box% sage ┌────────────────────────────────────────────────────────────────────┐ │ SageMath version 9.2, Release Date: 2020-10-24 │ │ Using Python 3.9.1. Type "help()" for help. │ └────────────────────────────────────────────────────────────────────┘ sage: R.<c1,c2,c3,c4,A,B,C,e1,e2,e3,e4,alpha> = PolynomialRing(QQ, order='lex') sage: G = ideal(-A+(c1+c2)*(c3+c4), -B+(c1+c3)*(c2+c4), -C+(c1+c4)*(c2+c3), -e1+c1+c2+c3+c4, -e2+c1*c2+c1*c3+c1*c4+c2*c3+c2*c4+c3*c4, -e3+c1*c2*c3+c1*c2*c4+c1*c3*c4+c2*c3*c4, -e4+c1*c2*c3*c4).groebner_basis() sage: ((A-alpha)*(B-alpha)*(C-alpha)).reduce(G) -e1^2*e4 + e1*e2*e3 - e1*e3*alpha - e2^2*alpha + 2*e2*alpha^2 - e3^2 + 4*e4*alpha - alpha^3 sage: G_ = ideal(-A+c1*c2+c3*c4, -B+c1*c3+c2*c4, -C+c1*c4+c2*c3, -e1+c1+c2+c3+c4, -e2+c1*c2+c1*c3+c1*c4+c2*c3+c2*c4+c3*c4, -e3+c1*c2*c3+c1*c2*c4+c1*c3*c4+c2*c3*c4, -e4+c1*c2*c3*c4).groebner_basis() sage: ((A-alpha)*(B-alpha)*(C-alpha)).reduce(G_) e1^2*e4 - e1*e3*alpha - 4*e2*e4 + e2*alpha^2 + e3^2 + 4*e4*alpha - alpha^3 sage:
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