def _autoreduce(G):
"""
Code from Toy Bucberger Algorithm Sympy
https://mattpap.github.io/masters-thesis/html/src/groebner.html
"""
G_red = []
for i, g in enumerate(G):
G[i] = normal_form(g, G[:i] + G[i + 1:])
if not G[i].isZero():
G_red.append(G[i])
return G_red
def test_normal_form_2(self):
x0, x1, x2, x3 = self.B.gens
_1 = self.B.one
_0 = self.B.zero
# Groebner basis for F should reduce to 0
G = [x0 + _1, x1 + _1, x2 + _1, x3 + _1]
p = x0 * x1 * x2 + x0 * x1 * x3 + x0 * x2 * x3 + x2
p_nf = normal_form(p, G)
self.assertEqual(_0, p_nf)
def buchberger(F, BRing):
"""Basic Buchberger algorithm with two criteria."""
if len(F) > 1:
G = autoreduce(F)
else:
G = F
k = len(G)
x = BRing.gens
# Fill pairs with negative indexes for field polynomials
pairs = [(i, j) for i in range(-len(BRing.gens), k) for j in range(len(G))]
# Matrix M with treated pairs
M = [[0 for i in range(len(G))] for j in range(len(G))]
count = 0
while pairs != []:
i, j = pairs.pop(0)
# case with field polynomial
if i < 0:
Gj_lm, xi = G[j].lm(), x[abs(i) - 1].lm()
if Gj_lm.isrelativelyprime(xi):
continue
s = G[j] * x[abs(i) - 1]
else:
M[i][j] = 1
if _criteria(i, j, M, G):
continue
p, q = G[i], G[j]
s = spoly(p, q)
h = normal_form(s, G)
if not h.isZero():
G.append(h)
pairs += [(i, k) for i in range(-len(BRing.gens), k)]
k += 1
# Extend M with new entry (h, g) forall g in G
for row in M:
row.append(0)
M.append([0 for i in range(len(G))])
count += 1
# Autoreduce
G_red = autoreduce(G)
return G_red
def autoreduce(F):
"""Autoreduce set of polynomials
K. Geddes, S. Czapor, G. Labahn
Algorithms for Computer Algebra (1992)
p. 448 ReduceSet(E)
"""
G = F.copy()
P = []
while G != []:
h = G.pop(-1)
h = normal_form(h, P)
if not h.isZero():
P_size = len(P)
i = 0
while i < P_size:
p = P[i]
if p.lm().isdivisible(h.lm()):
G.append(p)
P.remove(p)
P_size -= 1
else:
i += 1
P.append(h)
P_size = len(P)
i = 0
while i < P_size:
h = P.pop()
h = normal_form(h, P)
if h.isZero():
P_size -= 1
else:
P.append(h)
i += 1
return P
def test_normal_form_3(self):
x0, x1, x2, x3 = self.B.gens
_1 = self.B.one
F = [
x0 + x1 + x2 + x3, x0 * x1 + x0 * x3 + x1 * x2 + x2 * x3,
x0 * x1 * x2 + x0 * x1 * x3 + x0 * x2 * x3 + x1 * x2 * x3,
x0 * x1 * x2 * x3 + _1, x1 + x3
]
p = x1 * x2 * x3 + _1
p_nf = normal_form(p, F)
self.assertEqual(p_nf, x2 * x3 + _1)
def test_normal_form_5(self):
x1, x2, x3 = self.B.gens[:3]
_1 = self.B.one
s = x1 * x3 + x1 + x2 * x3 + x2 + x3 + _1
F = [
x1 * x2 * x3, x1 * x2 * x3 + x1 * x2, x1 * x2 * x3 + x1 * x3,
x1 * x2 * x3 + x1 * x2 + x1 * x3 + x1, x1 * x2 * x3 + x2 * x3,
x1 * x2 * x3 + x1 * x2 + x2 * x3 + x2,
x1 * x2 * x3 + x1 * x3 + x2 * x3 + x3,
x1 * x2 * x3 + x1 * x2 + x1 * x3 + x1 + x2 * x3 + x2 + x3 + _1,
x1 * x3, x2 * x3, x3, x1 * x2, x2, x1, _1
]
# Should reduce to 0
nf = normal_form(s, F)
self.assertTrue(nf.isZero())
def test_normal_form_4(self):
x1, x2, x3 = self.B.gens[:3]
_1 = self.B.one
F = [
x1 * x2 * x3, x1 * x2 * x3 + x1 * x2, x1 * x2 * x3 + x1 * x3,
x1 * x2 * x3 + x1 * x2 + x1 * x3 + x1, x1 * x2 * x3 + x2 * x3,
x1 * x2 * x3 + x1 * x2 + x2 * x3 + x2,
x1 * x2 * x3 + x1 * x3 + x2 * x3 + x3,
x1 * x2 * x3 + x1 * x2 + x1 * x3 + x1 + x2 * x3 + x2 + x3 + _1
]
# S-polynomial for p = x1*x2*x3 + x1*x2 and
# q = x1*x2*x3 + x1*x2 + x1*x3 + x1 + x2*x3 + x2 + x3 + 1
s = x1 * x3 + x1 + x2 * x3 + x2 + x3 + _1
nf = normal_form(s, F)
self.assertEqual(s, nf)