polynomials.gd

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##
#W  polynomials.gi              Manuel Delgado <[email protected]>
#W                          Pedro A. Garcia-Sanchez <[email protected]>
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##
#Y  Copyright 2015 by Manuel Delgado and Pedro Garcia-Sanchez
#Y  We adopt the copyright regulations of GAP as detailed in the
#Y  copyright notice in the GAP manual.
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#F NumericalSemigroupPolynomial(s,x)
## s is a numerical semigroup, and x a variable (or a value)
## returns the polynomial (1-x)\sum_{s\in S}x^s
##
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DeclareGlobalFunction("NumericalSemigroupPolynomial");

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#F IsNumericalSemigroupPolynomial(f) detects
## if there exists S a numerical semigroup such that P_S(x)=f
##
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DeclareGlobalFunction("IsNumericalSemigroupPolynomial");

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#F NumericalSemigroupFromNumericalSemigroupPolynomial(f) outputs
## a numerical semigroup S such that P_S(x)=f; error if no such
## S exists
##
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DeclareGlobalFunction("NumericalSemigroupFromNumericalSemigroupPolynomial");

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#F HilbertSeriesOfNumericalSemigroup(s,x)
## Computes the Hilber series of s in x : \sum_{s\in S}x^s
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DeclareGlobalFunction("HilbertSeriesOfNumericalSemigroup");

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#F Computes the Graeffe polynomial of p
##  see for instance [BD-cyclotomic]
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DeclareGlobalFunction("GraeffePolynomial");

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#F IsCyclotomicPolynomial(f) detects
## if f is a cyclotomic polynomial using the method explained in
## BD-cyclotomic
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DeclareGlobalFunction("IsCyclotomicPolynomial");

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#F IsKroneckerPolynomial(f) decides if
##   f is a Kronecker polynomial, that is,   a monic polynomial with integer coefficients
##   having all its roots in the unit circunference, equivalently, is a product of
##   cyclotomic polynomials
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DeclareGlobalFunction("IsKroneckerPolynomial");

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#P IsCyclotomicNumericalSemigroup(s)
## Checks if the polynomial fo s is Kronecker
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DeclareProperty("IsCyclotomicNumericalSemigroup",IsNumericalSemigroup);

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#F IsSelfReciprocalUnivariatePolynomial(p)
## Checks if the univariate polynomial p is selfreciprocal
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DeclareGlobalFunction("IsSelfReciprocalUnivariatePolynomial");

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# F SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity(f)
##  Computes the semigroup of values {mult(f,g) | g curve} of a plane curve
##   with one place at the infinity in the variables X(Rationals,1) and X(Rationals,2)
##  f must be monic on X(Rationals(2))
##  SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity(f,"all")
##    The same as above, but the output are the approximate roots and
##    delta-sequence
##
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DeclareGlobalFunction("SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity");

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#F IsDeltaSequence(l)
## tests whether or not l is a \delta-sequence (see for instancd [AGS14])
##
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DeclareGlobalFunction("IsDeltaSequence");

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#F DeltaSequencesWithFrobeniusNumber(f)
##   Computes  the list of delta-sequences with Frobenius number f
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DeclareGlobalFunction("DeltaSequencesWithFrobeniusNumber");

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#F CurveAssociatedToDeltaSequence(l)
##  computes the curve associated to a delta-sequence l in
##  the variables X(Rationals,1) and X(Rationals,2)
##  as explained in [AGS14]
##
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DeclareGlobalFunction("CurveAssociatedToDeltaSequence");


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#F SemigroupOfValuesOfCurve_Global(arg)
## Computes the semigroup of values of R=K[pols],
## that is, the set of possible degrees of polynomials in this ring
## pols is a set of polynomials in a single variable
## The semigroup of values is a numerical semigroup if l(K[x]/R) is finite
## If this length is not finite, the output is fail
## If the second argument "basis" is given, then the output is a basis B of
## R such that deg(B) minimally generates deg(R), and it is reduced
## If the second argument is an integer, then the output is a polynomial f in R
## with deg(f) that value (if there is none, then the output is fail)
## Implementation based in [AGSM14]
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DeclareGlobalFunction("SemigroupOfValuesOfCurve_Global");


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#F SemigroupOfValuesOfCurve_Local(arg)
## Computes the semigroup of values of R=K[pols],
## that is, the set of possible order of series in this ring
## pols is a set of polynomials in a single variable
## The semigroup of values is a numerical semigroup if l(K[[x]]/R) is finite
## If this length is not finite, the output is fail
## If the second argument "basis" is given, then the output is a basis B of
## R such that o(B) minimally generates o(R), and it is reduced
## If the second argument is an integer, then the output is a polynomial f in R
## with o(f) that value (if there is none, then the output is fail)
## Implementation based in [AGSM14]
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DeclareGlobalFunction("SemigroupOfValuesOfCurve_Local");

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#F GeneratorsModule_Global(A,M)
##
## A and M are lists of polynomials in the same variable
## Computes a basis of the ideal MK[A], that is, a set F such that
## deg(F) generates the ideal deg(MK[A]) of deg(K[A]), where deg
## stands for degree
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DeclareGlobalFunction("GeneratorsModule_Global");

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#F GeneratorsKahlerDifferentials(A)
##
## A synonym for GeneratorsModule_Global(A,M), with M the set of
## derivatives of the elements in A
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DeclareGlobalFunction("GeneratorsKahlerDifferentials");