def _rational_case(cls, poly, func):
"""Handle the rational function case. """
roots = symbols('r:%d' % poly.degree())
var, expr = func.variables[0], func.expr
f = sum(expr.subs(var, r) for r in roots)
p, q = together(f).as_numer_denom()
domain = QQ[roots]
p = p.expand()
q = q.expand()
try:
p = Poly(p, domain=domain, expand=False)
except GeneratorsNeeded:
p, p_coeff = None, (p, )
else:
p_monom, p_coeff = zip(*p.terms())
try:
q = Poly(q, domain=domain, expand=False)
except GeneratorsNeeded:
q, q_coeff = None, (q, )
else:
q_monom, q_coeff = zip(*q.terms())
coeffs, mapping = symmetrize(p_coeff + q_coeff, formal=True)
formulas, values = viete(poly, roots), []
for (sym, _), (_, val) in zip(mapping, formulas):
values.append((sym, val))
for i, (coeff, _) in enumerate(coeffs):
coeffs[i] = coeff.subs(values)
n = len(p_coeff)
p_coeff = coeffs[:n]
q_coeff = coeffs[n:]
if p is not None:
p = Poly(dict(zip(p_monom, p_coeff)), *p.gens).as_expr()
else:
(p, ) = p_coeff
if q is not None:
q = Poly(dict(zip(q_monom, q_coeff)), *q.gens).as_expr()
else:
(q, ) = q_coeff
return factor(p / q)
def _rational_case(cls, poly, func):
"""Handle the rational function case. """
roots = symbols('r:%d' % poly.degree())
var, expr = func.variables[0], func.expr
f = sum(expr.subs(var, r) for r in roots)
p, q = together(f).as_numer_denom()
domain = QQ[roots]
p = p.expand()
q = q.expand()
try:
p = Poly(p, domain=domain, expand=False)
except GeneratorsNeeded:
p, p_coeff = None, (p,)
else:
p_monom, p_coeff = zip(*p.terms())
try:
q = Poly(q, domain=domain, expand=False)
except GeneratorsNeeded:
q, q_coeff = None, (q,)
else:
q_monom, q_coeff = zip(*q.terms())
coeffs, mapping = symmetrize(p_coeff + q_coeff, formal=True)
formulas, values = viete(poly, roots), []
for (sym, _), (_, val) in zip(mapping, formulas):
values.append((sym, val))
for i, (coeff, _) in enumerate(coeffs):
coeffs[i] = coeff.subs(values)
n = len(p_coeff)
p_coeff = coeffs[:n]
q_coeff = coeffs[n:]
if p is not None:
p = Poly(dict(zip(p_monom, p_coeff)), *p.gens).as_expr()
else:
(p,) = p_coeff
if q is not None:
q = Poly(dict(zip(q_monom, q_coeff)), *q.gens).as_expr()
else:
(q,) = q_coeff
return factor(p/q)
def test_symmetrize():
assert symmetrize(0, x, y, z) == (0, 0)
assert symmetrize(1, x, y, z) == (1, 0)
s1 = x + y + z
s2 = x*y + x*z + y*z
s3 = x*y*z
assert symmetrize(1) == (1, 0)
assert symmetrize(1, formal=True) == (1, 0, [])
assert symmetrize(x) == (x, 0)
assert symmetrize(x + 1) == (x + 1, 0)
assert symmetrize(x, x, y) == (x + y, -y)
assert symmetrize(x + 1, x, y) == (x + y + 1, -y)
assert symmetrize(x, x, y, z) == (s1, -y - z)
assert symmetrize(x + 1, x, y, z) == (s1 + 1, -y - z)
assert symmetrize(x**2, x, y, z) == (s1**2 - 2*s2, -y**2 - z**2)
assert symmetrize(x**2 + y**2) == (-2*x*y + (x + y)**2, 0)
assert symmetrize(x**2 - y**2) == (-2*x*y + (x + y)**2, -2*y**2)
assert symmetrize(x**3 + y**2 + a*x**2 + b*y**3, x, y) == \
(-3*x*y*(x + y) - 2*a*x*y + a*(x + y)**2 + (x + y)**3, y**2*(1 - a) + y**3*(b - 1))
U = [u0, u1, u2] = symbols('u:3')
assert symmetrize(x + 1, x, y, z, formal=True, symbols=U) == \
(u0 + 1, -y - z, [(u0, x + y + z), (u1, x*y + x*z + y*z), (u2, x*y*z)])
assert symmetrize([1, 2, 3]) == [(1, 0), (2, 0), (3, 0)]
assert symmetrize([1, 2, 3], formal=True) == ([(1, 0), (2, 0), (3, 0)], [])
assert symmetrize([x + y, x - y]) == [(x + y, 0), (x + y, -2*y)]
def test_symmetrize():
assert symmetrize(0, x, y, z) == (0, 0)
assert symmetrize(1, x, y, z) == (1, 0)
s1 = x + y + z
s2 = x * y + x * z + y * z
s3 = x * y * z
assert symmetrize(1) == (1, 0)
assert symmetrize(1, formal=True) == (1, 0, [])
assert symmetrize(x) == (x, 0)
assert symmetrize(x + 1) == (x + 1, 0)
assert symmetrize(x, x, y) == (x + y, -y)
assert symmetrize(x + 1, x, y) == (x + y + 1, -y)
assert symmetrize(x, x, y, z) == (s1, -y - z)
assert symmetrize(x + 1, x, y, z) == (s1 + 1, -y - z)
assert symmetrize(x**2, x, y, z) == (s1**2 - 2 * s2, -y**2 - z**2)
assert symmetrize(x**2 + y**2) == (-2 * x * y + (x + y)**2, 0)
assert symmetrize(x**2 - y**2) == (-2 * x * y + (x + y)**2, -2 * y**2)
assert symmetrize(x**3 + y**2 + a*x**2 + b*y**3, x, y) == \
(-3*x*y*(x + y) - 2*a*x*y + a*(x + y)**2 + (x + y)**3, y**2*(1 - a) + y**3*(b - 1))
U = [u0, u1, u2] = symbols('u:3')
assert symmetrize(x + 1, x, y, z, formal=True, symbols=U) == \
(u0 + 1, -y - z, [(u0, x + y + z), (u1, x*y + x*z + y*z), (u2, x*y*z)])
assert symmetrize([1, 2, 3]) == [(1, 0), (2, 0), (3, 0)]
assert symmetrize([1, 2, 3], formal=True) == ([(1, 0), (2, 0), (3, 0)], [])
assert symmetrize([x + y, x - y]) == [(x + y, 0), (x + y, -2 * y)]
from sympy.polys.polyfuncs import symmetrize
from sympy import *
from sympy.polys.orderings import monomial_key
u1, u2, u3, u4 = symbols('u1 u2 u3 u4')
u = [u1, u2, u3, u4]
discriminant = 1
for i in range(4):
for j in range(i + 1, 4):
discriminant *= (u[i] - u[j]) * (u[i] - u[j])
print(latex(symmetrize(discriminant, formal=True)[0]))
for a in range(-10, 10):
if a != 0 and -8 % a == 0:
print("64 - 48 \cdot %d = %d" % (a, 64 - 48 * a))
x, y, z = symbols('x y z')
a = -8
b = 12
disc = 256 * b**3 - 27 * a**4
print("h(y) = %s" % latex(expand(y * y - disc)))
a = 0
b = 0
c = 8
from sympy.polys.polyfuncs import symmetrize
from sympy import *
from sympy.polys.orderings import monomial_key
x, y, z = symbols('u_1 u_2 u_3')
s1 = x + y + z
s2 = x * y + y * z + z * x
s3 = x * y * z
v = x * y * y + y * z * z + z * x * x
vv = y * x * x + x * z * z + z * y * y
#print("Problem 3(iii)")
#print(expand(v + vv - s1 * s2))
print("Problem 3(v)")
print(symmetrize(v * vv))
print(
expand(
symmetrize(v * vv)[0] -
(9 * s3**2 + s3 * s1**3 - 6 * s3 * s1 * s2 + s2**3)))