r:[9,25],min_max_diff_intvec:25,fine:19,find:[10,0,12,2,14,5,17,19,34,33,21,28,53,23,40,51,13,25,9],caruso:27,impact:52,coerc:[41,0,12,2,14,20,19,33,23,40,53,39,37,13],indexerror:[34,13],pretti:14,declare_r:13,e12:25,solut:[51,34,17,13,14],nvariabl:[53,17,2,13],is_cyclotomic_product:34,polynomialring_dense_padic_ring_fixed_mod:52,factor:[],bernstein:25,polynomialquotientringel:[37,19],notation:39,leading_coeffici:[0,34],express:[13,14,17,6,34,49],jeroen:[43,52],bd07:13,commutativ:[23,8,18,19],fastest:34,rt3:34,rt2:34,restart:25,toy_d_basi:51,other_ordering_first:13,univariate_r:8,arxiv:[5,8],compute_gb:14,next_spoli:13,factor_pad:[41,0,39],has_any_invers:20,ideal_pid:3,common:[26,0,43,2,14,20,5,41,47,33,21,35,39,24,34,13],certif:34,set:[10,12,8,2,14,45,5,17,19,34,21,28,52,6,23,7,51,40,25,13],dump:[29,12,2,14,42,17,18,19,21,28,52,23,40,34,13],emax:15,printout:14,decompos:14,mutabl:34,cartesian_product_iter:[17,34],see:[0,2,5,6,26,10,12,13,14,17,18,19,21,23,25,27,45,20,33,34,35,37,39,40,43,47,48,49,50,51,52,53],bard:17,kiran:[14,28,45,34,52,53],polish:[12,2],close:[5,34,9],chose:50,cr_with_categori:8,residue_class_degre:3,deleg:34,someth:[20,34],increase_precis:25,restructur:53,f25:19,coercionexcept:23,experi:39,altern:[13,14,6,23,15,25],blow:51,selmer_group:19,choose_degre:[8,18,13],x_10:[23,33],cr_str:17,induc:30,isol:[],easy_linear_factor:13,"_roots_univariate_polynomi":34,succeed:25,head_normal_form:13,red_tail:13,consumpt:13,solv:[10,2,14,17,19,34,13],subsample_vec_doctest:25,integral_closur:14,felip:48,polydict:[],both:[26,12,2,14,33,9,5,19,47,20,34,28,52,35,40,51,39,25,13],informatik:[29,13,53],last:[0,2,5,6,10,12,13,14,15,17,18,19,23,25,27,28,45,20,33,34,35,37,8,39,40,41,42,43,49,50,52,53],set_verbos:[10,7,21,14],isomorph:[37,8,19],sparse_it:15,context:[21,25,6],pdf:[21,13],cl_maximum_root:25,fieldid:[13,14],cuberoot2:53,cuberoot3:53,load:[29,12,2,14,42,17,18,19,21,28,52,23,40,34,13],"1e25":9,simpli:[50,34,35,53],compare_tuples_degrevlex:45,point:[13,14,36,34,52,40,51,25],slope_err:25,booleanpolynomialiter:13,residu:[3,20,34,52],param:13,suppli:[20,5,34,13],b_0:33,b_1:[23,33],zeta:[34,9,14],deceiv:[12,2,53],backend:[16,28,22],becom:[34,19,25],fractionfield:[23,34,52,53],help:[50,25],due:[29,12,9,14,36,34],empti:[17,13,15,5,36,50,39,25],implicit:[12,14],lift_x:14,akrita:25,as_float:25,is_primit:34,strategi:[41,0,13,40,7,51,50,25],b_j:25,cantor:34,combine_to_posit:15,neglex:45,monagan:32,imag:[30,9,19,13],gap:[39,25],coordin:25,func:20,demand:25,repetit:27,educ:[],stolen:10,look:[12,2,45,34,21,52,40,51,39,25,53],generic_pd:38,min_max_diff_doublevec:25,include_divisor:13,histor:14,"while":[14,45,17,32,21,7,51,40,25],cryptominisat:17,unifi:17,abov:[10,41,14,15,34,19,21,23,39,25],error:[41,0,27,14,49,34,17,47,33,21,6,23,25],fun:5,loos:52,loop:[34,9,25],bezout:0,larger:[34,21,25],readi:[18,6],coffici:35,echelon:[10,17,14],itself:[8,2,14,5,34,52,53,25,13],constantpolynomialsect:34,rif:[34,15,9,25,6],quadrat:14,hyperbola:14,pos_err:25,decor:14,irrelev:8,minim:[13,14,34,37,52,26],x61:6,belong:[2,14,5,19,33,34],x67:6,minu:[0,33],funni:[12,2,28],scalar_rmult:15,conflict:23,higher:[40,34,13,14,25],x83:6,wrap:12,flurri:17,is_exact:[18,52,28],optim:[10,12,2,16,34,21,38,7,51,25],x89:6,island:25,wherea:[39,22],moment:34,user:[13,14,32,21,22,38,15,25],coercion:[26,29,12,2,28,42,19,47,33,48,34,35,52,6,23,53,25,13],make_generic_polynomi:34,along:[17,52],syz:14,provabl:[39,25,14,53],specialis:[34,13],chang:[42,13,45,20,49,18,19,33,21,52,6,23,50,53,15,40,34,2],recent:[0,2,5,6,12,13,14,39,18,19,23,27,28,45,20,33,34,35,37,8,15,41,42,43,49,40,52,53],polynomialring_commut:52,lower:[35,34,39,25,14],ifft:36,lib:[47,12,35,26],discourag:23,entri:[10,13,15,5,17,32,34,52,6,39,25],icpss:14,pickl:[23,12,13],fractionfieldel:53,casteljau:25,expens:[50,25,14],radic:[32,34,14],mon:[12,2,20,13,15,53],a_7:23,salvi:14,a_4:33,a_3:23,a_1:23,real_root_interv:[41,0,39,25],as_etupl:[12,2,53],world:39,squarefree_decomposit:[41,0,34,35],cut:19,"_singular_init_":29,aschenbrenn:[50,40],also:[2,5,6,7,10,12,13,14,39,17,18,19,21,23,25,45,33,34,15,40,41,48,50,51],is_loc:45,ocean:25,nmonomi:17,stabilis:40,wordsize_r:25,theorem:[21,25],input:[0,2,5,6,7,26,10,12,13,14,9,15,17,18,19,21,23,24,25,27,28,45,20,32,33,34,35,37,39,40,41,8,47,48,50,51,52,53],subsequ:34,celement_:[47,35,26],greater_tuple_negwdeglex:45,interpolate_smallest_lex:13,marco:[34,2],march:30,stein:[42,14,28,45,20,36,19,34,37,15,52,53],format:[12,2,14,18,21,52,6,15,34,13],prefer:[41,7,51,35,25],subfield:14,is_groebner_basi:14,bib:21,minimal_el:13,polybori:[],bia:25,add_commut:12,ll_red_nf_noredsb_single_recursive_cal:13,kash:39,bit:[21,13,34,25,53,52,2],characterist:[41,0,8,2,14,45,15,5,34,18,19,37,21,35,52,23,39,25,53],print_mod:43,formal:[26,14,20,5,34,52,25,13],semi:14,resolv:[41,0,12,13,39,34,15],rgbcolor:34,collect:[10,14],princip:[0,14,3,19,40,52],"_mul_":[35,0,21,39,34],intvec_to_doublevec:25,mpoli:8,mpolynomial_libsingular:[5,2],variant:[7,17,52],bruin:[30,52],monomial_all_divisor:[12,2,28],compare_tuples_wdegrevlex:45,err_inc:25,often:[40,25,14],bouillaguet:[17,13],creation:[12,18,14,6],some:[0,17,2,5,6,7,10,12,13,14,39,36,19,21,23,25,27,45,20,32,33,34,15,40,41,50,52,53],back:[14,19,34,35,25,52],unspecifi:34,sampl:[19,14,25],e451:34,mulitvari:14,scienc:21,surpris:[5,51,40,14],"979365054e":34,brent:26,exterioralgebra:12,scale:[17,25],mult_bound:14,prot:[13,14],"_new_constant_poli":34,per:[10,13],substitut:[],mathemat:[12,2,14,34,23,40,53,27],larg:[21,14,34,35,39,25,53],prod:[47,34],proc:14,somehow:[14,45],run:[12,13,17,18,19,34,52,25],zzz:52,zzx:41,lose:19,laurentpolynomialfunctor:18,step:[33,50,40,9,25],weil:[17,14],squeez:[33,40],cyclotomicfield:[34,14,53],mk_ibpi:25,subtract:[36,15],hilbert:14,noncommutative_id:14,constraint:[21,25],transpos:34,memori:[15,13,52,6],primary_decomposit:14,reduced:13,require_field:14,radqi:14,t_1:10,etupl:[12,2,28,16,15,53],block:[8,13,45,17,18,6,25],s000:[17,13],remove_var:[8,18,13],doubl:[34,25],emphasi:[10,7,51],trianglfak:14,laurentpolynomi:20,interval_bernstein_polynomi:25,within:[34,25,6],contains_on:13,polynomialring_gener:52,intv:9,reducible_bi:13,ensur:[33,39,25,14,45],coppersmith:21,x_6000:33,a111:13,implic:13,inclus:[47,8,36,35,26],seger:[12,2,28],triangl:14,triangm:14,less_bit:25,element_class:[19,52],pseudoinvers:25,minimal_associated_prim:14,enabl:25,relprec:27,fast:[],custom:[34,52],arithmet:[41,21,34,52,23,51,15,25,9],includ:[41,13,45,3,19,20,34,6,27,53,50,25,9],suit:19,forward:34,nonsens:14,bradford:34,properli:[10,2,49],reorgan:28,sqrt3:48,sqrt2:[48,53],exhaustive_search:17,decomposit:[41,0,9,14,34,35,25],"87259191484932e":9,link:[47,35,26,25],translat:[40,13],"faug\u00e8r":14,groebnerbasi:[40,14],line:[34,17,25],info:14,cif:[34,9],consist:[10,12,2,14,17,19,34,52,23,24,25,13],is_mpolynomialid:14,caller:34,craig:[37,19],monoidel:13,horner:34,similar:25,alarm:34,polynomial_coeffici:15,curv:[34,39,14],nbit:21,constant:[41,0,12,8,2,20,5,34,48,21,35,53,39,25,13],negdeglex:45,euclidean_degre:[34,19],doesn:[10,2,14,45,15,34,6,38,39],repres:[29,12,2,14,20,15,5,27,53,39,25,13],"char":[5,8],plusinfin:19,bracket:[18,25,6],polygon:[34,27],nan:34,gf2x_buildirred_list:26,invalid:34,priori:17,nav:13,retract:19,grh:19,nice:15,deseri:2,n_variabl:13,create_object:[23,12],polynomialr:[0,2,5,6,7,10,29,12,13,14,17,18,19,21,23,24,28,45,20,34,35,37,8,41,42,43,47,49,51,52,53],william:[42,14,28,45,20,36,19,34,37,15,52,53],val_of_var:27,irrelevant_id:8,polynomialsequ:[17,14],clue:15,eliminate_linear_vari:17,algorithm:[],vice:52,"062854959141552e":34,x73:6,x71:6,discrimin:[41,0,43,2,19,21,39,34],depth:[8,52],dot:25,relative_bound:25,far:[23,25],bernstein_polynomi:25,hello:39,try_split:25,hampton:7,code:[12,2,14,17,53,25,13],partial:[41,0,2,14,5,21,34],edg:17,tx_sibibtex:21,x97:6,n_prime_div:34,ellips:14,ellipt:[34,39,14],"__getattr__":19,compare_tuples_invlex:45,cython:[29,13,34,22,38,15],simon:[12,2,45,19,33,34,23,50,40,52,13],sensit:[17,39,9,13],elsewher:34,send:[37,19],adjoin:[12,2,53],aris:[23,17,13],adleman:52,grevlex:45,berichte_itwm:13,is_idempot:19,bericht122:13,pseudo_divrem:0,uniquefactori:[23,12],volum:21,implicitli:14,ginv:14,relev:[10,12,52],tri:[23,33,39,14],greater_tuple_negdegrevlex:45,question:[34,15,25,14],joel:[41,0,12,2,15,52,53],michael:[32,12,13,14],fewer:[12,2,20,13,25,53],"try":[10,12,2,14,5,17,19,21,28,52,23,39,25,34],polynomialrealdens:[52,49],n_deg:35,maxsiz:35,nonunit:[42,34],rfld:34,prepar:50,defining_polynomi:34,pleas:[8,39,13],impli:[2,14,13],smaller:[27,14,33,40,25,13],euler:34,natur:[13,30,17,34,23,40,53],among:[10,25],flt:[12,2,28],fold:34,fast_when_fals:47,multi_polynomial_id:[7,13,14],d_deg:35,is_weighted_degree_ord:45,if_then_els:13,modular_composit:26,odd:[32,12],append:[13,14],compat:[8,2,3,5,34,13],index:[11,2,14,20,5,33,6,23,50,15,40,13],compar:[27,45,33,21,23,15],my_cmp:32,affin:14,access:[],from_ocean:25,integer_mod:[5,34],tensor_with_r:23,journal:10,can:[0,9,5,6,2,29,12,13,14,17,18,19,21,22,23,25,27,45,20,32,33,34,38,39,40,41,8,48,50,51,52,53],triangular_decomposit:14,is_finit:[8,18,19,52],greater_tuple_negdeglex:45,iteritem:20,len:[10,12,36,2,14,9,34,17,21,24,25,13],closur:[51,13,14],ibp:25,let:[42,9,14,45,5,17,34,33,21,50,51,39,40,25,13],lex:[12,8,2,14,45,20,5,17,18,53,33,6,23,7,51,24,13],vertic:[5,17,27],sinc:[2,5,6,13,39,17,18,19,21,22,23,25,27,20,33,34,37,15,50,41,43,40,51,52],e225:34,convert:[29,13,45,36,52,49,39,25,53],booleanpolynomialring_constructor:6,convers:[29,2,14,17,19,33,34,25,13],across:[13,25],oleksandr:12,genu:14,ctx:25,diff:[20,34,13],later:[23,40,5,50,14],converg:34,rdf:[34,18,52,25],typic:[17,52],miguel:[34,2],to_bernstein:25,brauer:38,chanc:50,degrevlex:[12,8,2,14,45,20,5,18,33,28,6,23,7,40,52,53],cryptolog:[17,21],greater_tuple_wdegrevlex:45,appli:[0,8,40,5,33,34,7,50,25],approxim:[52,34,19,25],submodul:40,elim_pol:10,elength:13,api:13,powerseriesr:[2,28,5,34,23,53],numer:[8,2,14,5,34,52,49,23,39,25,9],associated_prim:14,rational_root_bound:25,from:[],zip:[5,2,14,53],int_list:21,frob:[2,28],next:[2,14,15,5,19,33,34,37,23,50,39,40,25,13],multi_polynomial_ring_gener:[8,2,28,13],few:[10,20,34,2,13],save:[52,25],imprecis:34,commut:[12,8,14,45,36,18,19,6,23,7,15,40,52],remaind:[41,0,2,42,20,21,49,53,39,34,13],sort:[9,20,33,34,50,15,40],ans_sag:34,euclideandomain:6,benchmark:[41,0,14,15,24,25],bernstein_polynomial_factory_intlist:25,fglm:[13,14],rare:[39,22],roughli:25,x37:6,bf05:52,account:[40,13,14,45],variableblock:13,alia:[2,14,19],polynomial_pad:[1,43,27],cambridg:8,obvious:[34,25,14],thin:13,squarefre:[9,19,32,34,35,25],lcmt:15,previous_row:52,functool:14,proof:[41,0,14,8,2,28,18,19,52,53],control:[52,7,34,13,14],mpolynomialring_macaulay2_repr:28,make_etupl:15,process:[33,40,9,14,13],has_inverse_of:20,slim:13,pancratz:39,high:[9,34,49,38,15,25,13],xmin:34,locu:14,sought:34,hit:25,kwankyu:45,surfac:[51,14],polynomial_gf2x:[47,35,26,52],ingram:[5,8],gendictwithbas:23,lgap:25,mpolynomialring_gener:[8,2,28,13],cardin:[52,19,14],acycl:38,polynomial_quotient_ring_el:[37,19],instead:[41,0,13,14,15,32,34,52,39,25,53],sin:9,chri:17,symmetric_reduct:[50,40],compare_tuples_negwdeglex:45,usign:25,frac:[42,34,15,14],overridden:[5,34],singular:[],added_nam:52,s2s0:13,polynomial_generic_field:42,unit_group:19,has_constant_part:13,physic:[12,2,53],gcd:[26,0,2,20,5,41,19,47,33,48,21,35,27,39,34,13],alloc:[15,45],drop:34,essenti:[10,40,34,36],nativ:[34,13,45,26],bind:13,max_diamet:25,correspond:[13,14,17,19,33,34,37,52,6,23,51,39,25],lnc:17,element:[],issu:[14,19,32,34,15,53],vigkla:25,polynomialring_field_with_categori:52,allow:[12,2,14,5,33,34,6,23,50,53,40,25,13],retval:[9,25],llen:25,bpw06:17,reductor:[17,13],monomial_divid:[12,2,28],cyclotomic_polynomi:[32,34,52],schoolbook:52,wit:33,move:[17,15,34],"00000000000000e50":34,comma:[21,18,52,45,6],graph:[38,17,34],x24:14,d_basi:[40,51,14],chosen:[8,21,14,34,40,53,25,2],w211:17,w212:17,w213:17,brook:52,get_dc_log:25,alpha_10:23,nf3:13,mehlhorn:25,polynomial_ring_constructor:6,de_casteljau:25,therefor:[10,13,34,23,39,52],cusp:14,crash:[34,39,13],greater:[33,39,27,52,45],"__getitem__":[47,23,35,26],nonneg:0,python:[29,12,36,2,6,32,21,22,38,53,15,13],nvar:15,bfs04:14,overal:[12,25],dai:[5,8,25],mathematik:13,monomialord:45,lagrange_polynomi:52,front:33,padicr:27,unpickle_mpolynomial_libsingular:2,crt:21,trac:[0,1,2,5,6,9,12,13,14,39,17,18,19,23,27,45,32,34,37,15,41,43,49,52,53],anyth:[17,19,14],edit:[36,52],ntl_zz_px:21,tran:[40,14],addwithcarri:[12,2,28],nameerror:13,use_int:25,slide:25,mode:[39,25],monomials:45,laurent_polynomial_r:18,subset:[28,53],redsb:13,polynomial_zz_pex:47,intellig:[41,0],w14:18,w13:18,w12:18,w11:18,w10:18,x20:14,is_linearly_depend:10,our:[13,33,34,23,50,51,40,25],high_degre:18,special:[12,2,45,16,34,52,39,25,53],out:[41,0,12,13,5,19,34,37,25,53],variabl:[2,5,6,10,12,13,14,17,18,19,23,25,27,28,45,20,33,34,37,15,40,42,8,48,50,52,53],s_r:[5,2,53],pring:52,polynomial_modn_dense_ntl:[42,21,35,52],rep:28,root_field:34,polynomial_zz_px:47,ret:34,singular_str:45,categori:[12,13,5,18,19,34,6,52],n_node:13,suitabl:[23,50,25],rel:[],lattic:21,zxntl:6,deg_bound:[13,14],math:[40,14],polynomial_dense_modn_ntl_zz:[21,35,52],shrink_bp:25,random_el:[32,8,2,14,42,5,36,18,19,47,48,21,52,39,34,13],footprint:33,mpolynomialideal_magma_repr:14,royalord:45,polynomial_system:[17,13],katsura:[7,17,2,14],manipul:26,analysi:[34,52],vector_space_dimens:14,york:[7,14],macaulay_result:[5,8],releas:12,k001:[17,13],k000:[17,13],quo:[37,42,34,35,19],k002:[17,13],s_1:[5,2,53],mod_mon_set:13,raichev:17,transposit:40,via:[],could:[33,34,18,25,6],wdeglex:45,ask:[50,34,19],mac:[5,8],min_degre:52,counterpart:13,length:[36,13,14,45,15,34,17,21,35,6,37,39,25,27],enforc:15,tqblocklow:14,outsid:[19,25],geometri:[5,51,8],greater_tuple_neglex:45,ringlist:12,distinguish:[34,14],invlex:[29,45,18,53,6],"2007ff":17,parent_gen:18,infpoli:23,prmu:34,date:13,"_deriv":[20,5,34],jafferi:34,shortcut:14,facil:34,add_pd:38,"long":[41,0,2,14,33,34,39,25],laz92:10,finitefield:[5,8,52,53],unknown:[51,45,25,14,6],system:[10,8,13,14,17,34,23,40,51,52],wrapper:[41,0,12,13,21,23,25],attach:45,attack:[17,21],subsample_vec:25,termin:[23,33,40,21],fplll:21,"final":[50,5,34,13],travi:20,shallow:13,rsa:[17,21],"2864981197413038e":34,diminish:[33,50,40],exactli:[41,8,13,14,20,17,18,34,52,6,15,25,9],alarminterrupt:34,structur:[42,12,8,13,45,20,5,17,18,19,33,34,52,23,15,37],overflowerror:[15,39,2],squarefreedecomp:[41,0],sens:[33,40,34,53,45],bitsize_doctest:25,add_m_mul_q:2,plaintext:17,julian:34,arg:[8,26,14,20,5,17,18,19,47,21,35,52,23,34],deprec:34,peform:13,infinitepolynomialringfactori:23,booleconst:13,have:[0,17,9,5,6,2,12,13,14,15,36,18,19,22,23,24,25,27,20,33,34,35,37,39,40,51,52,53],ringel:[33,12],disjoint:[34,9,25],thesi:21,miscomput:14,polynomial_templ:[47,35,26],heft:45,min:25,mid:25,mix:6,which:[0,36,2,5,6,26,10,29,12,13,14,15,17,19,21,22,23,24,25,27,45,20,32,33,34,35,37,39,40,41,8,47,48,49,50,51,52,53],int64:49,days4schedul:25,singl:[6,17,18,32,34,52,22,8,40,15,25],maximum_root_first_lambda:25,pyx:[20,0,41,42],unless:[12,14,33,34,35,50,40,52],versa:52,coeffs_bits:25,discov:25,return_reductor:17,cipher:17,polynomialring_dense_padic_field_capped_rel:52,segment:[2,25],"class":[],pushout:52,homogen:[12,8,2,14,5,34,53,15,13],lee:45,laurent:[],dens:[],request:[43,13,17,34,52,6,8,39,25],inde:[40,13,14,19],infinite_polynomial_el:[23,33,50],univari:[],determin:[10,8,2,14,5,34,6,23,40,25],xavier:27,float32:49,fact:[10,13,14,32,6,23,40,52,9],x_0:33,spuriou:34,zpfm:1,npair:13,wikipedia:12,verbos:[7,17,21,14],intervals_disjoint:9,reviv:13,zero_id:52,rough:25,trivial:[14,19,33,34,51,52],homomorph:[],varnam:6,qbar:21,locat:[34,25],nois:9,b111:13,b112:13,forev:[9,25],should:[10,8,13,14,17,33,34,52,6,23,50,53,15,40,25,2],bardet:14,macaulay2_str:45,smallest:[10,13,5,17,34,50,52,26],suppos:[23,34,13,25],local:[17,25,14,45],karatsuba:[34,52],meant:15,next_prim:[47,21,52,34],contribut:[13,19],nontrivi:19,maximal_degre:17,overdetermin:14,complexintervalfield:[34,9],s1s1:13,s1s0:13,increas:[50,17,25,14],subalgorithm:[10,25],precision_absolut:27,booleset:13,smash:9,terminal_on:13,compare_tuples_neglex:45,mpolynomialideal_macaulay2_repr:14,w203:17,w202:17,y_5:33,y_4:[33,50],y_3:[33,50,40],y_2:[23,33,50,34,40],y_1:[23,33,50,34,40],y_0:[23,33,40,34],stuff:5,integr:[2,14,45,20,18,19,32,34,49,23,39,53],contain:[10,41,12,8,2,14,20,15,17,34,28,52,23,40,51,39,25,13],pippeng:38,polynomialring_field:52,view:[12,2,20,15,5,17,19,34,50,39,53],moduli:[21,53],modulo:[0,2,5,13,14,19,21,23,27,28,32,34,35,37,39,50,41,42,49,40,51,52],charpoli:[37,42,9],smooth:51,integraldomainel:42,is_integral_domain:[23,8,18,52,28],lshift_coeff:27,endomorph:34,temporarili:6,stack:[8,25],modulu:[42,19,21,35,52,37,34],gmail:[23,33,13],cole:52,twosid:[12,14],correctli:[0,8,2,17,15,52,13],repr_long:[17,8],ieee:[34,25],boundari:25,favor:34,written:[36,13,25],unusu:19,bruno:[42,14],bernstein_up:25,krick:14,theta:34,neither:20,polynomialring_dense_padic_field_gener:52,kei:[12,14,28,16,33,21,52,23,34,53],ll_red_nf_noredsb:13,notion:[7,51,13,14,53],moebiu:32,entir:[34,25],x53:6,useless:7,inp:13,parenthesi:15,mpolynomialring_polydict_with_categori:[33,2],addit:[36,14,45,34,19,21,52,23,7,15,40,25],c111:13,top_index:13,fileadmin:13,equal:[0,8,13,14,5,17,19,33,34,37,52,6,23,40,39,25,27],cmr05:17,etc:[10,34,15,9,25],instanc:[13,45,34,23,15,25],equat:[21,51,17,13,14],std_libsingular:24,subclas:[5,34],anti:12,x_n:13,max_scal:25,indici:13,walk:[23,13,14],is_prim:14,m10:6,laurentpolynomial_mpair:20,m12:6,respect:[10,12,8,2,14,45,20,40,5,36,33,34,52,23,50,53,24,25,13],"_gcd_univariate_polynomi":34,inverse_of_unit:[20,34,2,53],ntl_gf2x_linkag:[47,35,26],quit:[21,13,34],subset1:13,subset0:13,variableconstruct:13,addition:[16,14,45],tuple_weight:45,compon:[25,15,9,14,13],zz_pex:[],treat:[52,12,2,28],immedi:[13,25],mpolynomial_polydict:[5,53,15,45],mike:[23,33],presenc:13,assert:[34,2,52],x_4:[23,33,50,15],x_5:23,x_2:[13,33,23,50,15,40],togeth:[47,35,26,14],k102:[17,13],x_1:[23,33,50,40,13],gtz:14,present:[10,8,13,5,17,7,51],multi:[8,2,20,15,52,53],xx_2:23,"65220890052e":34,polynomial_ring_homomorph:30,is_spars:52,rbp:25,defin:[0,2,5,6,12,13,14,39,17,18,19,21,23,25,27,45,20,32,34,37,15,40,41,43,47,48,50,51,52,53],fault:2,k113:17,observ:[34,14],sequence_gener:17,helper:[13,25],monoid_class:13,almost:20,infinitepolynomialgen:23,"__reduce__":2,"_assign_nam":[18,6],archiv:17,rndu:25,amer:40,symmetricreductionstrategi:[50,40],noetherian:[23,40,18,14],complexfield:34,is_squarefre:[34,2],uniti:[34,36],cross:34,sqrt:[48,34,25],set_modulu:21,member:[12,2,13,15,25,53],multi_polynomi:[5,53,2,13],mk_context:25,whenev:[14,5,34,35,39,25],largest:[10,27,14,34,25,13],graded_part:13,new_base_r:[5,34],slope_rang:25,infer:[23,17],multi_polynomial_el:[5,15,53],"_rmul_":[35,0,21,39,34],ringwrap:12,http:[21,34,17,13,25],denot:[40,39,14],logic:13,upon:38,effect:[2,17,34,23,40,25,13],poly_repr:15,eurocrypt:21,resoltuion:14,noncanon:2,keep:[23,50,13,25],gmp:25,plural:[],off:[41,0,43,13,42,21,25],center:14,mention:[34,25],nevertheless:23,whole:[25,14],well:[12,43,2,14,5,17,32,33,34,28,49,23,8,40,53,39,13],thought:39,"__div__":53,exampl:[0,36,2,3,5,6,7,9,10,29,12,13,14,15,17,18,19,20,21,23,24,25,26,27,28,45,30,32,33,34,35,37,8,39,40,41,42,43,47,48,49,50,51,52,53],command:[40,17,24,14],choos:[8,27,21,6,34,13],position:[18,6],fail:[9,14,5,17,18,34,40,39,25,27],valuation_of_coeffici:27,"_xgcd_univariate_polynomi":34,usual:[8,13,14,33,35,23,50],sig_on_count:49,less:[0,36,2,15,34,17,21,52,37,39,25,13],"boolean":[],vec:13,obtain:[39,5,17,33,34,50,15],abar:13,heavili:38,increasingli:40,paul:26,glue:[47,35,26],downscal:25,interred_libsingular:24,"97936990747e":34,mohler:[41,0,12,2,15,52,53],unpickle_ncpolynomial_plur:12,ideal_gener:[40,14],cleanup:2,"1443916996305594e":34,polynomial_el:[41,0,1,43,26,42,47,21,35,52,49,39,34],match:[8,13,14,5,17,19,34,23,25],twostd:14,int32:49,interlac:15,max_degre:52,piec:25,five:6,know:[17,19,21,51,15,25],burden:52,denser:17,press:8,recurs:[10,13,5,36,52,38,25,53],singular_moreblock:45,arbitrarili:[21,25],tail:[13,14,17,33,50,40],like:[8,13,45,5,17,34,6,23,40,15,25],lost:34,schmitt:25,hilbert_polynomi:14,unord:17,necessari:[10,0,13,34,35,15],realfield:[34,25,49],martin:[10,29,12,2,14,42,17,47,34,28,52,26,53,35,7,51,15,24,45,13],all_gener:13,down_degree_it:25,page:[11,7,51,52],didn:15,is_const:[12,2,20,13,34,53],shea:[8,14],interact:[53,18,6],average_step:13,assume_squarefre:34,grassmann:12,polynomialring_singular_repr:[29,52,28],"export":36,degree_of_semi_regular:14,convex:5,proper:[25,14],guarante:[41,0,13,14,23,50,40,25],peter:[30,52],b_2:33,librari:[26,12,2,16,47,34,35,22,39,13],x18:14,lead:[41,0,12,2,14,33,53,50,17,40,19,27,34,35,42,7,51,39,24,13],avoid:[29,2,33,23,7,50,25,13],overlap:[23,25],multi_polynomial_libsingular:[5,2,52],commutativeringel:[37,5],estim:[34,9],ab2008:40,ab2007:[50,40],notail:50,degree_reduction_next_s:25,any_root:34,imaginari:9,usag:23,noisi:34,unsuccess:25,although:[23,32,15,25],warp_map:25,diagonal_matrix:34,quotientr:14,stage:25,continu:[33,50,40,25],about:[27,14,21,34,35,22,40,25],x13:[14,6],actual:[8,9,14,17,19,34,6,7,25,27],buchberger_improv:7,testsuit:[12,18,13,19],column:[10,25],ters:43,"1331077795295987e":34,commutativealgebra:[52,6],booleanmulact:13,constructor:[],fals:[0,1,2,5,6,26,10,29,12,13,14,9,15,17,18,19,23,25,27,28,45,20,32,33,34,35,37,8,39,40,41,42,43,47,48,50,52,53],discard:[42,21,34,49,27,25,9],own:[5,34,52],fglmstrategi:13,implicit_plot:14,convolut:[],ring_of_integ:34,automat:[21,2,14,13],connection_graph:17,infinitepolynomial_spars:33,weyl:[8,52],b_5:23,tqblockhigh:14,albrecht:[10,29,12,2,14,42,17,47,34,28,52,26,53,35,7,51,15,24,45,13],pointless:34,mere:23,merg:[10,23,25,33],realintervalfieldel:25,val:43,pictur:25,approx_bp:25,target_list:25,trigger:25,laigl:47,replac:[13,17,33,50,40,25],"var":[12,8,2,14,45,5,49,18,48,34,6,37,53,15,52,13],individu:[5,13],"function":[0,36,2,3,5,6,7,8,26,10,29,12,13,14,17,19,21,25,27,20,32,33,34,35,38,15,41,43,47,40,51,52,53],unpickle_mpolynomialring_generic_v1:8,sqrt2sqrt3:48,triangular:[],defining_id:[13,14],m02:6,laurentpolynomialring_gener:18,m01:6,overflow:[15,2],highest:[13,14,25],bug:[27,14,13,34,23,38,2],decomp:14,count:[20,5,34,25],succe:[34,9,25],made:[10,43,34,7,51,15,25,53],wise:[23,40],whether:[10,12,9,14,20,5,17,18,19,33,34,6,50,39,40,25,13],dml:25,displai:17,record:25,below:[42,8,21,14,13,34,35,49,40,25,2],limit:[21,34],force_etupl:15,otherwis:[10,41,12,2,14,45,20,15,5,17,32,34,28,52,27,53,39,50,25,13],problem:[36,13,14,17,34,52,23,25,53],scale_log2:25,chip:13,evalu:[13,14,49,32,34,6,23,53,52,2],"int":[21,15,32,34,52,49,23,39,25],dure:[25,13,14,19],pid:[],kdigit:21,zassenhau:34,implement:[],k112:17,k111:17,k110:17,ind:13,annihil:17,inf:34,codebas:17,probabl:[47,0,41,27,10],ccuddnavig:13,max_abs_doublevec:25,detail:[13,14,45,20,5,19,48,34,52,6,23,40,39,25],is_polynomialr:52,is_gen:[42,1,43,27,47,48,21,35,52,34,26],toy_buchberg:[10,7,17,14],bool:[12,2,14,15,18,33,34,6,50,39,40,13],futur:10,branch:13,varieti:[17,13,14,22],motsak:12,repeat:[27,19,34,52,40,25,9],complexintervalfieldel:9,fulli:[40,13],june:52,reduction_strategi:13,monic:[10,41,48,21,35,52,39,34],gauss_on_linear:13,involut:35,matrix:[10,42,12,8,2,45,5,17,18,34,21,37,25,9],"885780586188048e":34,polynomial_integer_dense_flint:[41,0,52],old_r:13,cyclotomic_valu:32,rule:[12,2,28,6,23,25],portion:25,auxiliari:12,you:[10,0,12,9,14,53,5,34,18,41,27,21,52,6,7,51,39,13,25,2],invari:[23,33,40,25],"_test_el":19,"8th":52},objtypes:{"0":"py:module","1":"py:function","2":"py:method","3":"py:class","4":"py:attribute","5":"py:exception"},objnames:{"0":["py","module","Python module"],"1":["py","function","Python function"],"2":["py","method","Python method"],"3":["py","class","Python class"],"4":["py","attribute","Python attribute"],"5":["py","exception","Python exception"]},filenames:["sage/rings/polynomial/polynomial_integer_dense_flint","sage/rings/polynomial/padics/polynomial_padic_flat","sage/rings/polynomial/multi_polynomial_libsingular","sage/rings/polynomial/ideal","polynomial_rings_laurent","sage/rings/polynomial/multi_polynomial","sage/rings/polynomial/polynomial_ring_constructor","sage/rings/polynomial/toy_buchberger","sage/rings/polynomial/multi_polynomial_ring_generic","sage/rings/polynomial/complex_roots","sage/rings/polynomial/toy_variety","index","sage/rings/polynomial/plural","sage/rings/polynomial/pbori","sage/rings/polynomial/multi_polynomial_ideal","sage/rings/polynomial/polydict","polynomial_rings_multivar","sage/rings/polynomial/multi_polynomial_sequence","sage/rings/polynomial/laurent_polynomial_ring","sage/rings/polynomial/polynomial_quotient_ring","sage/rings/polynomial/laurent_polynomial","sage/rings/polynomial/polynomial_modn_dense_ntl","polynomial_rings_univar","sage/rings/polynomial/infinite_polynomial_ring","sage/rings/polynomial/multi_polynomial_ideal_libsingular","sage/rings/polynomial/real_roots","sage/rings/polynomial/polynomial_gf2x","sage/rings/polynomial/padics/polynomial_padic_capped_relative_dense","sage/rings/polynomial/multi_polynomial_ring","sage/rings/polynomial/polynomial_singular_interface","sage/rings/polynomial/polynomial_ring_homomorphism","polynomial_rings_infinite","sage/rings/polynomial/cyclotomic","sage/rings/polynomial/infinite_polynomial_element","sage/rings/polynomial/polynomial_element","sage/rings/polynomial/polynomial_zmod_flint","sage/rings/polynomial/convolution","sage/rings/polynomial/polynomial_quotient_ring_element","sage/rings/polynomial/polynomial_compiled","sage/rings/polynomial/polynomial_rational_flint","sage/rings/polynomial/symmetric_ideal","sage/rings/polynomial/polynomial_integer_dense_ntl","sage/rings/polynomial/polynomial_element_generic","sage/rings/polynomial/padics/polynomial_padic","sage/rings/polynomial/polynomial_fateman","sage/rings/polynomial/term_order","polynomial_rings_toy_implementations","sage/rings/polynomial/polynomial_zz_pex","sage/rings/polynomial/polynomial_number_field","sage/rings/polynomial/polynomial_real_mpfr_dense","sage/rings/polynomial/symmetric_reduction","sage/rings/polynomial/toy_d_basis","sage/rings/polynomial/polynomial_ring","sage/rings/polynomial/multi_polynomial_element"],titles:["Dense univariate polynomials over <span class=\"math\">\\(\\ZZ\\)</span>, implemented using FLINT.","p-adic Flat Polynomials","Multivariate Polynomials via libSINGULAR","Ideals in Univariate Polynomial Rings.","Laurent Polynomials and Polynomial Rings","Base class for elements of multivariate polynomial rings","Constructors for polynomial rings","Educational Versions of Groebner Basis Algorithms.","Base class for multivariate polynomial rings","Isolate Complex Roots of Polynomials","Educational Versions of Groebner Basis Algorithms: Triangular Factorization.","Polynomial Rings","Noncommutative Polynomials via libSINGULAR/Plural","Boolean Polynomials","Ideals in multivariate polynomial rings.","PolyDict engine for generic multivariate polynomial rings","Multivariate Polynomials and Polynomial Rings","Polynomial Sequences","Ring of Laurent Polynomials","Quotients of Univariate Polynomial Rings","Elements of Laurent polynomial rings","Dense univariate polynomials over  <span class=\"math\">\\(\\ZZ/n\\ZZ\\)</span>, implemented using NTL.","Univariate Polynomials and Polynomial Rings","Infinite Polynomial Rings.","Direct low-level access to SINGULAR&#8217;s Groebner basis engine via libSINGULAR.","Isolate Real Roots of Real Polynomials","Univariate Polynomials over GF(2) via NTL&#8217;s GF2X.","p-adic Capped Relative Dense Polynomials","Multivariate Polynomial Rings over Generic Rings","Polynomial Interfaces to Singular","Ring homomorphisms from a polynomial ring to another ring","Infinite Polynomial Rings","Fast calculation of cyclotomic polynomials","Elements of Infinite Polynomial Rings","Univariate Polynomial Base Class","Dense univariate polynomials over <span class=\"math\">\\(\\ZZ/n\\ZZ\\)</span>, implemented using FLINT.","Generic Convolution","Elements of Quotients of Univariate Polynomial Rings","Polynomial Compilers","Univariate polynomials over <span class=\"math\">\\(\\QQ\\)</span> implemented via FLINT","Symmetric Ideals of Infinite Polynomial Rings","Dense univariate polynomials over <span class=\"math\">\\(\\ZZ\\)</span>, implemented using NTL.","Univariate Polynomials over domains and fields","Base class for generic <span class=\"math\">\\(p\\)</span>-adic polynomials","Polynomial multiplication by Kronecker substitution","Term orders","Educational Versions of Groebner Basis and Related Algorithms","Univariate Polynomials over GF(p^e) via NTL&#8217;s ZZ_pEX.","Univariate polynomials over number fields.","Dense univariate polynomials over <span class=\"math\">\\(\\RR\\)</span>, implemented using MPFR","Symmetric Reduction of Infinite Polynomials","Educational version of the <span class=\"math\">\\(d\\)</span>-Groebner Basis Algorithm over PIDs.","Univariate Polynomial Rings","Generic Multivariate Polynomials"],objects:{"sage.rings.polynomial.polynomial_gf2x.Polynomial_GF2X":{is_irreducible:[26,2,1,""],modular_composition:[26,2,1,""]},"sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_base_repr":{syzygy_module:[14,2,1,""]},"sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_field":{quo_rem:[42,2,1,""]},"sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal":{weil_restriction:[14,2,1,""],plot:[14,2,1,""],subs:[14,2,1,""],groebner_basis:[14,2,1,""],basis:[14,4,1,""],reduce:[14,2,1,""],gens:[14,2,1,""],random_element:[14,2,1,""],degree_of_semi_regularity:[14,2,1,""],groebner_fan:[14,2,1,""],change_ring:[14,2,1,""],is_homogeneous:[14,2,1,""],homogenize:[14,2,1,""]},"sage.rings.polynomial.pbori.FGLMStrategy":{main:[13,2,1,""]},"sage.rings.polynomial.real_roots.interval_bernstein_polynomial":{try_rand_split:[25,2,1,""],region:[25,2,1,""],variations:[25,2,1,""],region_width:[25,2,1,""],try_split:[25,2,1,""]},"sage.rings.polynomial.real_roots.island":{shrink_bp:[25,2,1,""],bp_done:[25,2,1,""],more_bits:[25,2,1,""],has_root:[25,2,1,""],done:[25,2,1,""],reset_root_width:[25,2,1,""],less_bits:[25,2,1,""],refine_recurse:[25,2,1,""],refine:[25,2,1,""]},"sage.rings.polynomial.polynomial_integer_dense_flint":{Polynomial_integer_dense_flint:[0,3,1,""]},"sage.rings.polynomial.infinite_polynomial_element":{InfinitePolynomial:[33,1,1,""],InfinitePolynomial_sparse:[33,3,1,""],InfinitePolynomial_dense:[33,3,1,""]},"sage.rings.polynomial.pbori.BooleConstant":{is_one:[13,2,1,""],has_constant_part:[13,2,1,""],variables:[13,2,1,""],is_zero:[13,2,1,""],is_constant:[13,2,1,""],deg:[13,2,1,""]},"sage.rings.polynomial.pbori.GroebnerStrategy":{add_as_you_wish:[13,2,1,""],top_sugar:[13,2,1,""],small_spolys_in_next_degree:[13,2,1,""],minimalize:[13,2,1,""],all_spolys_in_next_degree:[13,2,1,""],select:[13,2,1,""],some_spolys_in_next_degree:[13,2,1,""],suggest_plugin_variable:[13,2,1,""],variable_has_value:[13,2,1,""],clean_top_by_chain_criterion:[13,2,1,""],nf:[13,2,1,""],minimalize_and_tail_reduce:[13,2,1,""],contains_one:[13,2,1,""],all_generators:[13,2,1,""],add_generator:[13,2,1,""],add_generator_delayed:[13,2,1,""],reduction_strategy:[13,4,1,""],npairs:[13,2,1,""],ll_reduce_all:[13,2,1,""],symmGB_F2:[13,2,1,""],next_spoly:[13,2,1,""],faugere_step_dense:[13,2,1,""],implications:[13,2,1,""]},"sage.rings.polynomial.real_roots.context":{get_be_log:[25,2,1,""],get_dc_log:[25,2,1,""]},"sage.rings.polynomial.multi_polynomial_ring_generic.MPolynomialRing_generic":{remove_var:[8,2,1,""],completion:[8,2,1,""],macaulay_resultant:[8,2,1,""],repr_long:[8,2,1,""],irrelevant_ideal:[8,2,1,""],characteristic:[8,2,1,""],krull_dimension:[8,2,1,""],is_finite:[8,2,1,""],is_integral_domain:[8,2,1,""],random_element:[8,2,1,""],variable_names_recursive:[8,2,1,""],change_ring:[8,2,1,""],is_noetherian:[8,2,1,""],ngens:[8,2,1,""],weyl_algebra:[8,2,1,""],term_order:[8,2,1,""],construction:[8,2,1,""],univariate_ring:[8,2,1,""],gen:[8,2,1,""],is_field:[8,2,1,""]},"sage.rings.polynomial.multi_polynomial":{MPolynomial:[5,3,1,""],is_MPolynomial:[5,1,1,""]},"sage.rings.polynomial.term_order.TermOrder":{compare_tuples_wdegrevlex:[45,2,1,""],compare_tuples_matrix:[45,2,1,""],greater_tuple_wdegrevlex:[45,2,1,""],compare_tuples_wdeglex:[45,2,1,""],compare_tuples_negdeglex:[45,2,1,""],greater_tuple_lex:[45,2,1,""],singular_moreblocks:[45,2,1,""],greater_tuple_deglex:[45,2,1,""],compare_tuples_invlex:[45,2,1,""],compare_tuples_degrevlex:[45,2,1,""],compare_tuples_negwdeglex:[45,2,1,""],matrix:[45,2,1,""],singular_str:[45,2,1,""],greater_tuple_wdeglex:[45,2,1,""],macaulay2_str:[45,2,1,""],greater_tuple_matrix:[45,2,1,""],greater_tuple_degneglex:[45,2,1,""],weights:[45,2,1,""],greater_tuple_negwdeglex:[45,2,1,""],compare_tuples_block:[45,2,1,""],tuple_weight:[45,2,1,""],blocks:[45,2,1,""],greater_tuple_negdeglex:[45,2,1,""],greater_tuple_invlex:[45,2,1,""],compare_tuples_degneglex:[45,2,1,""],compare_tuples_negdegrevlex:[45,2,1,""],greater_tuple_neglex:[45,2,1,""],greater_tuple_negwdegrevlex:[45,2,1,""],is_local:[45,2,1,""],compare_tuples_negwdegrevlex:[45,2,1,""],is_global:[45,2,1,""],compare_tuples_neglex:[45,2,1,""],name:[45,2,1,""],greater_tuple_negdegrevlex:[45,2,1,""],greater_tuple_degrevlex:[45,2,1,""],is_block_order:[45,2,1,""],is_weighted_degree_order:[45,2,1,""],magma_str:[45,2,1,""],greater_tuple_block:[45,2,1,""],compare_tuples_lex:[45,2,1,""],compare_tuples_deglex:[45,2,1,""]},"sage.rings.polynomial.real_roots.warp_map":{to_ocean:[25,2,1,""],from_ocean:[25,2,1,""]},"sage.rings.polynomial.polynomial_quotient_ring":{PolynomialQuotientRing:[19,1,1,""],PolynomialQuotientRing_field:[19,3,1,""],PolynomialQuotientRing_domain:[19,3,1,""],is_PolynomialQuotientRing:[19,1,1,""],PolynomialQuotientRing_generic:[19,3,1,""]},"sage.rings.polynomial.multi_polynomial_ring_generic":{unpickle_MPolynomialRing_generic_v1:[8,1,1,""],MPolynomialRing_generic:[8,3,1,""],unpickle_MPolynomialRing_generic:[8,1,1,""],is_MPolynomialRing:[8,1,1,""]},"sage.rings.polynomial.polynomial_integer_dense_ntl.Polynomial_integer_dense_ntl":{quo_rem:[41,2,1,""],resultant:[41,2,1,""],gcd:[41,2,1,""],degree:[41,2,1,""],discriminant:[41,2,1,""],xgcd:[41,2,1,""],list:[41,2,1,""],real_root_intervals:[41,2,1,""],content:[41,2,1,""],factor_mod:[41,2,1,""],factor:[41,2,1,""],squarefree_decomposition:[41,2,1,""],lcm:[41,2,1,""],factor_padic:[41,2,1,""]},"sage.rings.polynomial.polynomial_zz_pex.Polynomial_ZZ_pEX":{shift:[47,2,1,""],resultant:[47,2,1,""],list:[47,2,1,""],is_irreducible:[47,2,1,""]},"sage.rings.polynomial.polynomial_ring.PolynomialRing_general":{krull_dimension:[52,2,1,""],is_integral_domain:[52,2,1,""],is_noetherian:[52,2,1,""],extend_variables:[52,2,1,""],completion:[52,2,1,""],variable_names_recursive:[52,2,1,""],gens_dict:[52,2,1,""],change_ring:[52,2,1,""],is_field:[52,2,1,""],parameter:[52,2,1,""],is_exact:[52,2,1,""],is_finite:[52,2,1,""],polynomials:[52,2,1,""],change_var:[52,2,1,""],monics:[52,2,1,""],random_element:[52,2,1,""],some_elements:[52,2,1,""],gen:[52,2,1,""],is_sparse:[52,2,1,""],cyclotomic_polynomial:[52,2,1,""],characteristic:[52,2,1,""],set_karatsuba_threshold:[52,2,1,""],ngens:[52,2,1,""],construction:[52,2,1,""],is_unique_factorization_domain:[52,2,1,""],base_extend:[52,2,1,""],karatsuba_threshold:[52,2,1,""]},"sage.rings.polynomial.polynomial_number_field.Polynomial_relative_number_field_dense":{gcd:[48,2,1,""]},"sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_finite_field":{irreducible_element:[52,2,1,""]},"sage.rings.polynomial.polynomial_ring_constructor":{BooleanPolynomialRing_constructor:[6,1,1,""],PolynomialRing:[6,1,1,""],polynomial_default_category:[6,1,1,""]},"sage.rings.polynomial.pbori.ReductionStrategy":{reduced_normal_form:[13,2,1,""],cheap_reductions:[13,2,1,""],can_rewrite:[13,2,1,""],nf:[13,2,1,""],head_normal_form:[13,2,1,""],add_generator:[13,2,1,""]},"sage.rings.polynomial.pbori.BooleanPolynomialRing":{remove_var:[13,2,1,""],cover_ring:[13,2,1,""],clone:[13,2,1,""],get_order_code:[13,2,1,""],defining_ideal:[13,2,1,""],ideal:[13,2,1,""],ngens:[13,2,1,""],one:[13,2,1,""],variable:[13,2,1,""],zero:[13,2,1,""],gens:[13,2,1,""],id:[13,2,1,""],random_element:[13,2,1,""],get_base_order_code:[13,2,1,""],n_variables:[13,2,1,""],gen:[13,2,1,""],has_degree_order:[13,2,1,""],interpolation_polynomial:[13,2,1,""]},"sage.rings.polynomial.multi_polynomial_ideal_libsingular":{interred_libsingular:[24,1,1,""],std_libsingular:[24,1,1,""],slimgb_libsingular:[24,1,1,""],kbase_libsingular:[24,1,1,""]},"sage.rings.polynomial.multi_polynomial_libsingular":{unpickle_MPolynomial_libsingular:[2,1,1,""],MPolynomial_libsingular:[2,3,1,""],unpickle_MPolynomialRing_libsingular:[2,1,1,""],MPolynomialRing_libsingular:[2,3,1,""]},"sage.rings.polynomial.padics":{polynomial_padic:[43,0,0,"-"],polynomial_padic_capped_relative_dense:[27,0,0,"-"],polynomial_padic_flat:[1,0,0,"-"]},"sage.rings.polynomial.multi_polynomial_element":{MPolynomial_polydict:[53,3,1,""],is_MPolynomial:[53,1,1,""],MPolynomial_element:[53,3,1,""],degree_lowest_rational_function:[53,1,1,""]},"sage.rings.polynomial.toy_buchberger":{LM:[7,1,1,""],buchberger:[7,1,1,""],update:[7,1,1,""],LT:[7,1,1,""],spol:[7,1,1,""],buchberger_improved:[7,1,1,""],LCM:[7,1,1,""],select:[7,1,1,""],inter_reduction:[7,1,1,""]},"sage.rings.polynomial.polydict":{make_PolyDict:[15,1,1,""],ETuple:[15,3,1,""],make_ETuple:[15,1,1,""],ETupleIter:[15,3,1,""],PolyDict:[15,3,1,""]},"sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_mod_n":{residue_field:[52,2,1,""],modulus:[52,2,1,""]},"sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_domain":{is_unit:[42,2,1,""]},"sage.rings.polynomial.polynomial_number_field":{Polynomial_relative_number_field_dense:[48,3,1,""],Polynomial_absolute_number_field_dense:[48,3,1,""]},"sage.rings.polynomial.complex_roots":{intervals_disjoint:[9,1,1,""],refine_root:[9,1,1,""],interval_roots:[9,1,1,""],complex_roots:[9,1,1,""]},"sage.rings.polynomial.cyclotomic":{cyclotomic_value:[32,1,1,""],cyclotomic_coeffs:[32,1,1,""],my_cmp:[32,1,1,""],bateman_bound:[32,1,1,""]},"sage.rings.polynomial.pbori.BooleanMonomialVariableIterator":{next:[13,2,1,""]},"sage.rings.polynomial.pbori.BooleanPolynomialIterator":{next:[13,2,1,""]},"sage.rings.polynomial.polynomial_ring.PolynomialRing_commutative":{quotient_by_principal_ideal:[52,2,1,""],weyl_algebra:[52,2,1,""]},"sage.rings.polynomial.pbori.BooleanMonomialIterator":{next:[13,2,1,""]},"sage.rings.polynomial.polynomial_quotient_ring_element":{PolynomialQuotientRingElement:[37,3,1,""]},"sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_mod_p":{irreducible_element:[52,2,1,""]},"sage.rings.polynomial.polynomial_rational_flint":{Polynomial_rational_flint:[39,3,1,""]},"sage.rings.polynomial.pbori.BooleSetIterator":{next:[13,2,1,""]},"sage.rings.polynomial.real_roots.bernstein_polynomial_factory":{lsign:[25,2,1,""],usign:[25,2,1,""]},"sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict":{is_univariate:[53,2,1,""],subs:[53,2,1,""],variables:[53,2,1,""],reduce:[53,2,1,""],resultant:[53,2,1,""],monomial_coefficient:[53,2,1,""],is_constant:[53,2,1,""],inverse_of_unit:[53,2,1,""],lc:[53,2,1,""],nvariables:[53,2,1,""],is_generator:[53,2,1,""],lm:[53,2,1,""],lt:[53,2,1,""],degrees:[53,2,1,""],factor:[53,2,1,""],is_homogeneous:[53,2,1,""],is_unit:[53,2,1,""],quo_rem:[53,2,1,""],total_degree:[53,2,1,""],coefficient:[53,2,1,""],degree:[53,2,1,""],integral:[53,2,1,""],is_term:[53,2,1,""],lift:[53,2,1,""],variable:[53,2,1,""],constant_coefficient:[53,2,1,""],is_monomial:[53,2,1,""],exponents:[53,2,1,""],monomials:[53,2,1,""],univariate_polynomial:[53,2,1,""],dict:[53,2,1,""]},"sage.rings.polynomial.real_roots.interval_bernstein_polynomial_integer":{down_degree_iter:[25,2,1,""],get_msb_bit:[25,2,1,""],slope_range:[25,2,1,""],downscale:[25,2,1,""],down_degree:[25,2,1,""],as_float:[25,2,1,""],de_casteljau:[25,2,1,""]},"sage.rings.polynomial.padics.polynomial_padic_flat":{Polynomial_padic_flat:[1,3,1,""]},"sage.rings.polynomial.infinite_polynomial_ring.GenDictWithBasering":{next:[23,2,1,""]},"sage.rings.polynomial.polynomial_ring.PolynomialRing_field":{divided_difference:[52,2,1,""],lagrange_polynomial:[52,2,1,""],fraction_field:[52,2,1,""]},"sage.rings.polynomial.real_roots.ocean":{find_roots:[25,2,1,""],approx_bp:[25,2,1,""],increase_precision:[25,2,1,""],all_done:[25,2,1,""],reset_root_width:[25,2,1,""],refine_all:[25,2,1,""],roots:[25,2,1,""]},"sage.rings.polynomial.plural":{ExteriorAlgebra:[12,1,1,""],NCPolynomial_plural:[12,3,1,""],new_CRing:[12,1,1,""],ExteriorAlgebra_plural:[12,3,1,""],unpickle_NCPolynomial_plural:[12,1,1,""],SCA:[12,1,1,""],NCPolynomialRing_plural:[12,3,1,""],G_AlgFactory:[12,3,1,""],new_NRing:[12,1,1,""],new_Ring:[12,1,1,""]},"sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_domain":{monomial_all_divisors:[28,2,1,""],monomial_reduce:[28,2,1,""],is_integral_domain:[28,2,1,""],monomial_lcm:[28,2,1,""],ideal:[28,2,1,""],is_field:[28,2,1,""],monomial_quotient:[28,2,1,""],monomial_divides:[28,2,1,""],monomial_pairwise_prime:[28,2,1,""]},"sage.rings.polynomial.polynomial_ring":{PolynomialRing_dense_padic_ring_capped_absolute:[52,3,1,""],PolynomialRing_dense_padic_field_generic:[52,3,1,""],PolynomialRing_dense_mod_p:[52,3,1,""],PolynomialRing_general:[52,3,1,""],PolynomialRing_dense_padic_ring_fixed_mod:[52,3,1,""],PolynomialRing_field:[52,3,1,""],is_PolynomialRing:[52,1,1,""],PolynomialRing_dense_padic_ring_lazy:[52,3,1,""],PolynomialRing_dense_finite_field:[52,3,1,""],PolynomialRing_dense_padic_field_lazy:[52,3,1,""],PolynomialRing_dense_padic_field_capped_relative:[52,3,1,""],PolynomialRing_integral_domain:[52,3,1,""],polygen:[52,1,1,""],polygens:[52,1,1,""],PolynomialRing_dense_mod_n:[52,3,1,""],PolynomialRing_commutative:[52,3,1,""],PolynomialRing_dense_padic_ring_generic:[52,3,1,""],PolynomialRing_dense_padic_ring_capped_relative:[52,3,1,""]},"sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_mpair":{coefficient:[20,2,1,""],has_any_inverse:[20,2,1,""],coefficients:[20,2,1,""],subs:[20,2,1,""],degree:[20,2,1,""],is_monomial:[20,2,1,""],has_inverse_of:[20,2,1,""],variables:[20,2,1,""],monomials:[20,2,1,""],is_univariate:[20,2,1,""],exponents:[20,2,1,""],dict:[20,2,1,""],constant_coefficient:[20,2,1,""],factor:[20,2,1,""],derivative:[20,2,1,""],is_unit:[20,2,1,""],diff:[20,2,1,""],monomial_coefficient:[20,2,1,""],univariate_polynomial:[20,2,1,""],differentiate:[20,2,1,""]},"sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_gf2e":{weil_restriction:[17,2,1,""]},"sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_generic":{polynomial_ring:[19,2,1,""],S_units:[19,2,1,""],base_ring:[19,2,1,""],S_class_group:[19,2,1,""],cover_ring:[19,2,1,""],krull_dimension:[19,2,1,""],ambient:[19,2,1,""],class_group:[19,2,1,""],number_field:[19,2,1,""],discriminant:[19,2,1,""],is_field:[19,2,1,""],units:[19,2,1,""],modulus:[19,2,1,""],selmer_group:[19,2,1,""],degree:[19,2,1,""],random_element:[19,2,1,""],Element:[19,4,1,""],retract:[19,2,1,""],construction:[19,2,1,""],cardinality:[19,2,1,""],gen:[19,2,1,""],characteristic:[19,2,1,""],ngens:[19,2,1,""],lift:[19,2,1,""],order:[19,2,1,""]},"sage.rings.polynomial.plural.NCPolynomialRing_plural":{monomial_all_divisors:[12,2,1,""],monomial_reduce:[12,2,1,""],is_commutative:[12,2,1,""],relations:[12,2,1,""],monomial_lcm:[12,2,1,""],ideal:[12,2,1,""],ngens:[12,2,1,""],monomial_quotient:[12,2,1,""],term_order:[12,2,1,""],monomial_divides:[12,2,1,""],monomial_pairwise_prime:[12,2,1,""],gen:[12,2,1,""],is_field:[12,2,1,""]},"sage.rings.polynomial.polynomial_element_generic":{Polynomial_generic_field:[42,3,1,""],Polynomial_generic_sparse:[42,3,1,""],Polynomial_generic_sparse_field:[42,3,1,""],Polynomial_generic_domain:[42,3,1,""],Polynomial_generic_dense_field:[42,3,1,""]},"sage.rings.polynomial.toy_d_basis":{LC:[51,1,1,""],gpol:[51,1,1,""],LM:[51,1,1,""],update:[51,1,1,""],spol:[51,1,1,""],d_basis:[51,1,1,""],select:[51,1,1,""]},"sage.rings.polynomial.real_roots.linear_map":{to_ocean:[25,2,1,""],from_ocean:[25,2,1,""]},"sage.rings.polynomial.pbori":{BooleanPolynomialIterator:[13,3,1,""],BooleanPolynomialEntry:[13,3,1,""],get_var_mapping:[13,1,1,""],BooleanPolynomial:[13,3,1,""],unpickle_BooleanPolynomialRing:[13,1,1,""],BooleanMonomialVariableIterator:[13,3,1,""],BooleanMonomialMonoid:[13,3,1,""],MonomialFactory:[13,3,1,""],BooleanMulAction:[13,3,1,""],contained_vars:[13,1,1,""],if_then_else:[13,1,1,""],ll_red_nf_noredsb_single_recursive_call:[13,1,1,""],interpolate_smallest_lex:[13,1,1,""],BooleanMonomialIterator:[13,3,1,""],VariableBlock:[13,3,1,""],BooleanPolynomialVectorIterator:[13,3,1,""],CCuddNavigator:[13,3,1,""],nf3:[13,1,1,""],BooleSetIterator:[13,3,1,""],MonomialConstruct:[13,3,1,""],zeros:[13,1,1,""],set_random_seed:[13,1,1,""],BooleanMonomial:[13,3,1,""],TermOrder_from_pb_order:[13,1,1,""],GroebnerStrategy:[13,3,1,""],BooleanPolynomialVector:[13,3,1,""],PolynomialConstruct:[13,3,1,""],top_index:[13,1,1,""],ll_red_nf_noredsb:[13,1,1,""],unpickle_BooleanPolynomial0:[13,1,1,""],mod_var_set:[13,1,1,""],interpolate:[13,1,1,""],red_tail:[13,1,1,""],recursively_insert:[13,1,1,""],mult_fact_sim_C:[13,1,1,""],ll_red_nf_redsb:[13,1,1,""],BooleanPolynomialRing:[13,3,1,""],BooleSet:[13,3,1,""],mod_mon_set:[13,1,1,""],random_set:[13,1,1,""],BooleConstant:[13,3,1,""],parallel_reduce:[13,1,1,""],VariableConstruct:[13,3,1,""],BooleanPolynomialIdeal:[13,3,1,""],map_every_x_to_x_plus_one:[13,1,1,""],substitute_variables:[13,1,1,""],unpickle_BooleanPolynomial:[13,1,1,""],ReductionStrategy:[13,3,1,""],gauss_on_polys:[13,1,1,""],VariableFactory:[13,3,1,""],FGLMStrategy:[13,3,1,""],PolynomialFactory:[13,3,1,""],add_up_polynomials:[13,1,1,""],easy_linear_factors:[13,1,1,""]},"sage.rings.polynomial.plural.G_AlgFactory":{create_key_and_extra_args:[12,2,1,""],create_object:[12,2,1,""]},"sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_domain":{is_finite:[19,2,1,""],field_extension:[19,2,1,""]},"sage.rings.polynomial.polydict.ETupleIter":{next:[15,2,1,""]},"sage.rings.polynomial.pbori.BooleanPolynomial":{set:[13,2,1,""],is_univariate:[13,2,1,""],subs:[13,2,1,""],reducible_by:[13,2,1,""],variables:[13,2,1,""],reduce:[13,2,1,""],is_equal:[13,2,1,""],lead_divisors:[13,2,1,""],elength:[13,2,1,""],map_every_x_to_x_plus_one:[13,2,1,""],constant:[13,2,1,""],is_constant:[13,2,1,""],has_constant_part:[13,2,1,""],monomial_coefficient:[13,2,1,""],is_singleton_or_pair:[13,2,1,""],lead:[13,2,1,""],nvariables:[13,2,1,""],lex_lead_deg:[13,2,1,""],is_singleton:[13,2,1,""],lm:[13,2,1,""],lt:[13,2,1,""],is_homogeneous:[13,2,1,""],is_zero:[13,2,1,""],graded_part:[13,2,1,""],total_degree:[13,2,1,""],zeros_in:[13,2,1,""],terms:[13,2,1,""],degree:[13,2,1,""],n_vars:[13,2,1,""],spoly:[13,2,1,""],navigation:[13,2,1,""],stable_hash:[13,2,1,""],variable:[13,2,1,""],is_unit:[13,2,1,""],constant_coefficient:[13,2,1,""],lex_lead:[13,2,1,""],first_term:[13,2,1,""],is_one:[13,2,1,""],is_pair:[13,2,1,""],ring:[13,2,1,""],n_nodes:[13,2,1,""],lead_deg:[13,2,1,""],monomials:[13,2,1,""],vars_as_monomial:[13,2,1,""],univariate_polynomial:[13,2,1,""],deg:[13,2,1,""]},"sage.rings.polynomial.polydict.PolyDict":{coefficient:[15,2,1,""],latex:[15,2,1,""],compare:[15,2,1,""],homogenize:[15,2,1,""],degree:[15,2,1,""],scalar_lmult:[15,2,1,""],max_exp:[15,2,1,""],list:[15,2,1,""],poly_repr:[15,2,1,""],polynomial_coefficient:[15,2,1,""],total_degree:[15,2,1,""],min_exp:[15,2,1,""],is_homogeneous:[15,2,1,""],scalar_rmult:[15,2,1,""],lcmt:[15,2,1,""],dict:[15,2,1,""],monomial_coefficient:[15,2,1,""],valuation:[15,2,1,""],exponents:[15,2,1,""],coefficients:[15,2,1,""]},"sage.rings.polynomial.multi_polynomial_ideal":{MPolynomialIdeal_magma_repr:[14,3,1,""],MPolynomialIdeal_macaulay2_repr:[14,3,1,""],is_MPolynomialIdeal:[14,1,1,""],NCPolynomialIdeal:[14,3,1,""],require_field:[14,4,1,""],MPolynomialIdeal_singular_repr:[14,3,1,""],MPolynomialIdeal:[14,3,1,""],MPolynomialIdeal_singular_base_repr:[14,3,1,""],RequireField:[14,3,1,""]},"sage.rings.polynomial.padics.polynomial_padic":{Polynomial_padic:[43,3,1,""]},"sage.rings.polynomial.real_roots.interval_bernstein_polynomial_float":{get_msb_bit:[25,2,1,""],de_casteljau:[25,2,1,""],slope_range:[25,2,1,""],as_float:[25,2,1,""]},"sage.rings.polynomial.polynomial_element.PolynomialBaseringInjection":{section:[34,2,1,""]},"sage.rings.polynomial.polynomial_element.Polynomial":{square:[34,2,1,""],base_ring:[34,2,1,""],"_sub_":[34,2,1,""],is_nilpotent:[34,2,1,""],list:[34,2,1,""],leading_coefficient:[34,2,1,""],is_square:[34,2,1,""],integral:[34,2,1,""],newton_raphson:[34,2,1,""],is_irreducible:[34,2,1,""],is_squarefree:[34,2,1,""],"_mul_":[34,2,1,""],"_mul_trunc_":[34,2,1,""],is_cyclotomic_product:[34,2,1,""],differentiate:[34,2,1,""],diff:[34,2,1,""],polynomial:[34,2,1,""],factor:[34,2,1,""],roots:[34,2,1,""],is_constant:[34,2,1,""],ord:[34,2,1,""],inverse_of_unit:[34,2,1,""],squarefree_decomposition:[34,2,1,""],variables:[34,2,1,""],plot:[34,2,1,""],truncate:[34,2,1,""],is_monic:[34,2,1,""],sylvester_matrix:[34,2,1,""],add_bigoh:[34,2,1,""],discriminant:[34,2,1,""],radical:[34,2,1,""],prec:[34,2,1,""],is_homogeneous:[34,2,1,""],content:[34,2,1,""],is_gen:[34,2,1,""],change_ring:[34,2,1,""],dict:[34,2,1,""],substitute:[34,2,1,""],is_zero:[34,2,1,""],map_coefficients:[34,2,1,""],derivative:[34,2,1,""],is_unit:[34,2,1,""],is_primitive:[34,2,1,""],exponents:[34,2,1,""],norm:[34,2,1,""],constant_coefficient:[34,2,1,""],"_rmul_":[34,2,1,""],denominator:[34,2,1,""],resultant:[34,2,1,""],degree:[34,2,1,""],root_field:[34,2,1,""],real_roots:[34,2,1,""],any_root:[34,2,1,""],is_term:[34,2,1,""],args:[34,2,1,""],padded_list:[34,2,1,""],homogenize:[34,2,1,""],subs:[34,2,1,""],change_variable_name:[34,2,1,""],newton_slopes:[34,2,1,""],"_lmul_":[34,2,1,""],lcm:[34,2,1,""],valuation:[34,2,1,""],coefficients:[34,2,1,""],gcd:[34,2,1,""],xgcd:[34,2,1,""],is_one:[34,2,1,""],reverse:[34,2,1,""],"_add_":[34,2,1,""],monic:[34,2,1,""],is_monomial:[34,2,1,""],shift:[34,2,1,""],numerator:[34,2,1,""],mod:[34,2,1,""],variable_name:[34,2,1,""],hamming_weight:[34,2,1,""],euclidean_degree:[34,2,1,""],is_cyclotomic:[34,2,1,""],inverse_mod:[34,2,1,""],coeffs:[34,2,1,""],base_extend:[34,2,1,""],splitting_field:[34,2,1,""],complex_roots:[34,2,1,""]},"sage.rings.polynomial.real_roots.bernstein_polynomial_factory_ar":{bernstein_polynomial:[25,2,1,""],coeffs_bitsize:[25,2,1,""]},"sage.rings.polynomial.multi_polynomial_sequence":{is_PolynomialSequence:[17,1,1,""],PolynomialSequence:[17,1,1,""],PolynomialSequence_gf2e:[17,3,1,""],PolynomialSequence_gf2:[17,3,1,""],PolynomialSequence_generic:[17,3,1,""]},"sage.rings.polynomial.polynomial_modn_dense_ntl":{make_element:[21,1,1,""],Polynomial_dense_mod_p:[21,3,1,""],Polynomial_dense_mod_n:[21,3,1,""],small_roots:[21,1,1,""],Polynomial_dense_modn_ntl_zz:[21,3,1,""],Polynomial_dense_modn_ntl_ZZ:[21,3,1,""]},"sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse":{degree:[42,2,1,""],shift:[42,2,1,""],list:[42,2,1,""],dict:[42,2,1,""],quo_rem:[42,2,1,""],valuation:[42,2,1,""],exponents:[42,2,1,""],coefficients:[42,2,1,""]},"sage.rings.polynomial.polynomial_gf2x.Polynomial_template":{get_cparent:[26,2,1,""],is_one:[26,2,1,""],truncate:[26,2,1,""],degree:[26,2,1,""],shift:[26,2,1,""],xgcd:[26,2,1,""],list:[26,2,1,""],is_gen:[26,2,1,""],is_zero:[26,2,1,""],quo_rem:[26,2,1,""],gcd:[26,2,1,""]},"sage.rings.polynomial.infinite_polynomial_element.InfinitePolynomial_sparse":{coefficient:[33,2,1,""],lc:[33,2,1,""],stretch:[33,2,1,""],symmetric_cancellation_order:[33,2,1,""],variables:[33,2,1,""],lm:[33,2,1,""],reduce:[33,2,1,""],tail:[33,2,1,""],is_unit:[33,2,1,""],squeezed:[33,2,1,""],lt:[33,2,1,""],footprint:[33,2,1,""],max_index:[33,2,1,""],ring:[33,2,1,""],polynomial:[33,2,1,""],gcd:[33,2,1,""]},"sage.rings.polynomial.polynomial_singular_interface":{can_convert_to_singular:[29,1,1,""],PolynomialRing_singular_repr:[29,3,1,""],Polynomial_singular_repr:[29,3,1,""]},"sage.rings.polynomial":{polynomial_ring:[52,0,0,"-"],laurent_polynomial_ring:[18,0,0,"-"],polynomial_ring_homomorphism:[30,0,0,"-"],convolution:[36,0,0,"-"],multi_polynomial_ring_generic:[8,0,0,"-"],infinite_polynomial_element:[33,0,0,"-"],polynomial_ring_constructor:[6,0,0,"-"],infinite_polynomial_ring:[23,0,0,"-"],polynomial_compiled:[38,0,0,"-"],polynomial_real_mpfr_dense:[49,0,0,"-"],polynomial_gf2x:[26,0,0,"-"],polynomial_modn_dense_ntl:[21,0,0,"-"],cyclotomic:[32,0,0,"-"],polynomial_zmod_flint:[35,0,0,"-"],pbori:[13,0,0,"-"],multi_polynomial:[5,0,0,"-"],polynomial_fateman:[44,0,0,"-"],polynomial_integer_dense_ntl:[41,0,0,"-"],multi_polynomial_libsingular:[2,0,0,"-"],ideal:[3,0,0,"-"],multi_polynomial_ideal_libsingular:[24,0,0,"-"],toy_buchberger:[7,0,0,"-"],symmetric_reduction:[50,0,0,"-"],polynomial_singular_interface:[29,0,0,"-"],plural:[12,0,0,"-"],polynomial_element:[34,0,0,"-"],toy_variety:[10,0,0,"-"],real_roots:[25,0,0,"-"],polynomial_zz_pex:[47,0,0,"-"],multi_polynomial_ring:[28,0,0,"-"],polynomial_quotient_ring:[19,0,0,"-"],multi_polynomial_ideal:[14,0,0,"-"],term_order:[45,0,0,"-"],polydict:[15,0,0,"-"],multi_polynomial_element:[53,0,0,"-"],polynomial_rational_flint:[39,0,0,"-"],polynomial_integer_dense_flint:[0,0,0,"-"],multi_polynomial_sequence:[17,0,0,"-"],polynomial_quotient_ring_element:[37,0,0,"-"],symmetric_ideal:[40,0,0,"-"],toy_d_basis:[51,0,0,"-"],laurent_polynomial:[20,0,0,"-"],polynomial_element_generic:[42,0,0,"-"],polynomial_number_field:[48,0,0,"-"],complex_roots:[9,0,0,"-"]},"sage.rings.polynomial.padics.polynomial_padic_capped_relative_dense.Polynomial_padic_capped_relative_dense":{precision_relative:[27,2,1,""],rshift_coeffs:[27,2,1,""],valuation_of_coefficient:[27,2,1,""],degree:[27,2,1,""],rescale:[27,2,1,""],xgcd:[27,2,1,""],list:[27,2,1,""],lshift_coeffs:[27,2,1,""],prec_degree:[27,2,1,""],content:[27,2,1,""],factor_mod:[27,2,1,""],disc:[27,2,1,""],lift:[27,2,1,""],newton_polygon:[27,2,1,""],newton_slopes:[27,2,1,""],precision_absolute:[27,2,1,""],quo_rem:[27,2,1,""],valuation:[27,2,1,""],hensel_lift:[27,2,1,""],reverse:[27,2,1,""]},"sage.rings.polynomial.polynomial_compiled":{CompiledPolynomialFunction:[38,3,1,""],pow_pd:[38,3,1,""],abc_pd:[38,3,1,""],binary_pd:[38,3,1,""],unary_pd:[38,3,1,""],mul_pd:[38,3,1,""],var_pd:[38,3,1,""],add_pd:[38,3,1,""],dummy_pd:[38,3,1,""],univar_pd:[38,3,1,""],sqr_pd:[38,3,1,""],generic_pd:[38,3,1,""],coeff_pd:[38,3,1,""]},"sage.rings.polynomial.padics.polynomial_padic_capped_relative_dense":{make_padic_poly:[27,1,1,""],Polynomial_padic_capped_relative_dense:[27,3,1,""]},"sage.rings.polynomial.symmetric_ideal":{SymmetricIdeal:[40,3,1,""]},"sage.rings.polynomial.polynomial_gf2x":{Polynomial_template:[26,3,1,""],make_element:[26,1,1,""],GF2X_BuildSparseIrred_list:[26,1,1,""],GF2X_BuildIrred_list:[26,1,1,""],GF2X_BuildRandomIrred_list:[26,1,1,""],Polynomial_GF2X:[26,3,1,""]},"sage.rings.polynomial.polynomial_zz_pex.Polynomial_template":{get_cparent:[47,2,1,""],is_one:[47,2,1,""],truncate:[47,2,1,""],degree:[47,2,1,""],shift:[47,2,1,""],xgcd:[47,2,1,""],list:[47,2,1,""],is_gen:[47,2,1,""],is_zero:[47,2,1,""],quo_rem:[47,2,1,""],gcd:[47,2,1,""]},"sage.rings.polynomial.polynomial_zmod_flint":{Polynomial_template:[35,3,1,""],Polynomial_zmod_flint:[35,3,1,""],make_element:[35,1,1,""]},"sage.rings.polynomial.pbori.BooleanMonomial":{index:[13,2,1,""],set:[13,2,1,""],gcd:[13,2,1,""],degree:[13,2,1,""],reducible_by:[13,2,1,""],variables:[13,2,1,""],iterindex:[13,2,1,""],stable_hash:[13,2,1,""],ring:[13,2,1,""],navigation:[13,2,1,""],divisors:[13,2,1,""],multiples:[13,2,1,""],deg:[13,2,1,""]},"sage.rings.polynomial.polynomial_integer_dense_ntl":{Polynomial_integer_dense_ntl:[41,3,1,""]},"sage.rings.polynomial.multi_polynomial_ring":{MPolynomialRing_macaulay2_repr:[28,3,1,""],MPolynomialRing_polydict_domain:[28,3,1,""],MPolynomialRing_polydict:[28,3,1,""]},"sage.rings.polynomial.pbori.BooleanPolynomialIdeal":{interreduced_basis:[13,2,1,""],groebner_basis:[13,2,1,""],reduce:[13,2,1,""],dimension:[13,2,1,""],variety:[13,2,1,""]},"sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint":{"_sub_":[0,2,1,""],resultant:[0,2,1,""],"_mul_":[0,2,1,""],"_mul_trunc_":[0,2,1,""],squarefree_decomposition:[0,2,1,""],"_rmul_":[0,2,1,""],gcd:[0,2,1,""],discriminant:[0,2,1,""],real_root_intervals:[0,2,1,""],content:[0,2,1,""],factor_mod:[0,2,1,""],is_zero:[0,2,1,""],factor:[0,2,1,""],quo_rem:[0,2,1,""],factor_padic:[0,2,1,""],disc:[0,2,1,""],degree:[0,2,1,""],xgcd:[0,2,1,""],pseudo_divrem:[0,2,1,""],"_lmul_":[0,2,1,""],lcm:[0,2,1,""],reverse:[0,2,1,""],"_add_":[0,2,1,""],list:[0,2,1,""],revert_series:[0,2,1,""]},"sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_p":{xgcd:[21,2,1,""],resultant:[21,2,1,""],gcd:[21,2,1,""],discriminant:[21,2,1,""]},"sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_n":{ntl_set_directly:[21,2,1,""],degree:[21,2,1,""],shift:[21,2,1,""],int_list:[21,2,1,""],list:[21,2,1,""],ntl_ZZ_pX:[21,2,1,""],small_roots:[21,2,1,""],quo_rem:[21,2,1,""]},"sage.rings.polynomial.polynomial_ring_homomorphism":{PolynomialRingHomomorphism_from_base:[30,3,1,""]},"sage.rings.polynomial.polydict.ETuple":{esub:[15,2,1,""],emin:[15,2,1,""],emax:[15,2,1,""],eadd:[15,2,1,""],nonzero_positions:[15,2,1,""],combine_to_positives:[15,2,1,""],emul:[15,2,1,""],nonzero_values:[15,2,1,""],common_nonzero_positions:[15,2,1,""],eadd_p:[15,2,1,""],reversed:[15,2,1,""],sparse_iter:[15,2,1,""]},"sage.rings.polynomial.pbori.PolynomialConstruct":{lead:[13,2,1,""]},"sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_generic":{remove_var:[18,2,1,""],completion:[18,2,1,""],krull_dimension:[18,2,1,""],is_exact:[18,2,1,""],characteristic:[18,2,1,""],polynomial_ring:[18,2,1,""],is_finite:[18,2,1,""],is_integral_domain:[18,2,1,""],ideal:[18,2,1,""],ngens:[18,2,1,""],change_ring:[18,2,1,""],is_noetherian:[18,2,1,""],is_field:[18,2,1,""],random_element:[18,2,1,""],term_order:[18,2,1,""],construction:[18,2,1,""],gen:[18,2,1,""],fraction_field:[18,2,1,""]},"sage.rings.polynomial.multi_polynomial_ideal.NCPolynomialIdeal":{std:[14,2,1,""],res:[14,2,1,""],syzygy_module:[14,2,1,""],reduce:[14,2,1,""],twostd:[14,2,1,""]},"sage.rings.polynomial.toy_variety":{linear_representation:[10,1,1,""],is_triangular:[10,1,1,""],triangular_factorization:[10,1,1,""],elim_pol:[10,1,1,""],is_linearly_dependent:[10,1,1,""],coefficient_matrix:[10,1,1,""]},"sage.rings.polynomial.infinite_polynomial_ring":{GenDictWithBasering:[23,3,1,""],InfinitePolynomialRingFactory:[23,3,1,""],InfiniteGenDict:[23,3,1,""],InfinitePolynomialGen:[23,3,1,""],InfinitePolynomialRing_sparse:[23,3,1,""],InfinitePolynomialRing_dense:[23,3,1,""]},"sage.rings.polynomial.padics.polynomial_padic.Polynomial_padic":{factor:[43,2,1,""]},"sage.rings.polynomial.infinite_polynomial_ring.InfinitePolynomialRing_sparse":{characteristic:[23,2,1,""],krull_dimension:[23,2,1,""],is_integral_domain:[23,2,1,""],gens_dict:[23,2,1,""],ngens:[23,2,1,""],one:[23,2,1,""],varname_cmp:[23,2,1,""],is_noetherian:[23,2,1,""],construction:[23,2,1,""],is_field:[23,2,1,""],tensor_with_ring:[23,2,1,""],order:[23,2,1,""],gen:[23,2,1,""]},"sage.rings.polynomial.laurent_polynomial_ring":{LaurentPolynomialRing_mpair:[18,3,1,""],LaurentPolynomialRing_univariate:[18,3,1,""],LaurentPolynomialRing_generic:[18,3,1,""],LaurentPolynomialRing:[18,1,1,""],is_LaurentPolynomialRing:[18,1,1,""]},"sage.rings.polynomial.real_roots":{PrecisionError:[25,5,1,""],bernstein_expand:[25,1,1,""],interval_bernstein_polynomial_integer:[25,3,1,""],maximum_root_local_max:[25,1,1,""],interval_bernstein_polynomial_float:[25,3,1,""],split_for_targets:[25,1,1,""],intvec_to_doublevec:[25,1,1,""],subsample_vec_doctest:[25,1,1,""],min_max_diff_intvec:[25,1,1,""],scale_intvec_var:[25,1,1,""],get_realfield_rndu:[25,1,1,""],bernstein_down:[25,1,1,""],mk_ibpi:[25,1,1,""],de_casteljau_doublevec:[25,1,1,""],taylor_shift1_intvec:[25,1,1,""],rr_gap:[25,3,1,""],mk_ibpf:[25,1,1,""],bernstein_polynomial_factory_ratlist:[25,3,1,""],precompute_degree_reduction_cache:[25,1,1,""],mk_context:[25,1,1,""],interval_bernstein_polynomial:[25,3,1,""],reverse_intvec:[25,1,1,""],linear_map:[25,3,1,""],cl_maximum_root_first_lambda:[25,1,1,""],wordsize_rational:[25,1,1,""],rational_root_bounds:[25,1,1,""],bernstein_up:[25,1,1,""],dprod_imatrow_vec:[25,1,1,""],pseudoinverse:[25,1,1,""],real_roots:[25,1,1,""],bernstein_polynomial_factory:[25,3,1,""],bernstein_polynomial_factory_intlist:[25,3,1,""],relative_bounds:[25,1,1,""],max_abs_doublevec:[25,1,1,""],to_bernstein_warp:[25,1,1,""],de_casteljau_intvec:[25,1,1,""],max_bitsize_intvec_doctest:[25,1,1,""],bernstein_polynomial_factory_ar:[25,3,1,""],min_max_delta_intvec:[25,1,1,""],island:[25,3,1,""],bitsize_doctest:[25,1,1,""],min_max_diff_doublevec:[25,1,1,""],degree_reduction_next_size:[25,1,1,""],ocean:[25,3,1,""],root_bounds:[25,1,1,""],maximum_root_first_lambda:[25,1,1,""],context:[25,3,1,""],cl_maximum_root_local_max:[25,1,1,""],warp_map:[25,3,1,""],cl_maximum_root:[25,1,1,""],to_bernstein:[25,1,1,""]},"sage.rings.polynomial.symmetric_ideal.SymmetricIdeal":{groebner_basis:[40,2,1,""],reduce:[40,2,1,""],symmetrisation:[40,2,1,""],symmetric_basis:[40,2,1,""],interreduction:[40,2,1,""],is_maximal:[40,2,1,""],squeezed:[40,2,1,""],normalisation:[40,2,1,""],interreduced_basis:[40,2,1,""]},"sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint":{"_rmul_":[35,2,1,""],reverse:[35,2,1,""],"_add_":[35,2,1,""],"_sub_":[35,2,1,""],"_mul_":[35,2,1,""],is_irreducible:[35,2,1,""],monic:[35,2,1,""],resultant:[35,2,1,""],rational_reconstruct:[35,2,1,""],revert_series:[35,2,1,""],"_mul_trunc_":[35,2,1,""],factor:[35,2,1,""],small_roots:[35,2,1,""],"_lmul_":[35,2,1,""],squarefree_decomposition:[35,2,1,""]},"sage.rings.polynomial.pbori.BooleanPolynomialVector":{append:[13,2,1,""]},"sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular":{monomial_all_divisors:[2,2,1,""],monomial_reduce:[2,2,1,""],ngens:[2,2,1,""],Element:[2,4,1,""],monomial_lcm:[2,2,1,""],ideal:[2,2,1,""],monomial_quotient:[2,2,1,""],monomial_divides:[2,2,1,""],monomial_pairwise_prime:[2,2,1,""],gen:[2,2,1,""]},"sage.rings.polynomial.infinite_polynomial_ring.InfinitePolynomialRingFactory":{create_key:[23,2,1,""],create_object:[23,2,1,""]},"sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_zz":{ntl_set_directly:[21,2,1,""],"_rmul_":[21,2,1,""],reverse:[21,2,1,""],"_add_":[21,2,1,""],truncate:[21,2,1,""],"_sub_":[21,2,1,""],shift:[21,2,1,""],int_list:[21,2,1,""],is_gen:[21,2,1,""],"_mul_":[21,2,1,""],"_mul_trunc_":[21,2,1,""],"_lmul_":[21,2,1,""],quo_rem:[21,2,1,""],valuation:[21,2,1,""],degree:[21,2,1,""]},"sage.rings.polynomial.polynomial_element":{Polynomial_generic_dense:[34,3,1,""],PolynomialBaseringInjection:[34,3,1,""],make_generic_polynomial:[34,1,1,""],Polynomial:[34,3,1,""],universal_discriminant:[34,1,1,""],is_Polynomial:[34,1,1,""],ConstantPolynomialSection:[34,3,1,""]},"sage.rings.polynomial.symmetric_reduction":{SymmetricReductionStrategy:[50,3,1,""]},"sage.rings.polynomial.multi_polynomial_element.MPolynomial_element":{change_ring:[53,2,1,""],element:[53,2,1,""]},"sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_macaulay2_repr":{is_exact:[28,2,1,""]},"sage.rings.polynomial.plural.NCPolynomial_plural":{coefficient:[12,2,1,""],total_degree:[12,2,1,""],lc:[12,2,1,""],degree:[12,2,1,""],is_monomial:[12,2,1,""],lm:[12,2,1,""],reduce:[12,2,1,""],is_homogeneous:[12,2,1,""],exponents:[12,2,1,""],lt:[12,2,1,""],dict:[12,2,1,""],is_zero:[12,2,1,""],degrees:[12,2,1,""],monomials:[12,2,1,""],is_constant:[12,2,1,""],monomial_coefficient:[12,2,1,""],constant_coefficient:[12,2,1,""]},"sage.rings.polynomial.ideal.Ideal_1poly_field":{residue_class_degree:[3,2,1,""],residue_field:[3,2,1,""]},"sage.rings.polynomial.real_roots.rr_gap":{region:[25,2,1,""]},"sage.rings.polynomial.polynomial_real_mpfr_dense":{PolynomialRealDense:[49,3,1,""],make_PolynomialRealDense:[49,1,1,""]},"sage.rings.polynomial.polynomial_number_field.Polynomial_absolute_number_field_dense":{gcd:[48,2,1,""]},"sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular":{is_univariate:[2,2,1,""],add_m_mul_q:[2,2,1,""],variables:[2,2,1,""],reduce:[2,2,1,""],resultant:[2,2,1,""],is_squarefree:[2,2,1,""],monomial_coefficient:[2,2,1,""],is_constant:[2,2,1,""],sub_m_mul_q:[2,2,1,""],subs:[2,2,1,""],gcd:[2,2,1,""],nvariables:[2,2,1,""],gradient:[2,2,1,""],lm:[2,2,1,""],lt:[2,2,1,""],dict:[2,2,1,""],inverse_of_unit:[2,2,1,""],is_zero:[2,2,1,""],factor:[2,2,1,""],is_homogeneous:[2,2,1,""],is_unit:[2,2,1,""],quo_rem:[2,2,1,""],exponents:[2,2,1,""],coefficient:[2,2,1,""],degree:[2,2,1,""],integral:[2,2,1,""],lift:[2,2,1,""],variable:[2,2,1,""],lcm:[2,2,1,""],constant_coefficient:[2,2,1,""],lc:[2,2,1,""],coefficients:[2,2,1,""],is_monomial:[2,2,1,""],discriminant:[2,2,1,""],numerator:[2,2,1,""],total_degree:[2,2,1,""],degrees:[2,2,1,""],monomials:[2,2,1,""],univariate_polynomial:[2,2,1,""]},"sage.rings.polynomial.real_roots.bernstein_polynomial_factory_intlist":{bernstein_polynomial:[25,2,1,""],coeffs_bitsize:[25,2,1,""]},"sage.rings.polynomial.laurent_polynomial":{LaurentPolynomial_univariate:[20,3,1,""],LaurentPolynomial_mpair:[20,3,1,""],LaurentPolynomial_generic:[20,3,1,""]},"sage.rings.polynomial.pbori.BooleanPolynomialEntry":{p:[13,4,1,""]},"sage.rings.polynomial.pbori.BooleanPolynomialVectorIterator":{next:[13,2,1,""]},"sage.rings.polynomial.ideal":{Ideal_1poly_field:[3,3,1,""]},"sage.rings.polynomial.pbori.CCuddNavigator":{terminal_one:[13,2,1,""],then_branch:[13,2,1,""],constant:[13,2,1,""],else_branch:[13,2,1,""],value:[13,2,1,""]},"sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_ZZ":{truncate:[21,2,1,""],degree:[21,2,1,""],shift:[21,2,1,""],list:[21,2,1,""],is_gen:[21,2,1,""],quo_rem:[21,2,1,""],valuation:[21,2,1,""],reverse:[21,2,1,""]},"sage.rings.polynomial.real_roots.bernstein_polynomial_factory_ratlist":{bernstein_polynomial:[25,2,1,""],coeffs_bitsize:[25,2,1,""]},"sage.rings.polynomial.multi_polynomial.MPolynomial":{jacobian_ideal:[5,2,1,""],args:[5,2,1,""],sylvester_matrix:[5,2,1,""],truncate:[5,2,1,""],gradient:[5,2,1,""],is_generator:[5,2,1,""],denominator:[5,2,1,""],numerator:[5,2,1,""],content:[5,2,1,""],map_coefficients:[5,2,1,""],change_ring:[5,2,1,""],lift:[5,2,1,""],is_homogeneous:[5,2,1,""],inverse_mod:[5,2,1,""],weighted_degree:[5,2,1,""],polynomial:[5,2,1,""],derivative:[5,2,1,""],macaulay_resultant:[5,2,1,""],newton_polytope:[5,2,1,""],coefficients:[5,2,1,""],homogenize:[5,2,1,""]},"sage.rings.polynomial.pbori.BooleSet":{divisors_of:[13,2,1,""],set:[13,2,1,""],subset1:[13,2,1,""],divide:[13,2,1,""],vars:[13,2,1,""],union:[13,2,1,""],multiples_of:[13,2,1,""],include_divisors:[13,2,1,""],stable_hash:[13,2,1,""],minimal_elements:[13,2,1,""],n_nodes:[13,2,1,""],empty:[13,2,1,""],size_double:[13,2,1,""],intersect:[13,2,1,""],diff:[13,2,1,""],subset0:[13,2,1,""],ring:[13,2,1,""],navigation:[13,2,1,""],change:[13,2,1,""],cartesian_product:[13,2,1,""]},"sage.rings.polynomial.polynomial_quotient_ring_element.PolynomialQuotientRingElement":{fcp:[37,2,1,""],matrix:[37,2,1,""],trace:[37,2,1,""],minpoly:[37,2,1,""],list:[37,2,1,""],field_extension:[37,2,1,""],lift:[37,2,1,""],charpoly:[37,2,1,""],is_unit:[37,2,1,""],norm:[37,2,1,""]},"sage.rings.polynomial.polynomial_element.Polynomial_generic_dense":{truncate:[34,2,1,""],degree:[34,2,1,""],shift:[34,2,1,""],list:[34,2,1,""],quo_rem:[34,2,1,""],constant_coefficient:[34,2,1,""]},"sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_gf2":{eliminate_linear_variables:[17,2,1,""],solve:[17,2,1,""],reduced:[17,2,1,""]},"sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic":{connected_components:[17,2,1,""],algebraic_dependence:[17,2,1,""],subs:[17,2,1,""],groebner_basis:[17,2,1,""],universe:[17,2,1,""],variables:[17,2,1,""],nvariables:[17,2,1,""],connection_graph:[17,2,1,""],nparts:[17,2,1,""],reduced:[17,2,1,""],is_groebner:[17,2,1,""],ideal:[17,2,1,""],parts:[17,2,1,""],part:[17,2,1,""],maximal_degree:[17,2,1,""],ring:[17,2,1,""],coefficient_matrix:[17,2,1,""],monomials:[17,2,1,""],nmonomials:[17,2,1,""]},"sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_univariate":{degree:[20,2,1,""],polynomial_construction:[20,2,1,""],variables:[20,2,1,""],integral:[20,2,1,""],is_constant:[20,2,1,""],inverse_of_unit:[20,2,1,""],residue:[20,2,1,""],gcd:[20,2,1,""],change_ring:[20,2,1,""],dict:[20,2,1,""],coefficients:[20,2,1,""],is_zero:[20,2,1,""],variable_name:[20,2,1,""],derivative:[20,2,1,""],is_unit:[20,2,1,""],quo_rem:[20,2,1,""],exponents:[20,2,1,""],truncate:[20,2,1,""],factor:[20,2,1,""],valuation:[20,2,1,""],constant_coefficient:[20,2,1,""],is_monomial:[20,2,1,""],shift:[20,2,1,""]},"sage.rings.polynomial.polynomial_real_mpfr_dense.PolynomialRealDense":{reverse:[49,2,1,""],degree:[49,2,1,""],list:[49,2,1,""],truncate_abs:[49,2,1,""],integral:[49,2,1,""],change_ring:[49,2,1,""],shift:[49,2,1,""],truncate:[49,2,1,""],quo_rem:[49,2,1,""]},"sage.rings.polynomial.polynomial_zmod_flint.Polynomial_template":{degree:[35,2,1,""],get_cparent:[35,2,1,""],is_one:[35,2,1,""],gcd:[35,2,1,""],truncate:[35,2,1,""],shift:[35,2,1,""],xgcd:[35,2,1,""],list:[35,2,1,""],is_gen:[35,2,1,""],is_zero:[35,2,1,""],quo_rem:[35,2,1,""]},"sage.rings.polynomial.term_order":{termorder_from_singular:[45,1,1,""],TermOrder:[45,3,1,""]},"sage.rings.polynomial.polynomial_zz_pex":{Polynomial_template:[47,3,1,""],Polynomial_ZZ_pEX:[47,3,1,""],Polynomial_ZZ_pX:[47,3,1,""],make_element:[47,1,1,""]},"sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint":{"_sub_":[39,2,1,""],is_irreducible:[39,2,1,""],resultant:[39,2,1,""],"_mul_":[39,2,1,""],"_mul_trunc_":[39,2,1,""],real_root_intervals:[39,2,1,""],"_rmul_":[39,2,1,""],gcd:[39,2,1,""],discriminant:[39,2,1,""],galois_group:[39,2,1,""],factor_mod:[39,2,1,""],is_zero:[39,2,1,""],quo_rem:[39,2,1,""],factor_padic:[39,2,1,""],disc:[39,2,1,""],numerator:[39,2,1,""],degree:[39,2,1,""],xgcd:[39,2,1,""],denominator:[39,2,1,""],"_lmul_":[39,2,1,""],lcm:[39,2,1,""],is_one:[39,2,1,""],reverse:[39,2,1,""],"_add_":[39,2,1,""],hensel_lift:[39,2,1,""],list:[39,2,1,""],revert_series:[39,2,1,""],truncate:[39,2,1,""]},"sage.rings.polynomial.infinite_polynomial_ring.InfinitePolynomialRing_dense":{polynomial_ring:[23,2,1,""],construction:[23,2,1,""],tensor_with_ring:[23,2,1,""]},"sage.rings.polynomial.pbori.PolynomialFactory":{lead:[13,2,1,""]},"sage.rings.polynomial.convolution":{convolution:[36,1,1,""]},"sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_field":{base_field:[19,2,1,""],complex_embeddings:[19,2,1,""]},"sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_repr":{radical:[14,2,1,""],primary_decomposition:[14,2,1,""],associated_primes:[14,2,1,""],plot:[14,2,1,""],variety:[14,2,1,""],syzygy_module:[14,2,1,""],transformed_basis:[14,2,1,""],hilbert_series:[14,2,1,""],saturation:[14,2,1,""],integral_closure:[14,2,1,""],elimination_ideal:[14,2,1,""],interreduced_basis:[14,2,1,""],intersection:[14,2,1,""],minimal_associated_primes:[14,2,1,""],complete_primary_decomposition:[14,2,1,""],primary_decomposition_complete:[14,2,1,""],hilbert_numerator:[14,2,1,""],is_prime:[14,2,1,""],vector_space_dimension:[14,2,1,""],normal_basis:[14,2,1,""],dimension:[14,2,1,""],basis_is_groebner:[14,2,1,""],triangular_decomposition:[14,2,1,""],genus:[14,2,1,""],hilbert_polynomial:[14,2,1,""],quotient:[14,2,1,""]},"sage.rings.polynomial.pbori.BooleanMonomialMonoid":{gen:[13,2,1,""],ngens:[13,2,1,""],gens:[13,2,1,""]},"sage.rings.polynomial.symmetric_reduction.SymmetricReductionStrategy":{reset:[50,2,1,""],tailreduce:[50,2,1,""],reduce:[50,2,1,""],gens:[50,2,1,""],setgens:[50,2,1,""],add_generator:[50,2,1,""]}},titleterms:{origin:13,compil:38,domain:42,via:[12,2,47,39,24,26],polybori:13,over:[41,0,26,28,42,47,48,21,35,49,51,39],cyclotom:32,direct:24,indic:11,"boolean":13,tabl:11,ring:[11,52,8,28,14,33,3,18,5,22,4,19,31,30,20,6,23,40,15,16,37],complex:9,todo:[30,12,34,8],reduct:50,homomorph:30,infinit:[23,33,50,40,31],note:13,from:30,anoth:30,convolut:36,ideal:[3,40,14],fast:32,gf2x:26,access:[13,24],field:[42,48],substitut:44,version:[10,7,51,46],interfac:[29,13],low:24,rel:27,factor:10,zz_pex:47,symmetr:[50,40],plural:12,gener:[15,36,43,28,53],noncommut:12,real:25,flat:1,singular:[29,24],sequenc:17,groebner:[10,7,51,24,46],relat:46,kroneck:44,triangular:10,isol:[9,25],polynomi:[0,1,2,3,4,5,6,8,9,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,27,28,29,30,31,32,33,34,35,37,38,39,40,41,42,43,44,47,48,49,50,52,53],adic:[1,43,27],flint:[35,0,39],ntl:[47,41,21,26],engin:[15,24],multipl:44,educ:[10,7,51,46],"class":[8,5,17,43,34],polydict:15,basi:[10,7,51,24,46],term:45,laurent:[20,18,4],specif:13,algorithm:[10,7,51,46],level:24,dens:[41,0,27,21,35,49],pid:51,cap:27,libsingular:[12,2,24],element:[20,33,5,37],calcul:32,base:[8,5,34,43],univari:[41,0,26,42,3,49,19,47,48,21,35,52,22,37,39,34],number:48,constructor:6,implement:[41,0,13,21,35,49,39],mpfr:49,root:[9,25],order:45,multivari:[14,8,2,28,5,15,16,53],quotient:[37,19]}})