t1,t2,t3,t4,p,q=var('t1,t2,t3,t4,p,q') m = matrix(SR,[[t1^4+p*t1^2,t1^3+q*t1^2,t1,1], [t2^4+p*t2^2,t2^3+q*t2^2,t2,1], [t3^4+p*t3^2,t3^3+q*t3^2,t3,1], [t4^4+p*t4^2,t4^3+q*t4^2,t4,1]]) (m.determinant().factor()/((t1-t2)*(t1-t3)*(t1-t4)*(t2-t3)*(t2-t4)*(t3-t4))).show()
qt1+qt2+t1t2+qt3+t1t3+t2t3+qt4+t1t4+t2t4+t3t4−p
p,q,r=var('p,q,r') m = matrix(SR,[[2,0,0,1], [0,1,0,r], [0,0,2,r^2], [p,q,r,0]]) (m.determinant()/(-2)).show()
r3+2qr+p
t1,t2,t3,t4,p,q,r=var('t1,t2,t3,t4,p,q,r') m = matrix(SR,[[t1^4+p*t1,t1^3-q*t1,t1^2+r*t1,1], [t2^4+p*t2,t2^3-q*t2,t2^2+r*t2,1], [t3^4+p*t3,t3^3-q*t3,t3^2+r*t3,1], [t4^4+p*t4,t4^3-q*t4,t4^2+r*t4,1]]) (m.determinant().factor()/((t1-t2)*(t1-t3)*(t1-t4)*(t2-t3)*(t2-t4)*(t3-t4))).show()
rt1t2+rt1t3+rt2t3+t1t2t3+rt1t4+rt2t4+t1t2t4+rt3t4+t1t3t4+t2t3t4+qt1+qt2+qt3+qt4+p
p,q,r,s=var('p,q,r,s') #a14,a23,a24,a33,a34,a44 m = matrix(SR,[[1,1,0,0,0,0], [2*s,0,2,1,0,0], [0,r,s,0,1,0], [0,2*q,0,2*r,2*s,1], [p,0,q,0,r,s], [2*p*s,2*q*r,2*q*s,r^2,2*r*s,s^2]]) (m.determinant().factor()/2).show()
(r3−2qrs+ps2−q2+pr)(s2+r)
p,q,r,s=var('p,q,r,s') #a13,a22,a24,a33,a34,a44 m = matrix(SR,[[2,1,0,0,0,0], [2*r,0,2,1,0,0], [0,q,s,0,1,0], [2*p,0,0,2*r,2*s,1], [0,0,q,0,r,s], [2*p*r,q^2,2*q*s,r^2,2*r*s,s^2]]) (m.determinant().factor()/2).show()
(r3−2qrs+ps2−q2+pr)(rs+q)
t1,t2,t3,t4,p,q,r,s=var('t1,t2,t3,t4,p,q,r,s') m = matrix(SR,[[t1^4-p,t1^3+q,t1^2-r,t1+s], [t2^4-p,t2^3+q,t2^2-r,t2+s], [t3^4-p,t3^3+q,t3^2-r,t3+s], [t4^4-p,t4^3+q,t4^2-r,t4+s]]) (m.determinant().factor()/((t1-t2)*(t1-t3)*(t1-t4)*(t2-t3)*(t2-t4)*(t3-t4))).show()
st1t2t3+st1t2t4+st1t3t4+st2t3t4+t1t2t3t4+rt1t2+rt1t3+rt2t3+rt1t4+rt2t4+rt3t4+qt1+qt2+qt3+qt4+p
p,q,r,s=var('p,q,r,s') #a13,a14,a24,a33,a34,a44 m = matrix(SR,[[-2*r,2*s,2,1,0,0], [-2*q,r,s,0,1,0], [-2*p,-2*q,0,-2*r,2*s,1], [0,-p,q,0,-r,s], [2*(p*r-q^2),-2*(p*s-q*r),2*q*s,r^2,-2*r*s,s^2]]) minors = m.minors(5) for i in minors: i.factor().show()
4(r3−2qrs+ps2+q2−pr)(q2−pr)
−2(r3−2qrs+ps2+q2−pr)(qr−ps)
4(r3−2qrs+ps2+q2−pr)(qs−p)
−2(r3−2qrs+ps2+q2−pr)(r2−2qs+p)
−2(r3−2qrs+ps2+q2−pr)(rs−q)
2(r3−2qrs+ps2+q2−pr)(s2−r)
R.<p,q,r,s,b,d> = PolynomialRing(QQ,6) I = R * [r - s*(b+d) + b*d, q - s*(b^2+b*d+d^2) + b*d*(b+d), p - s*(b^3+b^2*d+b*d^2+d^3) + b*d*(b^2+b*d+d^2)] I.elimination_ideal([b,d])
Ideal (r^3 - 2*q*r*s + p*s^2 + q^2 - p*r) of Multivariate Polynomial Ring in p, q, r, s, b, d over Rational Field
# rational space quartic with coordinates of degrees 4,3,2,1 t,t1,t2,t3,t4=var('t,t1,t2,t3,t4') p, q, r, s = var('p,q,r,s') p1 = t^4 - p p2 = t^3 + q p3 = t^2 - r p4 = t + s # q(x) describes the space quartic as x varies q(x) = [p1.substitute(t==x), p2.substitute(t==x), p3.substitute(t==x), p4.substitute(t==x)] # 4 distinct points t1, t2, t3, t4 are coplanar iff the following matrix is singular m = matrix(SR,[q(t1), q(t2), q(t3), q(t4)]) F=(m.determinant().factor())/((t1-t2)*(t1-t3)*(t1-t4)*(t2-t3)*(t2-t4)*(t3-t4)) print("The symmetric expression F is") F.show() print("Solve F=0 for t4 and set it equal to f") f=solve(F,t4)[0].rhs(); f.show() print("Quotient of the partial derivatives of f with respect to t2 and t3:") g = (f.diff(t2)/f.diff(t3)).simplify_full(); g.show() print("The above expression should be independent of t1, so take the partial derivative wrt t1, take the numerator of that and factorize:") h = g.diff(t1).numerator().factor(); h.show() print("Since t1, t2, t3 are arbitrary in some small ball, the first and last factors are non-zero generically, and we obtain that the catalecticant vanishes.") g.diff(t1).denominator().factor().show()
The symmetric expression F is
st1t2t3+st1t2t4+st1t3t4+st2t3t4+t1t2t3t4+rt1t2+rt1t3+rt2t3+rt1t4+rt2t4+rt3t4+qt1+qt2+qt3+qt4+p
Solve F=0 for t4 and set it equal to f
−rt1+(st1+r)t2+(st1+(s+t1)t2+r)t3+qqt1+(rt1+q)t2+(rt1+(st1+r)t2+q)t3+p
Quotient of the partial derivatives of f with respect to t2 and t3:
(r2−qs)t12+((s2−r)t12+r2−qs+(rs−q)t1)t22+q2−pr+(qr−ps)t1+((rs−q)t12+qr−ps+(r2−p)t1)t2(r2−qs)t12+((s2−r)t12+r2−qs+(rs−q)t1)t32+q2−pr+(qr−ps)t1+((rs−q)t12+qr−ps+(r2−p)t1)t3
The above expression should be independent of t1, so take the partial derivative wrt t1, take the numerator of that and factorize:
(st12t2+st12t3+2st1t2t3+t12t2t3+rt12+2rt1t2+2rt1t3+rt2t3+2qt1+qt2+qt3+p)(r3−2qrs+ps2+q2−pr)(t2−t3)
Since t1, t2, t3 are arbitrary in some small ball, the first and last factors are non-zero generically, and we obtain that the catalecticant vanishes.
(s2t12t22+rst12t2+rst1t22−rt12t22+r2t12−qst12+r2t1t2−qt12t2+r2t22−qst22−qt1t22+qrt1−pst1+qrt2−pst2−pt1t2+q2−pr)2
R.<p,q,r,s,b,d> = PolynomialRing(QQ,6) I = R * [b*d, b*d*(b+d), 1 + b*d*(b^2+b*d+d^2)] I.elimination_ideal([b,d])
Ideal (1) of Multivariate Polynomial Ring in p, q, r, s, b, d over Rational Field