Gröbner bases (Sage)

# Fateman's benchmark: the product f*(f + 1) for f = (1 + x + y + z + t)^n
R.<x,y,z,t> = QQ[]
f = (1 + x + y + z + t)^12
g = f*(f + 1)
len(g.dict())

20475

# gcd and factorization (Wang's EEZ: leading coefficients distributed, Hensel lifting)
F = g.factor()
[(h.degree(), e) for h, e in F]

[(1, 12), (4, 1), (8, 1)]

R.<x,y,z> = QQ[]
h = (x^2*y + 3*x - z^3)*(2*x*y*z - 7*y^2 + 1)^2*(x - y)
h.factor(), gcd(h, (x - y)^3*(x^2*y + 3*x - z^3))

((-x + y) * (-x^2*y + z^3 - 3*x) * (2*x*y*z - 7*y^2 + 1)^2, x^3*y - x^2*y^2 - x*z^3 + y*z^3 + 3*x^2 - 3*x*y)

# Over a prime field: a bivariate stage with recombination, then Hensel lifting
S.<a,b,c> = GF(7)[]
k = (a^2*b - c + 3)*(b*c^2 + a + 1)
(k^2*(a + b)).factor()

(a + b) * (-a^2*b + c - 3)^2 * (b*c^2 + a + 1)^2

# The standard benchmark families (sage.rings.ideal.Cyclic, Katsura)
P = PolynomialRing(QQ, 'x', 4)
sage.rings.ideal.Cyclic(P), sage.rings.ideal.Katsura(P)

(Ideal (x0 + x1 + x2 + x3, x0*x1 + x1*x2 + x0*x3 + x2*x3, x0*x1*x2 + x0*x1*x3 + x0*x2*x3 + x1*x2*x3, x0*x1*x2*x3 - 1) of Multivariate Polynomial Ring in x0, x1, x2, x3 over Rational Field, Ideal (x0 + 2*x1 + 2*x2 + 2*x3 - 1, x0^2 + 2*x1^2 + 2*x2^2 + 2*x3^2 - x0, 2*x0*x1 + 2*x1*x2 + 2*x2*x3 - x1, x1^2 + 2*x0*x2 + 2*x1*x3 - x2) of Multivariate Polynomial Ring in x0, x1, x2, x3 over Rational Field)

# Gröbner bases by F4 over GF(32003): the number of elements of each reduced basis
def basis_size(I):
    return len(I.groebner_basis())
for n in [5, 6, 7]:
    print('cyclic-%d' % n, basis_size(sage.rings.ideal.Cyclic(PolynomialRing(GF(32003), 'x', n))))
for n in [6, 7]:
    print('katsura-%d' % n, basis_size(sage.rings.ideal.Katsura(PolynomialRing(GF(32003), 'x', n + 1))))

cyclic-5 20
cyclic-6 45
cyclic-7 209

katsura-6 41
katsura-7 74

# Over QQ: one F4 run over QQ, each matrix reduced modulo primes < 2^31 and
# reconstructed (CRT, rational reconstruction); then, Sage's proof.polynomial()
# being on, a proof that all S-pairs reduce to 0 (a certificate from more
# eliminations modulo primes and a size bound; natively on all cores)
for n in [5, 6]:
    print('cyclic-%d' % n, basis_size(sage.rings.ideal.Cyclic(PolynomialRing(QQ, 'x', n))))
for n in [5, 6, 7]:
    print('katsura-%d' % n, basis_size(sage.rings.ideal.Katsura(PolynomialRing(QQ, 'x', n + 1))))

cyclic-5 20
cyclic-6 45

katsura-5 22
katsura-6 41

katsura-7 74

# proof.polynomial(False) skips the exact check (as Magma does): the basis is then
# wrong only if a reconstruction was accepted too early (each has two primes to spare)
with proof.WithProof('polynomial', False):
    print('cyclic-7', basis_size(sage.rings.ideal.Cyclic(PolynomialRing(QQ, 'x', 7))))
    print('katsura-7', basis_size(sage.rings.ideal.Katsura(PolynomialRing(QQ, 'x', 8))))

cyclic-7 209

katsura-7 74

# Lex bases of zero-dimensional ideals: the degrevlex basis by F4, then FGLM
# (linear algebra in the quotient); katsura-8's eliminant has degree 2^8
for n, K in [(8, GF(32003)), (5, QQ)]:
    B = sage.rings.ideal.Katsura(PolynomialRing(K, 'x', n + 1, order='lex')).groebner_basis()
    print('katsura-%d over %s:' % (n, K), len(B), 'elements, eliminant of degree', B[-1].degree())

katsura-8 over Finite Field of size 32003: 9 elements, eliminant of degree 256

katsura-5 over Rational Field: 6 elements, eliminant of degree 32

# A lexicographic basis and the points
T.<x,y,z> = PolynomialRing(QQ, order='lex')
J = T.ideal(x^2 + y*z - 2, y^2 - x*z + 1, z^2 - x*y - 3)
J.groebner_basis()[-1]

z^8 - 15/2*z^6 + 83/4*z^4 - 113/4*z^2 + 121/8

U.<u,v> = QQ[]
U.ideal((u - 2)*(u + 3)*(u - 5), v*(v - 1)).variety()

[{v: 1, u: 5}, {v: 1, u: 2}, {v: 1, u: -3}, {v: 0, u: 5}, {v: 0, u: 2}, {v: 0, u: -3}]