def solve_reduced_system(system, entry=False):
"""Recursively solves reduced polynomial systems. """
basis = poly_groebner(system)
if len(basis) == 1 and basis[0].is_one:
if not entry:
return []
else:
return None
univariate = filter(is_univariate, basis)
if len(univariate) == 1:
f = univariate.pop()
else:
raise PolynomialError("Not a zero-dimensional system")
zeros = roots(Poly(f, f.symbols[-1])).keys()
if not zeros:
return []
if len(basis) == 1:
return [ [zero] for zero in zeros ]
solutions = []
for zero in zeros:
new_system = []
for poly in basis[:-1]:
eq = poly.evaluate((poly.symbols[-1], zero))
if not eq.is_zero:
new_system.append(eq)
for solution in solve_reduced_system(new_system):
solutions.append(solution + [zero])
return solutions
def solve_reduced_system(system, entry=False):
"""Recursively solves reduced polynomial systems. """
basis = poly_groebner(system)
if len(basis) == 1 and basis[0].is_one:
if not entry:
return []
else:
return None
univariate = filter(is_univariate, basis)
if len(univariate) == 1:
f = univariate.pop()
else:
raise PolynomialError("Not a zero-dimensional system")
zeros = roots(Poly(f, f.symbols[-1])).keys()
if not zeros:
return []
if len(basis) == 1:
return [[zero] for zero in zeros]
solutions = []
for zero in zeros:
new_system = []
for poly in basis[:-1]:
eq = poly.evaluate((poly.symbols[-1], zero))
if not eq.is_zero:
new_system.append(eq)
for solution in solve_reduced_system(new_system):
solutions.append(solution + [zero])
return solutions
def test_poly_groebner():
assert poly_groebner(0, x) == [Poly((), x)]
assert poly_groebner(x*y, x) == [Poly(x, x)]
assert poly_groebner(x*y, z) == [Poly(1, z)]
assert poly_groebner((x**2 + 2*x*y**2, x*y + 2*y**3 - 1), y, x, order='lex') == \
[Poly(y**3 - Rational(1,2), y, x, order='lex'),
Poly(x, y, x, order='lex')]
assert poly_groebner((y-x**2, z-x**3), y, z, x, order='lex') == \
[Poly(-x**2+y, y, z, x, order='lex'),
Poly(z-x**3, y, z, x, order='lex')]
assert poly_groebner((x**3-2*x*y, x**2*y-2*y**2+x), x, y, order='grlex') == \
[Poly(x**2, x, y, order='grlex'),
Poly(x*y, x, y, order='grlex'),
Poly(y**2-x/2, x, y, order='grlex')]
def test_poly_groebner():
assert poly_groebner(0, x) == [Poly((), x)]
assert poly_groebner(x*y, x) == [Poly(x, x)]
assert poly_groebner(x*y, z) == [Poly(1, z)]
assert poly_groebner((x**2 + 2*x*y**2, x*y + 2*y**3 - 1), y, x, order='lex') == \
[Poly(y**3 - Rational(1,2), y, x, order='lex'),
Poly(x, y, x, order='lex')]
assert poly_groebner((y-x**2, z-x**3), y, z, x, order='lex') == \
[Poly(-x**2+y, y, z, x, order='lex'),
Poly(z-x**3, y, z, x, order='lex')]
assert poly_groebner((x**3-2*x*y, x**2*y-2*y**2+x), x, y, order='grlex') == \
[Poly(x**2, x, y, order='grlex'),
Poly(x*y, x, y, order='grlex'),
Poly(y**2-x/2, x, y, order='grlex')]
