MLPY/Lib/site-packages/sympy/polys/tests/test_rings.py

"""Test sparse polynomials. """

from functools import reduce
from operator import add, mul

from sympy.polys.rings import ring, xring, sring, PolyRing, PolyElement
from sympy.polys.fields import field, FracField
from sympy.polys.densebasic import ninf
from sympy.polys.domains import ZZ, QQ, RR, FF, EX
from sympy.polys.orderings import lex, grlex
from sympy.polys.polyerrors import GeneratorsError, \
    ExactQuotientFailed, MultivariatePolynomialError, CoercionFailed

from sympy.testing.pytest import raises
from sympy.core import Symbol, symbols
from sympy.core.singleton import S
from sympy.core.numbers import pi
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt

def test_PolyRing___init__():
    x, y, z, t = map(Symbol, "xyzt")

    assert len(PolyRing("x,y,z", ZZ, lex).gens) == 3
    assert len(PolyRing(x, ZZ, lex).gens) == 1
    assert len(PolyRing(("x", "y", "z"), ZZ, lex).gens) == 3
    assert len(PolyRing((x, y, z), ZZ, lex).gens) == 3
    assert len(PolyRing("", ZZ, lex).gens) == 0
    assert len(PolyRing([], ZZ, lex).gens) == 0

    raises(GeneratorsError, lambda: PolyRing(0, ZZ, lex))

    assert PolyRing("x", ZZ[t], lex).domain == ZZ[t]
    assert PolyRing("x", 'ZZ[t]', lex).domain == ZZ[t]
    assert PolyRing("x", PolyRing("t", ZZ, lex), lex).domain == ZZ[t]

    raises(GeneratorsError, lambda: PolyRing("x", PolyRing("x", ZZ, lex), lex))

    _lex = Symbol("lex")
    assert PolyRing("x", ZZ, lex).order == lex
    assert PolyRing("x", ZZ, _lex).order == lex
    assert PolyRing("x", ZZ, 'lex').order == lex

    R1 = PolyRing("x,y", ZZ, lex)
    R2 = PolyRing("x,y", ZZ, lex)
    R3 = PolyRing("x,y,z", ZZ, lex)

    assert R1.x == R1.gens[0]
    assert R1.y == R1.gens[1]
    assert R1.x == R2.x
    assert R1.y == R2.y
    assert R1.x != R3.x
    assert R1.y != R3.y

def test_PolyRing___hash__():
    R, x, y, z = ring("x,y,z", QQ)
    assert hash(R)

def test_PolyRing___eq__():
    assert ring("x,y,z", QQ)[0] == ring("x,y,z", QQ)[0]
    assert ring("x,y,z", QQ)[0] is ring("x,y,z", QQ)[0]

    assert ring("x,y,z", QQ)[0] != ring("x,y,z", ZZ)[0]
    assert ring("x,y,z", QQ)[0] is not ring("x,y,z", ZZ)[0]

    assert ring("x,y,z", ZZ)[0] != ring("x,y,z", QQ)[0]
    assert ring("x,y,z", ZZ)[0] is not ring("x,y,z", QQ)[0]

    assert ring("x,y,z", QQ)[0] != ring("x,y", QQ)[0]
    assert ring("x,y,z", QQ)[0] is not ring("x,y", QQ)[0]

    assert ring("x,y", QQ)[0] != ring("x,y,z", QQ)[0]
    assert ring("x,y", QQ)[0] is not ring("x,y,z", QQ)[0]

def test_PolyRing_ring_new():
    R, x, y, z = ring("x,y,z", QQ)

    assert R.ring_new(7) == R(7)
    assert R.ring_new(7*x*y*z) == 7*x*y*z

    f = x**2 + 2*x*y + 3*x + 4*z**2 + 5*z + 6

    assert R.ring_new([[[1]], [[2], [3]], [[4, 5, 6]]]) == f
    assert R.ring_new({(2, 0, 0): 1, (1, 1, 0): 2, (1, 0, 0): 3, (0, 0, 2): 4, (0, 0, 1): 5, (0, 0, 0): 6}) == f
    assert R.ring_new([((2, 0, 0), 1), ((1, 1, 0), 2), ((1, 0, 0), 3), ((0, 0, 2), 4), ((0, 0, 1), 5), ((0, 0, 0), 6)]) == f

    R, = ring("", QQ)
    assert R.ring_new([((), 7)]) == R(7)

def test_PolyRing_drop():
    R, x,y,z = ring("x,y,z", ZZ)

    assert R.drop(x) == PolyRing("y,z", ZZ, lex)
    assert R.drop(y) == PolyRing("x,z", ZZ, lex)
    assert R.drop(z) == PolyRing("x,y", ZZ, lex)

    assert R.drop(0) == PolyRing("y,z", ZZ, lex)
    assert R.drop(0).drop(0) == PolyRing("z", ZZ, lex)
    assert R.drop(0).drop(0).drop(0) == ZZ

    assert R.drop(1) == PolyRing("x,z", ZZ, lex)

    assert R.drop(2) == PolyRing("x,y", ZZ, lex)
    assert R.drop(2).drop(1) == PolyRing("x", ZZ, lex)
    assert R.drop(2).drop(1).drop(0) == ZZ

    raises(ValueError, lambda: R.drop(3))
    raises(ValueError, lambda: R.drop(x).drop(y))

def test_PolyRing___getitem__():
    R, x,y,z = ring("x,y,z", ZZ)

    assert R[0:] == PolyRing("x,y,z", ZZ, lex)
    assert R[1:] == PolyRing("y,z", ZZ, lex)
    assert R[2:] == PolyRing("z", ZZ, lex)
    assert R[3:] == ZZ

def test_PolyRing_is_():
    R = PolyRing("x", QQ, lex)

    assert R.is_univariate is True
    assert R.is_multivariate is False

    R = PolyRing("x,y,z", QQ, lex)

    assert R.is_univariate is False
    assert R.is_multivariate is True

    R = PolyRing("", QQ, lex)

    assert R.is_univariate is False
    assert R.is_multivariate is False

def test_PolyRing_add():
    R, x = ring("x", ZZ)
    F = [ x**2 + 2*i + 3 for i in range(4) ]

    assert R.add(F) == reduce(add, F) == 4*x**2 + 24

    R, = ring("", ZZ)

    assert R.add([2, 5, 7]) == 14

def test_PolyRing_mul():
    R, x = ring("x", ZZ)
    F = [ x**2 + 2*i + 3 for i in range(4) ]

    assert R.mul(F) == reduce(mul, F) == x**8 + 24*x**6 + 206*x**4 + 744*x**2 + 945

    R, = ring("", ZZ)

    assert R.mul([2, 3, 5]) == 30

def test_PolyRing_symmetric_poly():
    R, x, y, z, t = ring("x,y,z,t", ZZ)

    raises(ValueError, lambda: R.symmetric_poly(-1))
    raises(ValueError, lambda: R.symmetric_poly(5))

    assert R.symmetric_poly(0) == R.one
    assert R.symmetric_poly(1) == x + y + z + t
    assert R.symmetric_poly(2) == x*y + x*z + x*t + y*z + y*t + z*t
    assert R.symmetric_poly(3) == x*y*z + x*y*t + x*z*t + y*z*t
    assert R.symmetric_poly(4) == x*y*z*t

def test_sring():
    x, y, z, t = symbols("x,y,z,t")

    R = PolyRing("x,y,z", ZZ, lex)
    assert sring(x + 2*y + 3*z) == (R, R.x + 2*R.y + 3*R.z)

    R = PolyRing("x,y,z", QQ, lex)
    assert sring(x + 2*y + z/3) == (R, R.x + 2*R.y + R.z/3)
    assert sring([x, 2*y, z/3]) == (R, [R.x, 2*R.y, R.z/3])

    Rt = PolyRing("t", ZZ, lex)
    R = PolyRing("x,y,z", Rt, lex)
    assert sring(x + 2*t*y + 3*t**2*z, x, y, z) == (R, R.x + 2*Rt.t*R.y + 3*Rt.t**2*R.z)

    Rt = PolyRing("t", QQ, lex)
    R = PolyRing("x,y,z", Rt, lex)
    assert sring(x + t*y/2 + t**2*z/3, x, y, z) == (R, R.x + Rt.t*R.y/2 + Rt.t**2*R.z/3)

    Rt = FracField("t", ZZ, lex)
    R = PolyRing("x,y,z", Rt, lex)
    assert sring(x + 2*y/t + t**2*z/3, x, y, z) == (R, R.x + 2*R.y/Rt.t + Rt.t**2*R.z/3)

    r = sqrt(2) - sqrt(3)
    R, a = sring(r, extension=True)
    assert R.domain == QQ.algebraic_field(sqrt(2) + sqrt(3))
    assert R.gens == ()
    assert a == R.domain.from_sympy(r)

def test_PolyElement___hash__():
    R, x, y, z = ring("x,y,z", QQ)
    assert hash(x*y*z)

def test_PolyElement___eq__():
    R, x, y = ring("x,y", ZZ, lex)

    assert ((x*y + 5*x*y) == 6) == False
    assert ((x*y + 5*x*y) == 6*x*y) == True
    assert (6 == (x*y + 5*x*y)) == False
    assert (6*x*y == (x*y + 5*x*y)) == True

    assert ((x*y - x*y) == 0) == True
    assert (0 == (x*y - x*y)) == True

    assert ((x*y - x*y) == 1) == False
    assert (1 == (x*y - x*y)) == False

    assert ((x*y - x*y) == 1) == False
    assert (1 == (x*y - x*y)) == False

    assert ((x*y + 5*x*y) != 6) == True
    assert ((x*y + 5*x*y) != 6*x*y) == False
    assert (6 != (x*y + 5*x*y)) == True
    assert (6*x*y != (x*y + 5*x*y)) == False

    assert ((x*y - x*y) != 0) == False
    assert (0 != (x*y - x*y)) == False

    assert ((x*y - x*y) != 1) == True
    assert (1 != (x*y - x*y)) == True

    assert R.one == QQ(1, 1) == R.one
    assert R.one == 1 == R.one

    Rt, t = ring("t", ZZ)
    R, x, y = ring("x,y", Rt)

    assert (t**3*x/x == t**3) == True
    assert (t**3*x/x == t**4) == False

def test_PolyElement__lt_le_gt_ge__():
    R, x, y = ring("x,y", ZZ)

    assert R(1) < x < x**2 < x**3
    assert R(1) <= x <= x**2 <= x**3

    assert x**3 > x**2 > x > R(1)
    assert x**3 >= x**2 >= x >= R(1)

def test_PolyElement__str__():
    x, y = symbols('x, y')

    for dom in [ZZ, QQ, ZZ[x], ZZ[x,y], ZZ[x][y]]:
        R, t = ring('t', dom)
        assert str(2*t**2 + 1) == '2*t**2 + 1'

    for dom in [EX, EX[x]]:
        R, t = ring('t', dom)
        assert str(2*t**2 + 1) == 'EX(2)*t**2 + EX(1)'

def test_PolyElement_copy():
    R, x, y, z = ring("x,y,z", ZZ)

    f = x*y + 3*z
    g = f.copy()

    assert f == g
    g[(1, 1, 1)] = 7
    assert f != g

def test_PolyElement_as_expr():
    R, x, y, z = ring("x,y,z", ZZ)
    f = 3*x**2*y - x*y*z + 7*z**3 + 1

    X, Y, Z = R.symbols
    g = 3*X**2*Y - X*Y*Z + 7*Z**3 + 1

    assert f != g
    assert f.as_expr() == g

    U, V, W = symbols("u,v,w")
    g = 3*U**2*V - U*V*W + 7*W**3 + 1

    assert f != g
    assert f.as_expr(U, V, W) == g

    raises(ValueError, lambda: f.as_expr(X))

    R, = ring("", ZZ)
    assert R(3).as_expr() == 3

def test_PolyElement_from_expr():
    x, y, z = symbols("x,y,z")
    R, X, Y, Z = ring((x, y, z), ZZ)

    f = R.from_expr(1)
    assert f == 1 and isinstance(f, R.dtype)

    f = R.from_expr(x)
    assert f == X and isinstance(f, R.dtype)

    f = R.from_expr(x*y*z)
    assert f == X*Y*Z and isinstance(f, R.dtype)

    f = R.from_expr(x*y*z + x*y + x)
    assert f == X*Y*Z + X*Y + X and isinstance(f, R.dtype)

    f = R.from_expr(x**3*y*z + x**2*y**7 + 1)
    assert f == X**3*Y*Z + X**2*Y**7 + 1 and isinstance(f, R.dtype)

    r, F = sring([exp(2)])
    f = r.from_expr(exp(2))
    assert f == F[0] and isinstance(f, r.dtype)

    raises(ValueError, lambda: R.from_expr(1/x))
    raises(ValueError, lambda: R.from_expr(2**x))
    raises(ValueError, lambda: R.from_expr(7*x + sqrt(2)))

    R, = ring("", ZZ)
    f = R.from_expr(1)
    assert f == 1 and isinstance(f, R.dtype)

def test_PolyElement_degree():
    R, x,y,z = ring("x,y,z", ZZ)

    assert ninf == float('-inf')

    assert R(0).degree() is ninf
    assert R(1).degree() == 0
    assert (x + 1).degree() == 1
    assert (2*y**3 + z).degree() == 0
    assert (x*y**3 + z).degree() == 1
    assert (x**5*y**3 + z).degree() == 5

    assert R(0).degree(x) is ninf
    assert R(1).degree(x) == 0
    assert (x + 1).degree(x) == 1
    assert (2*y**3 + z).degree(x) == 0
    assert (x*y**3 + z).degree(x) == 1
    assert (7*x**5*y**3 + z).degree(x) == 5

    assert R(0).degree(y) is ninf
    assert R(1).degree(y) == 0
    assert (x + 1).degree(y) == 0
    assert (2*y**3 + z).degree(y) == 3
    assert (x*y**3 + z).degree(y) == 3
    assert (7*x**5*y**3 + z).degree(y) == 3

    assert R(0).degree(z) is ninf
    assert R(1).degree(z) == 0
    assert (x + 1).degree(z) == 0
    assert (2*y**3 + z).degree(z) == 1
    assert (x*y**3 + z).degree(z) == 1
    assert (7*x**5*y**3 + z).degree(z) == 1

    R, = ring("", ZZ)
    assert R(0).degree() is ninf
    assert R(1).degree() == 0

def test_PolyElement_tail_degree():
    R, x,y,z = ring("x,y,z", ZZ)

    assert R(0).tail_degree() is ninf
    assert R(1).tail_degree() == 0
    assert (x + 1).tail_degree() == 0
    assert (2*y**3 + x**3*z).tail_degree() == 0
    assert (x*y**3 + x**3*z).tail_degree() == 1
    assert (x**5*y**3 + x**3*z).tail_degree() == 3

    assert R(0).tail_degree(x) is ninf
    assert R(1).tail_degree(x) == 0
    assert (x + 1).tail_degree(x) == 0
    assert (2*y**3 + x**3*z).tail_degree(x) == 0
    assert (x*y**3 + x**3*z).tail_degree(x) == 1
    assert (7*x**5*y**3 + x**3*z).tail_degree(x) == 3

    assert R(0).tail_degree(y) is ninf
    assert R(1).tail_degree(y) == 0
    assert (x + 1).tail_degree(y) == 0
    assert (2*y**3 + x**3*z).tail_degree(y) == 0
    assert (x*y**3 + x**3*z).tail_degree(y) == 0
    assert (7*x**5*y**3 + x**3*z).tail_degree(y) == 0

    assert R(0).tail_degree(z) is ninf
    assert R(1).tail_degree(z) == 0
    assert (x + 1).tail_degree(z) == 0
    assert (2*y**3 + x**3*z).tail_degree(z) == 0
    assert (x*y**3 + x**3*z).tail_degree(z) == 0
    assert (7*x**5*y**3 + x**3*z).tail_degree(z) == 0

    R, = ring("", ZZ)
    assert R(0).tail_degree() is ninf
    assert R(1).tail_degree() == 0

def test_PolyElement_degrees():
    R, x,y,z = ring("x,y,z", ZZ)

    assert R(0).degrees() == (ninf, ninf, ninf)
    assert R(1).degrees() == (0, 0, 0)
    assert (x**2*y + x**3*z**2).degrees() == (3, 1, 2)

def test_PolyElement_tail_degrees():
    R, x,y,z = ring("x,y,z", ZZ)

    assert R(0).tail_degrees() == (ninf, ninf, ninf)
    assert R(1).tail_degrees() == (0, 0, 0)
    assert (x**2*y + x**3*z**2).tail_degrees() == (2, 0, 0)

def test_PolyElement_coeff():
    R, x, y, z = ring("x,y,z", ZZ, lex)
    f = 3*x**2*y - x*y*z + 7*z**3 + 23

    assert f.coeff(1) == 23
    raises(ValueError, lambda: f.coeff(3))

    assert f.coeff(x) == 0
    assert f.coeff(y) == 0
    assert f.coeff(z) == 0

    assert f.coeff(x**2*y) == 3
    assert f.coeff(x*y*z) == -1
    assert f.coeff(z**3) == 7

    raises(ValueError, lambda: f.coeff(3*x**2*y))
    raises(ValueError, lambda: f.coeff(-x*y*z))
    raises(ValueError, lambda: f.coeff(7*z**3))

    R, = ring("", ZZ)
    assert R(3).coeff(1) == 3

def test_PolyElement_LC():
    R, x, y = ring("x,y", QQ, lex)
    assert R(0).LC == QQ(0)
    assert (QQ(1,2)*x).LC == QQ(1, 2)
    assert (QQ(1,4)*x*y + QQ(1,2)*x).LC == QQ(1, 4)

def test_PolyElement_LM():
    R, x, y = ring("x,y", QQ, lex)
    assert R(0).LM == (0, 0)
    assert (QQ(1,2)*x).LM == (1, 0)
    assert (QQ(1,4)*x*y + QQ(1,2)*x).LM == (1, 1)

def test_PolyElement_LT():
    R, x, y = ring("x,y", QQ, lex)
    assert R(0).LT == ((0, 0), QQ(0))
    assert (QQ(1,2)*x).LT == ((1, 0), QQ(1, 2))
    assert (QQ(1,4)*x*y + QQ(1,2)*x).LT == ((1, 1), QQ(1, 4))

    R, = ring("", ZZ)
    assert R(0).LT == ((), 0)
    assert R(1).LT == ((), 1)

def test_PolyElement_leading_monom():
    R, x, y = ring("x,y", QQ, lex)
    assert R(0).leading_monom() == 0
    assert (QQ(1,2)*x).leading_monom() == x
    assert (QQ(1,4)*x*y + QQ(1,2)*x).leading_monom() == x*y

def test_PolyElement_leading_term():
    R, x, y = ring("x,y", QQ, lex)
    assert R(0).leading_term() == 0
    assert (QQ(1,2)*x).leading_term() == QQ(1,2)*x
    assert (QQ(1,4)*x*y + QQ(1,2)*x).leading_term() == QQ(1,4)*x*y

def test_PolyElement_terms():
    R, x,y,z = ring("x,y,z", QQ)
    terms = (x**2/3 + y**3/4 + z**4/5).terms()
    assert terms == [((2,0,0), QQ(1,3)), ((0,3,0), QQ(1,4)), ((0,0,4), QQ(1,5))]

    R, x,y = ring("x,y", ZZ, lex)
    f = x*y**7 + 2*x**2*y**3

    assert f.terms() == f.terms(lex) == f.terms('lex') == [((2, 3), 2), ((1, 7), 1)]
    assert f.terms(grlex) == f.terms('grlex') == [((1, 7), 1), ((2, 3), 2)]

    R, x,y = ring("x,y", ZZ, grlex)
    f = x*y**7 + 2*x**2*y**3

    assert f.terms() == f.terms(grlex) == f.terms('grlex') == [((1, 7), 1), ((2, 3), 2)]
    assert f.terms(lex) == f.terms('lex') == [((2, 3), 2), ((1, 7), 1)]

    R, = ring("", ZZ)
    assert R(3).terms() == [((), 3)]

def test_PolyElement_monoms():
    R, x,y,z = ring("x,y,z", QQ)
    monoms = (x**2/3 + y**3/4 + z**4/5).monoms()
    assert monoms == [(2,0,0), (0,3,0), (0,0,4)]

    R, x,y = ring("x,y", ZZ, lex)
    f = x*y**7 + 2*x**2*y**3

    assert f.monoms() == f.monoms(lex) == f.monoms('lex') == [(2, 3), (1, 7)]
    assert f.monoms(grlex) == f.monoms('grlex') == [(1, 7), (2, 3)]

    R, x,y = ring("x,y", ZZ, grlex)
    f = x*y**7 + 2*x**2*y**3

    assert f.monoms() == f.monoms(grlex) == f.monoms('grlex') == [(1, 7), (2, 3)]
    assert f.monoms(lex) == f.monoms('lex') == [(2, 3), (1, 7)]

def test_PolyElement_coeffs():
    R, x,y,z = ring("x,y,z", QQ)
    coeffs = (x**2/3 + y**3/4 + z**4/5).coeffs()
    assert coeffs == [QQ(1,3), QQ(1,4), QQ(1,5)]

    R, x,y = ring("x,y", ZZ, lex)
    f = x*y**7 + 2*x**2*y**3

    assert f.coeffs() == f.coeffs(lex) == f.coeffs('lex') == [2, 1]
    assert f.coeffs(grlex) == f.coeffs('grlex') == [1, 2]

    R, x,y = ring("x,y", ZZ, grlex)
    f = x*y**7 + 2*x**2*y**3

    assert f.coeffs() == f.coeffs(grlex) == f.coeffs('grlex') == [1, 2]
    assert f.coeffs(lex) == f.coeffs('lex') == [2, 1]

def test_PolyElement___add__():
    Rt, t = ring("t", ZZ)
    Ruv, u,v = ring("u,v", ZZ)
    Rxyz, x,y,z = ring("x,y,z", Ruv)

    assert dict(x + 3*y) == {(1, 0, 0): 1, (0, 1, 0): 3}

    assert dict(u + x) == dict(x + u) == {(1, 0, 0): 1, (0, 0, 0): u}
    assert dict(u + x*y) == dict(x*y + u) == {(1, 1, 0): 1, (0, 0, 0): u}
    assert dict(u + x*y + z) == dict(x*y + z + u) == {(1, 1, 0): 1, (0, 0, 1): 1, (0, 0, 0): u}

    assert dict(u*x + x) == dict(x + u*x) == {(1, 0, 0): u + 1}
    assert dict(u*x + x*y) == dict(x*y + u*x) == {(1, 1, 0): 1, (1, 0, 0): u}
    assert dict(u*x + x*y + z) == dict(x*y + z + u*x) == {(1, 1, 0): 1, (0, 0, 1): 1, (1, 0, 0): u}

    raises(TypeError, lambda: t + x)
    raises(TypeError, lambda: x + t)
    raises(TypeError, lambda: t + u)
    raises(TypeError, lambda: u + t)

    Fuv, u,v = field("u,v", ZZ)
    Rxyz, x,y,z = ring("x,y,z", Fuv)

    assert dict(u + x) == dict(x + u) == {(1, 0, 0): 1, (0, 0, 0): u}

    Rxyz, x,y,z = ring("x,y,z", EX)

    assert dict(EX(pi) + x*y*z) == dict(x*y*z + EX(pi)) == {(1, 1, 1): EX(1), (0, 0, 0): EX(pi)}

def test_PolyElement___sub__():
    Rt, t = ring("t", ZZ)
    Ruv, u,v = ring("u,v", ZZ)
    Rxyz, x,y,z = ring("x,y,z", Ruv)

    assert dict(x - 3*y) == {(1, 0, 0): 1, (0, 1, 0): -3}

    assert dict(-u + x) == dict(x - u) == {(1, 0, 0): 1, (0, 0, 0): -u}
    assert dict(-u + x*y) == dict(x*y - u) == {(1, 1, 0): 1, (0, 0, 0): -u}
    assert dict(-u + x*y + z) == dict(x*y + z - u) == {(1, 1, 0): 1, (0, 0, 1): 1, (0, 0, 0): -u}

    assert dict(-u*x + x) == dict(x - u*x) == {(1, 0, 0): -u + 1}
    assert dict(-u*x + x*y) == dict(x*y - u*x) == {(1, 1, 0): 1, (1, 0, 0): -u}
    assert dict(-u*x + x*y + z) == dict(x*y + z - u*x) == {(1, 1, 0): 1, (0, 0, 1): 1, (1, 0, 0): -u}

    raises(TypeError, lambda: t - x)
    raises(TypeError, lambda: x - t)
    raises(TypeError, lambda: t - u)
    raises(TypeError, lambda: u - t)

    Fuv, u,v = field("u,v", ZZ)
    Rxyz, x,y,z = ring("x,y,z", Fuv)

    assert dict(-u + x) == dict(x - u) == {(1, 0, 0): 1, (0, 0, 0): -u}

    Rxyz, x,y,z = ring("x,y,z", EX)

    assert dict(-EX(pi) + x*y*z) == dict(x*y*z - EX(pi)) == {(1, 1, 1): EX(1), (0, 0, 0): -EX(pi)}

def test_PolyElement___mul__():
    Rt, t = ring("t", ZZ)
    Ruv, u,v = ring("u,v", ZZ)
    Rxyz, x,y,z = ring("x,y,z", Ruv)

    assert dict(u*x) == dict(x*u) == {(1, 0, 0): u}

    assert dict(2*u*x + z) == dict(x*2*u + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1}
    assert dict(u*2*x + z) == dict(2*x*u + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1}
    assert dict(2*u*x + z) == dict(x*2*u + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1}
    assert dict(u*x*2 + z) == dict(x*u*2 + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1}

    assert dict(2*u*x*y + z) == dict(x*y*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
    assert dict(u*2*x*y + z) == dict(2*x*y*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
    assert dict(2*u*x*y + z) == dict(x*y*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
    assert dict(u*x*y*2 + z) == dict(x*y*u*2 + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}

    assert dict(2*u*y*x + z) == dict(y*x*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
    assert dict(u*2*y*x + z) == dict(2*y*x*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
    assert dict(2*u*y*x + z) == dict(y*x*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
    assert dict(u*y*x*2 + z) == dict(y*x*u*2 + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}

    assert dict(3*u*(x + y) + z) == dict((x + y)*3*u + z) == {(1, 0, 0): 3*u, (0, 1, 0): 3*u, (0, 0, 1): 1}

    raises(TypeError, lambda: t*x + z)
    raises(TypeError, lambda: x*t + z)
    raises(TypeError, lambda: t*u + z)
    raises(TypeError, lambda: u*t + z)

    Fuv, u,v = field("u,v", ZZ)
    Rxyz, x,y,z = ring("x,y,z", Fuv)

    assert dict(u*x) == dict(x*u) == {(1, 0, 0): u}

    Rxyz, x,y,z = ring("x,y,z", EX)

    assert dict(EX(pi)*x*y*z) == dict(x*y*z*EX(pi)) == {(1, 1, 1): EX(pi)}

def test_PolyElement___truediv__():
    R, x,y,z = ring("x,y,z", ZZ)

    assert (2*x**2 - 4)/2 == x**2 - 2
    assert (2*x**2 - 3)/2 == x**2

    assert (x**2 - 1).quo(x) == x
    assert (x**2 - x).quo(x) == x - 1

    assert (x**2 - 1)/x == x - x**(-1)
    assert (x**2 - x)/x == x - 1
    assert (x**2 - 1)/(2*x) == x/2 - x**(-1)/2

    assert (x**2 - 1).quo(2*x) == 0
    assert (x**2 - x)/(x - 1) == (x**2 - x).quo(x - 1) == x


    R, x,y,z = ring("x,y,z", ZZ)
    assert len((x**2/3 + y**3/4 + z**4/5).terms()) == 0

    R, x,y,z = ring("x,y,z", QQ)
    assert len((x**2/3 + y**3/4 + z**4/5).terms()) == 3

    Rt, t = ring("t", ZZ)
    Ruv, u,v = ring("u,v", ZZ)
    Rxyz, x,y,z = ring("x,y,z", Ruv)

    assert dict((u**2*x + u)/u) == {(1, 0, 0): u, (0, 0, 0): 1}
    raises(TypeError, lambda: u/(u**2*x + u))

    raises(TypeError, lambda: t/x)
    raises(TypeError, lambda: x/t)
    raises(TypeError, lambda: t/u)
    raises(TypeError, lambda: u/t)

    R, x = ring("x", ZZ)
    f, g = x**2 + 2*x + 3, R(0)

    raises(ZeroDivisionError, lambda: f.div(g))
    raises(ZeroDivisionError, lambda: divmod(f, g))
    raises(ZeroDivisionError, lambda: f.rem(g))
    raises(ZeroDivisionError, lambda: f % g)
    raises(ZeroDivisionError, lambda: f.quo(g))
    raises(ZeroDivisionError, lambda: f / g)
    raises(ZeroDivisionError, lambda: f.exquo(g))

    R, x, y = ring("x,y", ZZ)
    f, g = x*y + 2*x + 3, R(0)

    raises(ZeroDivisionError, lambda: f.div(g))
    raises(ZeroDivisionError, lambda: divmod(f, g))
    raises(ZeroDivisionError, lambda: f.rem(g))
    raises(ZeroDivisionError, lambda: f % g)
    raises(ZeroDivisionError, lambda: f.quo(g))
    raises(ZeroDivisionError, lambda: f / g)
    raises(ZeroDivisionError, lambda: f.exquo(g))

    R, x = ring("x", ZZ)

    f, g = x**2 + 1, 2*x - 4
    q, r = R(0), x**2 + 1

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    f, g = 3*x**3 + x**2 + x + 5, 5*x**2 - 3*x + 1
    q, r = R(0), f

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    f, g = 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1, x**2 + 2*x + 3
    q, r = 5*x**2 - 6*x, 20*x + 1

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    f, g = 5*x**5 + 4*x**4 + 3*x**3 + 2*x**2 + x, x**4 + 2*x**3 + 9
    q, r = 5*x - 6, 15*x**3 + 2*x**2 - 44*x + 54

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    R, x = ring("x", QQ)

    f, g = x**2 + 1, 2*x - 4
    q, r = x/2 + 1, R(5)

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    f, g = 3*x**3 + x**2 + x + 5, 5*x**2 - 3*x + 1
    q, r = QQ(3, 5)*x + QQ(14, 25), QQ(52, 25)*x + QQ(111, 25)

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    R, x,y = ring("x,y", ZZ)

    f, g = x**2 - y**2, x - y
    q, r = x + y, R(0)

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    assert f.exquo(g) == q

    f, g = x**2 + y**2, x - y
    q, r = x + y, 2*y**2

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    f, g = x**2 + y**2, -x + y
    q, r = -x - y, 2*y**2

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    f, g = x**2 + y**2, 2*x - 2*y
    q, r = R(0), f

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    R, x,y = ring("x,y", QQ)

    f, g = x**2 - y**2, x - y
    q, r = x + y, R(0)

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    assert f.exquo(g) == q

    f, g = x**2 + y**2, x - y
    q, r = x + y, 2*y**2

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    f, g = x**2 + y**2, -x + y
    q, r = -x - y, 2*y**2

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

    f, g = x**2 + y**2, 2*x - 2*y
    q, r = x/2 + y/2, 2*y**2

    assert f.div(g) == divmod(f, g) == (q, r)
    assert f.rem(g) == f % g == r
    assert f.quo(g) == f / g == q
    raises(ExactQuotientFailed, lambda: f.exquo(g))

def test_PolyElement___pow__():
    R, x = ring("x", ZZ, grlex)
    f = 2*x + 3

    assert f**0 == 1
    assert f**1 == f
    raises(ValueError, lambda: f**(-1))

    assert x**(-1) == x**(-1)

    assert f**2 == f._pow_generic(2) == f._pow_multinomial(2) == 4*x**2 + 12*x + 9
    assert f**3 == f._pow_generic(3) == f._pow_multinomial(3) == 8*x**3 + 36*x**2 + 54*x + 27
    assert f**4 == f._pow_generic(4) == f._pow_multinomial(4) == 16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81
    assert f**5 == f._pow_generic(5) == f._pow_multinomial(5) == 32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243

    R, x,y,z = ring("x,y,z", ZZ, grlex)
    f = x**3*y - 2*x*y**2 - 3*z + 1
    g = x**6*y**2 - 4*x**4*y**3 - 6*x**3*y*z + 2*x**3*y + 4*x**2*y**4 + 12*x*y**2*z - 4*x*y**2 + 9*z**2 - 6*z + 1

    assert f**2 == f._pow_generic(2) == f._pow_multinomial(2) == g

    R, t = ring("t", ZZ)
    f = -11200*t**4 - 2604*t**2 + 49
    g = 15735193600000000*t**16 + 14633730048000000*t**14 + 4828147466240000*t**12 \
      + 598976863027200*t**10 + 3130812416256*t**8 - 2620523775744*t**6 \
      + 92413760096*t**4 - 1225431984*t**2 + 5764801

    assert f**4 == f._pow_generic(4) == f._pow_multinomial(4) == g

def test_PolyElement_div():
    R, x = ring("x", ZZ, grlex)

    f = x**3 - 12*x**2 - 42
    g = x - 3

    q = x**2 - 9*x - 27
    r = -123

    assert f.div([g]) == ([q], r)

    R, x = ring("x", ZZ, grlex)
    f = x**2 + 2*x + 2
    assert f.div([R(1)]) == ([f], 0)

    R, x = ring("x", QQ, grlex)
    f = x**2 + 2*x + 2
    assert f.div([R(2)]) == ([QQ(1,2)*x**2 + x + 1], 0)

    R, x,y = ring("x,y", ZZ, grlex)
    f = 4*x**2*y - 2*x*y + 4*x - 2*y + 8

    assert f.div([R(2)]) == ([2*x**2*y - x*y + 2*x - y + 4], 0)
    assert f.div([2*y]) == ([2*x**2 - x - 1], 4*x + 8)

    f = x - 1
    g = y - 1

    assert f.div([g]) == ([0], f)

    f = x*y**2 + 1
    G = [x*y + 1, y + 1]

    Q = [y, -1]
    r = 2

    assert f.div(G) == (Q, r)

    f = x**2*y + x*y**2 + y**2
    G = [x*y - 1, y**2 - 1]

    Q = [x + y, 1]
    r = x + y + 1

    assert f.div(G) == (Q, r)

    G = [y**2 - 1, x*y - 1]

    Q = [x + 1, x]
    r = 2*x + 1

    assert f.div(G) == (Q, r)

    R, = ring("", ZZ)
    assert R(3).div(R(2)) == (0, 3)

    R, = ring("", QQ)
    assert R(3).div(R(2)) == (QQ(3, 2), 0)

def test_PolyElement_rem():
    R, x = ring("x", ZZ, grlex)

    f = x**3 - 12*x**2 - 42
    g = x - 3
    r = -123

    assert f.rem([g]) == f.div([g])[1] == r

    R, x,y = ring("x,y", ZZ, grlex)

    f = 4*x**2*y - 2*x*y + 4*x - 2*y + 8

    assert f.rem([R(2)]) == f.div([R(2)])[1] == 0
    assert f.rem([2*y]) == f.div([2*y])[1] == 4*x + 8

    f = x - 1
    g = y - 1

    assert f.rem([g]) == f.div([g])[1] == f

    f = x*y**2 + 1
    G = [x*y + 1, y + 1]
    r = 2

    assert f.rem(G) == f.div(G)[1] == r

    f = x**2*y + x*y**2 + y**2
    G = [x*y - 1, y**2 - 1]
    r = x + y + 1

    assert f.rem(G) == f.div(G)[1] == r

    G = [y**2 - 1, x*y - 1]
    r = 2*x + 1

    assert f.rem(G) == f.div(G)[1] == r

def test_PolyElement_deflate():
    R, x = ring("x", ZZ)

    assert (2*x**2).deflate(x**4 + 4*x**2 + 1) == ((2,), [2*x, x**2 + 4*x + 1])

    R, x,y = ring("x,y", ZZ)

    assert R(0).deflate(R(0)) == ((1, 1), [0, 0])
    assert R(1).deflate(R(0)) == ((1, 1), [1, 0])
    assert R(1).deflate(R(2)) == ((1, 1), [1, 2])
    assert R(1).deflate(2*y) == ((1, 1), [1, 2*y])
    assert (2*y).deflate(2*y) == ((1, 1), [2*y, 2*y])
    assert R(2).deflate(2*y**2) == ((1, 2), [2, 2*y])
    assert (2*y**2).deflate(2*y**2) == ((1, 2), [2*y, 2*y])

    f = x**4*y**2 + x**2*y + 1
    g = x**2*y**3 + x**2*y + 1

    assert f.deflate(g) == ((2, 1), [x**2*y**2 + x*y + 1, x*y**3 + x*y + 1])

def test_PolyElement_clear_denoms():
    R, x,y = ring("x,y", QQ)

    assert R(1).clear_denoms() == (ZZ(1), 1)
    assert R(7).clear_denoms() == (ZZ(1), 7)

    assert R(QQ(7,3)).clear_denoms() == (3, 7)
    assert R(QQ(7,3)).clear_denoms() == (3, 7)

    assert (3*x**2 + x).clear_denoms() == (1, 3*x**2 + x)
    assert (x**2 + QQ(1,2)*x).clear_denoms() == (2, 2*x**2 + x)

    rQQ, x,t = ring("x,t", QQ, lex)
    rZZ, X,T = ring("x,t", ZZ, lex)

    F = [x - QQ(17824537287975195925064602467992950991718052713078834557692023531499318507213727406844943097,413954288007559433755329699713866804710749652268151059918115348815925474842910720000)*t**7
           - QQ(4882321164854282623427463828745855894130208215961904469205260756604820743234704900167747753,12936071500236232304854053116058337647210926633379720622441104650497671088840960000)*t**6
           - QQ(36398103304520066098365558157422127347455927422509913596393052633155821154626830576085097433,25872143000472464609708106232116675294421853266759441244882209300995342177681920000)*t**5
           - QQ(168108082231614049052707339295479262031324376786405372698857619250210703675982492356828810819,58212321751063045371843239022262519412449169850208742800984970927239519899784320000)*t**4
           - QQ(5694176899498574510667890423110567593477487855183144378347226247962949388653159751849449037,1617008937529529038106756639507292205901365829172465077805138081312208886105120000)*t**3
           - QQ(154482622347268833757819824809033388503591365487934245386958884099214649755244381307907779,60637835157357338929003373981523457721301218593967440417692678049207833228942000)*t**2
           - QQ(2452813096069528207645703151222478123259511586701148682951852876484544822947007791153163,2425513406294293557160134959260938308852048743758697616707707121968313329157680)*t
           - QQ(34305265428126440542854669008203683099323146152358231964773310260498715579162112959703,202126117191191129763344579938411525737670728646558134725642260164026110763140),
         t**8 + QQ(693749860237914515552,67859264524169150569)*t**7
              + QQ(27761407182086143225024,610733380717522355121)*t**6
              + QQ(7785127652157884044288,67859264524169150569)*t**5
              + QQ(36567075214771261409792,203577793572507451707)*t**4
              + QQ(36336335165196147384320,203577793572507451707)*t**3
              + QQ(7452455676042754048000,67859264524169150569)*t**2
              + QQ(2593331082514399232000,67859264524169150569)*t
              + QQ(390399197427343360000,67859264524169150569)]

    G = [3725588592068034903797967297424801242396746870413359539263038139343329273586196480000*X -
         160420835591776763325581422211936558925462474417709511019228211783493866564923546661604487873*T**7 -
         1406108495478033395547109582678806497509499966197028487131115097902188374051595011248311352864*T**6 -
         5241326875850889518164640374668786338033653548841427557880599579174438246266263602956254030352*T**5 -
         10758917262823299139373269714910672770004760114329943852726887632013485035262879510837043892416*T**4 -
         13119383576444715672578819534846747735372132018341964647712009275306635391456880068261130581248*T**3 -
         9491412317016197146080450036267011389660653495578680036574753839055748080962214787557853941760*T**2 -
         3767520915562795326943800040277726397326609797172964377014046018280260848046603967211258368000*T -
         632314652371226552085897259159210286886724229880266931574701654721512325555116066073245696000,
         610733380717522355121*T**8 +
         6243748742141230639968*T**7 +
         27761407182086143225024*T**6 +
         70066148869420956398592*T**5 +
         109701225644313784229376*T**4 +
         109009005495588442152960*T**3 +
         67072101084384786432000*T**2 +
         23339979742629593088000*T +
         3513592776846090240000]

    assert [ f.clear_denoms()[1].set_ring(rZZ) for f in F ] == G

def test_PolyElement_cofactors():
    R, x, y = ring("x,y", ZZ)

    f, g = R(0), R(0)
    assert f.cofactors(g) == (0, 0, 0)

    f, g = R(2), R(0)
    assert f.cofactors(g) == (2, 1, 0)

    f, g = R(-2), R(0)
    assert f.cofactors(g) == (2, -1, 0)

    f, g = R(0), R(-2)
    assert f.cofactors(g) == (2, 0, -1)

    f, g = R(0), 2*x + 4
    assert f.cofactors(g) == (2*x + 4, 0, 1)

    f, g = 2*x + 4, R(0)
    assert f.cofactors(g) == (2*x + 4, 1, 0)

    f, g = R(2), R(2)
    assert f.cofactors(g) == (2, 1, 1)

    f, g = R(-2), R(2)
    assert f.cofactors(g) == (2, -1, 1)

    f, g = R(2), R(-2)
    assert f.cofactors(g) == (2, 1, -1)

    f, g = R(-2), R(-2)
    assert f.cofactors(g) == (2, -1, -1)

    f, g = x**2 + 2*x + 1, R(1)
    assert f.cofactors(g) == (1, x**2 + 2*x + 1, 1)

    f, g = x**2 + 2*x + 1, R(2)
    assert f.cofactors(g) == (1, x**2 + 2*x + 1, 2)

    f, g = 2*x**2 + 4*x + 2, R(2)
    assert f.cofactors(g) == (2, x**2 + 2*x + 1, 1)

    f, g = R(2), 2*x**2 + 4*x + 2
    assert f.cofactors(g) == (2, 1, x**2 + 2*x + 1)

    f, g = 2*x**2 + 4*x + 2, x + 1
    assert f.cofactors(g) == (x + 1, 2*x + 2, 1)

    f, g = x + 1, 2*x**2 + 4*x + 2
    assert f.cofactors(g) == (x + 1, 1, 2*x + 2)

    R, x, y, z, t = ring("x,y,z,t", ZZ)

    f, g = t**2 + 2*t + 1, 2*t + 2
    assert f.cofactors(g) == (t + 1, t + 1, 2)

    f, g = z**2*t**2 + 2*z**2*t + z**2 + z*t + z, t**2 + 2*t + 1
    h, cff, cfg = t + 1, z**2*t + z**2 + z, t + 1

    assert f.cofactors(g) == (h, cff, cfg)
    assert g.cofactors(f) == (h, cfg, cff)

    R, x, y = ring("x,y", QQ)

    f = QQ(1,2)*x**2 + x + QQ(1,2)
    g = QQ(1,2)*x + QQ(1,2)

    h = x + 1

    assert f.cofactors(g) == (h, g, QQ(1,2))
    assert g.cofactors(f) == (h, QQ(1,2), g)

    R, x, y = ring("x,y", RR)

    f = 2.1*x*y**2 - 2.1*x*y + 2.1*x
    g = 2.1*x**3
    h = 1.0*x

    assert f.cofactors(g) == (h, f/h, g/h)
    assert g.cofactors(f) == (h, g/h, f/h)

def test_PolyElement_gcd():
    R, x, y = ring("x,y", QQ)

    f = QQ(1,2)*x**2 + x + QQ(1,2)
    g = QQ(1,2)*x + QQ(1,2)

    assert f.gcd(g) == x + 1

def test_PolyElement_cancel():
    R, x, y = ring("x,y", ZZ)

    f = 2*x**3 + 4*x**2 + 2*x
    g = 3*x**2 + 3*x
    F = 2*x + 2
    G = 3

    assert f.cancel(g) == (F, G)

    assert (-f).cancel(g) == (-F, G)
    assert f.cancel(-g) == (-F, G)

    R, x, y = ring("x,y", QQ)

    f = QQ(1,2)*x**3 + x**2 + QQ(1,2)*x
    g = QQ(1,3)*x**2 + QQ(1,3)*x
    F = 3*x + 3
    G = 2

    assert f.cancel(g) == (F, G)

    assert (-f).cancel(g) == (-F, G)
    assert f.cancel(-g) == (-F, G)

    Fx, x = field("x", ZZ)
    Rt, t = ring("t", Fx)

    f = (-x**2 - 4)/4*t
    g = t**2 + (x**2 + 2)/2

    assert f.cancel(g) == ((-x**2 - 4)*t, 4*t**2 + 2*x**2 + 4)

def test_PolyElement_max_norm():
    R, x, y = ring("x,y", ZZ)

    assert R(0).max_norm() == 0
    assert R(1).max_norm() == 1

    assert (x**3 + 4*x**2 + 2*x + 3).max_norm() == 4

def test_PolyElement_l1_norm():
    R, x, y = ring("x,y", ZZ)

    assert R(0).l1_norm() == 0
    assert R(1).l1_norm() == 1

    assert (x**3 + 4*x**2 + 2*x + 3).l1_norm() == 10

def test_PolyElement_diff():
    R, X = xring("x:11", QQ)

    f = QQ(288,5)*X[0]**8*X[1]**6*X[4]**3*X[10]**2 + 8*X[0]**2*X[2]**3*X[4]**3 +2*X[0]**2 - 2*X[1]**2

    assert f.diff(X[0]) == QQ(2304,5)*X[0]**7*X[1]**6*X[4]**3*X[10]**2 + 16*X[0]*X[2]**3*X[4]**3 + 4*X[0]
    assert f.diff(X[4]) == QQ(864,5)*X[0]**8*X[1]**6*X[4]**2*X[10]**2 + 24*X[0]**2*X[2]**3*X[4]**2
    assert f.diff(X[10]) == QQ(576,5)*X[0]**8*X[1]**6*X[4]**3*X[10]

def test_PolyElement___call__():
    R, x = ring("x", ZZ)
    f = 3*x + 1

    assert f(0) == 1
    assert f(1) == 4

    raises(ValueError, lambda: f())
    raises(ValueError, lambda: f(0, 1))

    raises(CoercionFailed, lambda: f(QQ(1,7)))

    R, x,y = ring("x,y", ZZ)
    f = 3*x + y**2 + 1

    assert f(0, 0) == 1
    assert f(1, 7) == 53

    Ry = R.drop(x)

    assert f(0) == Ry.y**2 + 1
    assert f(1) == Ry.y**2 + 4

    raises(ValueError, lambda: f())
    raises(ValueError, lambda: f(0, 1, 2))

    raises(CoercionFailed, lambda: f(1, QQ(1,7)))
    raises(CoercionFailed, lambda: f(QQ(1,7), 1))
    raises(CoercionFailed, lambda: f(QQ(1,7), QQ(1,7)))

def test_PolyElement_evaluate():
    R, x = ring("x", ZZ)
    f = x**3 + 4*x**2 + 2*x + 3

    r = f.evaluate(x, 0)
    assert r == 3 and not isinstance(r, PolyElement)

    raises(CoercionFailed, lambda: f.evaluate(x, QQ(1,7)))

    R, x, y, z = ring("x,y,z", ZZ)
    f = (x*y)**3 + 4*(x*y)**2 + 2*x*y + 3

    r = f.evaluate(x, 0)
    assert r == 3 and isinstance(r, R.drop(x).dtype)
    r = f.evaluate([(x, 0), (y, 0)])
    assert r == 3 and isinstance(r, R.drop(x, y).dtype)
    r = f.evaluate(y, 0)
    assert r == 3 and isinstance(r, R.drop(y).dtype)
    r = f.evaluate([(y, 0), (x, 0)])
    assert r == 3 and isinstance(r, R.drop(y, x).dtype)

    r = f.evaluate([(x, 0), (y, 0), (z, 0)])
    assert r == 3 and not isinstance(r, PolyElement)

    raises(CoercionFailed, lambda: f.evaluate([(x, 1), (y, QQ(1,7))]))
    raises(CoercionFailed, lambda: f.evaluate([(x, QQ(1,7)), (y, 1)]))
    raises(CoercionFailed, lambda: f.evaluate([(x, QQ(1,7)), (y, QQ(1,7))]))

def test_PolyElement_subs():
    R, x = ring("x", ZZ)
    f = x**3 + 4*x**2 + 2*x + 3

    r = f.subs(x, 0)
    assert r == 3 and isinstance(r, R.dtype)

    raises(CoercionFailed, lambda: f.subs(x, QQ(1,7)))

    R, x, y, z = ring("x,y,z", ZZ)
    f = x**3 + 4*x**2 + 2*x + 3

    r = f.subs(x, 0)
    assert r == 3 and isinstance(r, R.dtype)
    r = f.subs([(x, 0), (y, 0)])
    assert r == 3 and isinstance(r, R.dtype)

    raises(CoercionFailed, lambda: f.subs([(x, 1), (y, QQ(1,7))]))
    raises(CoercionFailed, lambda: f.subs([(x, QQ(1,7)), (y, 1)]))
    raises(CoercionFailed, lambda: f.subs([(x, QQ(1,7)), (y, QQ(1,7))]))

def test_PolyElement_symmetrize():
    R, x, y = ring("x,y", ZZ)

    # Homogeneous, symmetric
    f = x**2 + y**2
    sym, rem, m = f.symmetrize()
    assert rem == 0
    assert sym.compose(m) + rem == f

    # Homogeneous, asymmetric
    f = x**2 - y**2
    sym, rem, m = f.symmetrize()
    assert rem != 0
    assert sym.compose(m) + rem == f

    # Inhomogeneous, symmetric
    f = x*y + 7
    sym, rem, m = f.symmetrize()
    assert rem == 0
    assert sym.compose(m) + rem == f

    # Inhomogeneous, asymmetric
    f = y + 7
    sym, rem, m = f.symmetrize()
    assert rem != 0
    assert sym.compose(m) + rem == f

    # Constant
    f = R.from_expr(3)
    sym, rem, m = f.symmetrize()
    assert rem == 0
    assert sym.compose(m) + rem == f

    # Constant constructed from sring
    R, f = sring(3)
    sym, rem, m = f.symmetrize()
    assert rem == 0
    assert sym.compose(m) + rem == f

def test_PolyElement_compose():
    R, x = ring("x", ZZ)
    f = x**3 + 4*x**2 + 2*x + 3

    r = f.compose(x, 0)
    assert r == 3 and isinstance(r, R.dtype)

    assert f.compose(x, x) == f
    assert f.compose(x, x**2) == x**6 + 4*x**4 + 2*x**2 + 3

    raises(CoercionFailed, lambda: f.compose(x, QQ(1,7)))

    R, x, y, z = ring("x,y,z", ZZ)
    f = x**3 + 4*x**2 + 2*x + 3

    r = f.compose(x, 0)
    assert r == 3 and isinstance(r, R.dtype)
    r = f.compose([(x, 0), (y, 0)])
    assert r == 3 and isinstance(r, R.dtype)

    r = (x**3 + 4*x**2 + 2*x*y*z + 3).compose(x, y*z**2 - 1)
    q = (y*z**2 - 1)**3 + 4*(y*z**2 - 1)**2 + 2*(y*z**2 - 1)*y*z + 3
    assert r == q and isinstance(r, R.dtype)

def test_PolyElement_is_():
    R, x,y,z = ring("x,y,z", QQ)

    assert (x - x).is_generator == False
    assert (x - x).is_ground == True
    assert (x - x).is_monomial == True
    assert (x - x).is_term == True

    assert (x - x + 1).is_generator == False
    assert (x - x + 1).is_ground == True
    assert (x - x + 1).is_monomial == True
    assert (x - x + 1).is_term == True

    assert x.is_generator == True
    assert x.is_ground == False
    assert x.is_monomial == True
    assert x.is_term == True

    assert (x*y).is_generator == False
    assert (x*y).is_ground == False
    assert (x*y).is_monomial == True
    assert (x*y).is_term == True

    assert (3*x).is_generator == False
    assert (3*x).is_ground == False
    assert (3*x).is_monomial == False
    assert (3*x).is_term == True

    assert (3*x + 1).is_generator == False
    assert (3*x + 1).is_ground == False
    assert (3*x + 1).is_monomial == False
    assert (3*x + 1).is_term == False

    assert R(0).is_zero is True
    assert R(1).is_zero is False

    assert R(0).is_one is False
    assert R(1).is_one is True

    assert (x - 1).is_monic is True
    assert (2*x - 1).is_monic is False

    assert (3*x + 2).is_primitive is True
    assert (4*x + 2).is_primitive is False

    assert (x + y + z + 1).is_linear is True
    assert (x*y*z + 1).is_linear is False

    assert (x*y + z + 1).is_quadratic is True
    assert (x*y*z + 1).is_quadratic is False

    assert (x - 1).is_squarefree is True
    assert ((x - 1)**2).is_squarefree is False

    assert (x**2 + x + 1).is_irreducible is True
    assert (x**2 + 2*x + 1).is_irreducible is False

    _, t = ring("t", FF(11))

    assert (7*t + 3).is_irreducible is True
    assert (7*t**2 + 3*t + 1).is_irreducible is False

    _, u = ring("u", ZZ)
    f = u**16 + u**14 - u**10 - u**8 - u**6 + u**2

    assert f.is_cyclotomic is False
    assert (f + 1).is_cyclotomic is True

    raises(MultivariatePolynomialError, lambda: x.is_cyclotomic)

    R, = ring("", ZZ)
    assert R(4).is_squarefree is True
    assert R(6).is_irreducible is True

def test_PolyElement_drop():
    R, x,y,z = ring("x,y,z", ZZ)

    assert R(1).drop(0).ring == PolyRing("y,z", ZZ, lex)
    assert R(1).drop(0).drop(0).ring == PolyRing("z", ZZ, lex)
    assert isinstance(R(1).drop(0).drop(0).drop(0), R.dtype) is False

    raises(ValueError, lambda: z.drop(0).drop(0).drop(0))
    raises(ValueError, lambda: x.drop(0))

def test_PolyElement_coeff_wrt():
    R, x, y, z = ring("x, y, z", ZZ)

    p = 4*x**3 + 5*y**2 + 6*y**2*z + 7
    assert p.coeff_wrt(1, 2) == 6*z + 5 # using generator index
    assert p.coeff_wrt(x, 3) == 4 # using generator

    p = 2*x**4 + 3*x*y**2*z + 10*y**2 + 10*x*z**2
    assert p.coeff_wrt(x, 1) == 3*y**2*z + 10*z**2
    assert p.coeff_wrt(y, 2) == 3*x*z + 10

    p = 4*x**2 + 2*x*y + 5
    assert p.coeff_wrt(z, 1) == R(0)
    assert p.coeff_wrt(y, 2) == R(0)

def test_PolyElement_prem():
    R, x, y = ring("x, y", ZZ)

    f, g = x**2 + x*y, 2*x + 2
    assert f.prem(g) == -4*y + 4 # first generator is chosen by default if it is not given

    f, g = x**2 + 1, 2*x - 4
    assert f.prem(g) == f.prem(g, x) == 20
    assert f.prem(g, 1) == R(0)

    f, g = x*y + 2*x + 1, x + y
    assert f.prem(g) == -y**2 - 2*y + 1
    assert f.prem(g, 1) == f.prem(g, y) == -x**2 + 2*x + 1

    raises(ZeroDivisionError, lambda: f.prem(R(0)))

def test_PolyElement_pdiv():
    R, x, y = ring("x,y", ZZ)

    f, g = x**4 + 5*x**3 + 7*x**2, 2*x**2 + 3
    assert f.pdiv(g) == f.pdiv(g, x) == (4*x**2 + 20*x + 22, -60*x - 66)

    f, g = x**2 - y**2, x - y
    assert f.pdiv(g) == f.pdiv(g, 0) == (x + y, 0)

    f, g = x*y + 2*x + 1, x + y
    assert f.pdiv(g) == (y + 2, -y**2 - 2*y + 1)
    assert f.pdiv(g, y) == f.pdiv(g, 1) == (x + 1, -x**2 + 2*x + 1)

    assert R(0).pdiv(g) == (0, 0)
    raises(ZeroDivisionError, lambda: f.prem(R(0)))

def test_PolyElement_pquo():
    R, x, y = ring("x, y", ZZ)

    f, g = x**4 - 4*x**2*y + 4*y**2, x**2 - 2*y
    assert f.pquo(g) == f.pquo(g, x) == x**2 - 2*y
    assert f.pquo(g, y) == 4*x**2 - 8*y + 4

    f, g = x**4 - y**4, x**2 - y**2
    assert f.pquo(g) == f.pquo(g, 0) == x**2 + y**2

def test_PolyElement_pexquo():
    R, x, y = ring("x, y", ZZ)

    f, g = x**2 - y**2, x - y
    assert f.pexquo(g) == f.pexquo(g, x) == x + y
    assert f.pexquo(g, y) == f.pexquo(g, 1) == x + y + 1

    f, g = x**2 + 3*x + 6, x + 2
    raises(ExactQuotientFailed, lambda: f.pexquo(g))

def test_PolyElement_gcdex():
    _, x = ring("x", QQ)

    f, g = 2*x, x**2 - 16
    s, t, h = x/32, -QQ(1, 16), 1

    assert f.half_gcdex(g) == (s, h)
    assert f.gcdex(g) == (s, t, h)

def test_PolyElement_subresultants():
    R, x, y = ring("x, y", ZZ)

    f, g = x**2*y + x*y, x + y # degree(f, x) > degree(g, x)
    h = y**3 - y**2
    assert f.subresultants(g) == [f, g, h] # first generator is chosen default

    # generator index or generator is given
    assert f.subresultants(g, 0) ==  f.subresultants(g, x) == [f, g, h]

    assert f.subresultants(g, y) == [x**2*y + x*y, x + y, x**3 + x**2]

    f, g = 2*x - y, x**2 + 2*y + x # degree(f, x) < degree(g, x)
    assert f.subresultants(g) == [x**2 + x + 2*y, 2*x - y, y**2 + 10*y]

    f, g = R(0), y**3 - y**2 # f = 0
    assert f.subresultants(g) == [y**3 - y**2, 1]

    f, g = x**2*y + x*y, R(0) # g = 0
    assert f.subresultants(g) == [x**2*y + x*y, 1]

    f, g = R(0), R(0) # f = 0 and g = 0
    assert f.subresultants(g) == [0, 0]

    f, g = x**2 + x, x**2 + x # f and g are same polynomial
    assert f.subresultants(g) == [x**2 + x, x**2 + x]

def test_PolyElement_resultant():
    _, x = ring("x", ZZ)
    f, g, h = x**2 - 2*x + 1, x**2 - 1, 0

    assert f.resultant(g) == h

def test_PolyElement_discriminant():
    _, x = ring("x", ZZ)
    f, g = x**3 + 3*x**2 + 9*x - 13, -11664

    assert f.discriminant() == g

    F, a, b, c = ring("a,b,c", ZZ)
    _, x = ring("x", F)

    f, g = a*x**2 + b*x + c, b**2 - 4*a*c

    assert f.discriminant() == g

def test_PolyElement_decompose():
    _, x = ring("x", ZZ)

    f = x**12 + 20*x**10 + 150*x**8 + 500*x**6 + 625*x**4 - 2*x**3 - 10*x + 9
    g = x**4 - 2*x + 9
    h = x**3 + 5*x

    assert g.compose(x, h) == f
    assert f.decompose() == [g, h]

def test_PolyElement_shift():
    _, x = ring("x", ZZ)
    assert (x**2 - 2*x + 1).shift(2) == x**2 + 2*x + 1
    assert (x**2 - 2*x + 1).shift_list([2]) == x**2 + 2*x + 1

    R, x, y = ring("x, y", ZZ)
    assert (x*y).shift_list([1, 2]) == (x+1)*(y+2)

    raises(MultivariatePolynomialError, lambda: (x*y).shift(1))

def test_PolyElement_sturm():
    F, t = field("t", ZZ)
    _, x = ring("x", F)

    f = 1024/(15625*t**8)*x**5 - 4096/(625*t**8)*x**4 + 32/(15625*t**4)*x**3 - 128/(625*t**4)*x**2 + F(1)/62500*x - F(1)/625

    assert f.sturm() == [
        x**3 - 100*x**2 + t**4/64*x - 25*t**4/16,
        3*x**2 - 200*x + t**4/64,
        (-t**4/96 + F(20000)/9)*x + 25*t**4/18,
        (-9*t**12 - 11520000*t**8 - 3686400000000*t**4)/(576*t**8 - 245760000*t**4 + 26214400000000),
    ]

def test_PolyElement_gff_list():
    _, x = ring("x", ZZ)

    f = x**5 + 2*x**4 - x**3 - 2*x**2
    assert f.gff_list() == [(x, 1), (x + 2, 4)]

    f = x*(x - 1)**3*(x - 2)**2*(x - 4)**2*(x - 5)
    assert f.gff_list() == [(x**2 - 5*x + 4, 1), (x**2 - 5*x + 4, 2), (x, 3)]

def test_PolyElement_norm():
    k = QQ
    K = QQ.algebraic_field(sqrt(2))
    sqrt2 = K.unit
    _, X, Y = ring("x,y", k)
    _, x, y = ring("x,y", K)

    assert (x*y + sqrt2).norm() == X**2*Y**2 - 2

def test_PolyElement_sqf_norm():
    R, x = ring("x", QQ.algebraic_field(sqrt(3)))
    X = R.to_ground().x

    assert (x**2 - 2).sqf_norm() == ([1], x**2 - 2*sqrt(3)*x + 1, X**4 - 10*X**2 + 1)

    R, x = ring("x", QQ.algebraic_field(sqrt(2)))
    X = R.to_ground().x

    assert (x**2 - 3).sqf_norm() == ([1], x**2 - 2*sqrt(2)*x - 1, X**4 - 10*X**2 + 1)

def test_PolyElement_sqf_list():
    _, x = ring("x", ZZ)

    f = x**5 - x**3 - x**2 + 1
    g = x**3 + 2*x**2 + 2*x + 1
    h = x - 1
    p = x**4 + x**3 - x - 1

    assert f.sqf_part() == p
    assert f.sqf_list() == (1, [(g, 1), (h, 2)])

def test_issue_18894():
    items = [S(3)/16 + sqrt(3*sqrt(3) + 10)/8, S(1)/8 + 3*sqrt(3)/16, S(1)/8 + 3*sqrt(3)/16, -S(3)/16 + sqrt(3*sqrt(3) + 10)/8]
    R, a = sring(items, extension=True)
    assert R.domain == QQ.algebraic_field(sqrt(3)+sqrt(3*sqrt(3)+10))
    assert R.gens == ()
    result = []
    for item in items:
        result.append(R.domain.from_sympy(item))
    assert a == result

def test_PolyElement_factor_list():
    _, x = ring("x", ZZ)

    f = x**5 - x**3 - x**2 + 1

    u = x + 1
    v = x - 1
    w = x**2 + x + 1

    assert f.factor_list() == (1, [(u, 1), (v, 2), (w, 1)])


def test_issue_21410():
    R, x = ring('x', FF(2))
    p = x**6 + x**5 + x**4 + x**3 + 1
    assert p._pow_multinomial(4) == x**24 + x**20 + x**16 + x**12 + 1