Beispiel #1
0
def dup_factor_list(f, K0):
    """Factor univariate polynomials into irreducibles in `K[x]`. """
    j, f = dup_terms_gcd(f, K0)
    cont, f = dup_primitive(f, K0)

    if K0.is_FiniteField:
        coeff, factors = dup_gf_factor(f, K0)
    elif K0.is_Algebraic:
        coeff, factors = dup_ext_factor(f, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dup_convert(f, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.is_Field:
            K = K0.get_ring()

            denom, f = dup_clear_denoms(f, K0, K)
            f = dup_convert(f, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            coeff, factors = dup_zz_factor(f, K)
        elif K.is_Poly:
            f, u = dmp_inject(f, 0, K)

            coeff, factors = dmp_factor_list(f, u, K.dom)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, u, K), k)

            coeff = K.convert(coeff, K.dom)
        else:  # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.is_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K, K0), k)

            coeff = K0.convert(coeff, K)
            coeff = K0.quo(coeff, denom)

            if K0_inexact:
                for i, (f, k) in enumerate(factors):
                    max_norm = dup_max_norm(f, K0)
                    f = dup_quo_ground(f, max_norm, K0)
                    f = dup_convert(f, K0, K0_inexact)
                    factors[i] = (f, k)
                    coeff = K0.mul(coeff, K0.pow(max_norm, k))

                coeff = K0_inexact.convert(coeff, K0)
                K0 = K0_inexact

    if j:
        factors.insert(0, ([K0.one, K0.zero], j))

    return coeff*cont, _sort_factors(factors)
Beispiel #2
0
def dup_factor_list(f, K0):
    """Factor univariate polynomials into irreducibles in `K[x]`. """
    j, f = dup_terms_gcd(f, K0)
    cont, f = dup_primitive(f, K0)

    if K0.is_FiniteField:
        coeff, factors = dup_gf_factor(f, K0)
    elif K0.is_Algebraic:
        coeff, factors = dup_ext_factor(f, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dup_convert(f, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.is_Field:
            K = K0.get_ring()

            denom, f = dup_clear_denoms(f, K0, K)
            f = dup_convert(f, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            coeff, factors = dup_zz_factor(f, K)
        elif K.is_Poly:
            f, u = dmp_inject(f, 0, K)

            coeff, factors = dmp_factor_list(f, u, K.dom)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, u, K), k)

            coeff = K.convert(coeff, K.dom)
        else:  # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.is_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K, K0), k)

            coeff = K0.convert(coeff, K)
            coeff = K0.quo(coeff, denom)

            if K0_inexact:
                for i, (f, k) in enumerate(factors):
                    max_norm = dup_max_norm(f, K0)
                    f = dup_quo_ground(f, max_norm, K0)
                    f = dup_convert(f, K0, K0_inexact)
                    factors[i] = (f, k)
                    coeff = K0.mul(coeff, K0.pow(max_norm, k))

                coeff = K0_inexact.convert(coeff, K0)
                K0 = K0_inexact

    if j:
        factors.insert(0, ([K0.one, K0.zero], j))

    return coeff*cont, _sort_factors(factors)
Beispiel #3
0
def dup_factor_list(f, K0, **args):
    """Factor polynomials into irreducibles in `K[x]`. """
    if not K0.has_CharacteristicZero:  # pragma: no cover
        raise DomainError('only characteristic zero allowed')

    if K0.is_Algebraic:
        coeff, factors = dup_ext_factor(f, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dup_convert(f, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.has_Field:
            K = K0.get_ring()

            denom, f = dup_ground_to_ring(f, K0, K)
            f = dup_convert(f, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            coeff, factors = dup_zz_factor(f, K, **args)
        elif K.is_Poly:
            f, u = dmp_inject(f, 0, K)

            coeff, factors = dmp_factor_list(f, u, K.dom, **args)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, u, K), k)

            coeff = K.convert(coeff, K.dom)
        else:  # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.has_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K, K0), k)

            coeff = K0.convert(coeff, K)
            denom = K0.convert(denom, K)

            coeff = K0.quo(coeff, denom)

        if K0_inexact is not None:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K0, K0_inexact), k)

            coeff = K0_inexact.convert(coeff, K0)

    if not args.get('include', False):
        return coeff, factors
    else:
        if not factors:
            return [(dup_strip([coeff]), 1)]
        else:
            g = dup_mul_ground(factors[0][0], coeff, K)
            return [(g, factors[0][1])] + factors[1:]
Beispiel #4
0
def dup_factor_list(f, K0, **args):
    """Factor polynomials into irreducibles in `K[x]`. """
    if not K0.has_CharacteristicZero: # pragma: no cover
        raise DomainError('only characteristic zero allowed')

    if K0.is_Algebraic:
        coeff, factors = dup_ext_factor(f, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dup_convert(f, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.has_Field:
            K = K0.get_ring()

            denom, f = dup_ground_to_ring(f, K0, K)
            f = dup_convert(f, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            coeff, factors = dup_zz_factor(f, K, **args)
        elif K.is_Poly:
            f, u = dmp_inject(f, 0, K)

            coeff, factors = dmp_factor_list(f, u, K.dom, **args)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, u, K), k)

            coeff = K.convert(coeff, K.dom)
        else: # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.has_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K, K0), k)

            coeff = K0.convert(coeff, K)
            denom = K0.convert(denom, K)

            coeff = K0.quo(coeff, denom)

        if K0_inexact is not None:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K0, K0_inexact), k)

            coeff = K0_inexact.convert(coeff, K0)

    if not args.get('include', False):
        return coeff, factors
    else:
        if not factors:
            return [(dup_strip([coeff]), 1)]
        else:
            g = dup_mul_ground(factors[0][0], coeff, K)
            return [(g, factors[0][1])] + factors[1:]
Beispiel #5
0
def dup_factor_list(f, K0):
    """Factor polynomials into irreducibles in `K[x]`. """
    j, f = dup_terms_gcd(f, K0)

    if not K0.has_CharacteristicZero:
        coeff, factors = dup_gf_factor(f, K0)
    elif K0.is_Algebraic:
        coeff, factors = dup_ext_factor(f, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dup_convert(f, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.has_Field:
            K = K0.get_ring()

            denom, f = dup_clear_denoms(f, K0, K)
            f = dup_convert(f, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            coeff, factors = dup_zz_factor(f, K)
        elif K.is_Poly:
            f, u = dmp_inject(f, 0, K)

            coeff, factors = dmp_factor_list(f, u, K.dom)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, u, K), k)

            coeff = K.convert(coeff, K.dom)
        else: # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.has_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K, K0), k)

            coeff = K0.convert(coeff, K)
            denom = K0.convert(denom, K)

            coeff = K0.quo(coeff, denom)

        if K0_inexact is not None:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K0, K0_inexact), k)

            coeff = K0_inexact.convert(coeff, K0)

    if j:
        factors.insert(0, ([K0.one, K0.zero], j))

    return coeff, _sort_factors(factors)
Beispiel #6
0
def dup_factor_list(f, K0):
    """Factor polynomials into irreducibles in `K[x]`. """
    j, f = dup_terms_gcd(f, K0)

    if not K0.has_CharacteristicZero:
        coeff, factors = dup_gf_factor(f, K0)
    elif K0.is_Algebraic:
        coeff, factors = dup_ext_factor(f, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dup_convert(f, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.has_Field:
            K = K0.get_ring()

            denom, f = dup_clear_denoms(f, K0, K)
            f = dup_convert(f, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            coeff, factors = dup_zz_factor(f, K)
        elif K.is_Poly:
            f, u = dmp_inject(f, 0, K)

            coeff, factors = dmp_factor_list(f, u, K.dom)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, u, K), k)

            coeff = K.convert(coeff, K.dom)
        else:  # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.has_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K, K0), k)

            coeff = K0.convert(coeff, K)
            denom = K0.convert(denom, K)

            coeff = K0.quo(coeff, denom)

        if K0_inexact is not None:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dup_convert(f, K0, K0_inexact), k)

            coeff = K0_inexact.convert(coeff, K0)

    if j:
        factors.insert(0, ([K0.one, K0.zero], j))

    return coeff, _sort_factors(factors)
def dmp_norm(f, u, K):
    """
    Norm of ``f`` in ``K[X1, ..., Xn]``, often not square-free.
    """
    if not K.is_Algebraic:
        raise DomainError("ground domain must be algebraic")

    g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
    h, _ = dmp_inject(f, u, K, front=True)

    return dmp_resultant(g, h, u + 1, K.dom)
Beispiel #8
0
def dmp_norm(f, u, K):
    """
    Norm of ``f`` in ``K[X1, ..., Xn]``, often not square-free.
    """
    if not K.is_Algebraic:
        raise DomainError("ground domain must be algebraic")

    g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
    h, _ = dmp_inject(f, u, K, front=True)

    return dmp_resultant(g, h, u + 1, K.dom)
def dmp_sqf_norm(f, u, K):
    """
    Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.

    Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
    is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> from sympy import I

    >>> K = QQ.algebraic_field(I)
    >>> R, x, y = ring("x,y", K)
    >>> _, X, Y = ring("x,y", QQ)

    >>> s, f, r = R.dmp_sqf_norm(x*y + y**2)

    >>> s == 1
    True
    >>> f == x*y + y**2 + K([QQ(-1), QQ(0)])*y
    True
    >>> r == X**2*Y**2 + 2*X*Y**3 + Y**4 + Y**2
    True

    """
    if not u:
        return dup_sqf_norm(f, K)

    if not K.is_Algebraic:
        raise DomainError("ground domain must be algebraic")

    g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
    F = dmp_raise([K.one, -K.unit], u, 0, K)

    s = 0

    while True:
        h, _ = dmp_inject(f, u, K, front=True)
        r = dmp_resultant(g, h, u + 1, K.dom)

        if dmp_sqf_p(r, u, K.dom):
            break
        else:
            f, s = dmp_compose(f, F, u, K), s + 1

    return s, f, r
Beispiel #10
0
def dmp_sqf_norm(f, u, K):
    """
    Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.

    Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
    is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> from sympy import I

    >>> K = QQ.algebraic_field(I)
    >>> R, x, y = ring("x,y", K)
    >>> _, X, Y = ring("x,y", QQ)

    >>> s, f, r = R.dmp_sqf_norm(x*y + y**2)

    >>> s == 1
    True
    >>> f == x*y + y**2 + K([QQ(-1), QQ(0)])*y
    True
    >>> r == X**2*Y**2 + 2*X*Y**3 + Y**4 + Y**2
    True

    """
    if not u:
        return dup_sqf_norm(f, K)

    if not K.is_Algebraic:
        raise DomainError("ground domain must be algebraic")

    g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
    F = dmp_raise([K.one, -K.unit], u, 0, K)

    s = 0

    while True:
        h, _ = dmp_inject(f, u, K, front=True)
        r = dmp_resultant(g, h, u + 1, K.dom)

        if dmp_sqf_p(r, u, K.dom):
            break
        else:
            f, s = dmp_compose(f, F, u, K), s + 1

    return s, f, r
Beispiel #11
0
def dmp_sqf_norm(f, u, K):
    """
    Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.

    Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
    is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.

    Examples
    ========

    >>> from sympy import I
    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.sqfreetools import dmp_sqf_norm

    >>> K = QQ.algebraic_field(I)

    >>> s, f, r = dmp_sqf_norm([[K(1), K(0)], [K(1), K(0), K(0)]], 1, K)

    >>> s == 1
    True
    >>> f == [[K(1), K(0)], [K(1), K([QQ(-1), QQ(0)]), K(0)]]
    True
    >>> r == [[1, 0, 0], [2, 0, 0, 0], [1, 0, 1, 0, 0]]
    True

    """
    if not u:
        return dup_sqf_norm(f, K)

    if not K.is_Algebraic:
        raise DomainError("ground domain must be algebraic")

    g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
    F = dmp_raise([K.one, -K.unit], u, 0, K)

    s = 0

    while True:
        h, _ = dmp_inject(f, u, K, front=True)
        r = dmp_resultant(g, h, u + 1, K.dom)

        if dmp_sqf_p(r, u, K.dom):
            break
        else:
            f, s = dmp_compose(f, F, u, K), s + 1

    return s, f, r
Beispiel #12
0
def dmp_sqf_norm(f, u, K):
    """
    Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.

    Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
    is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.

    Examples
    ========

    >>> from sympy import I
    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.sqfreetools import dmp_sqf_norm

    >>> K = QQ.algebraic_field(I)

    >>> s, f, r = dmp_sqf_norm([[K(1), K(0)], [K(1), K(0), K(0)]], 1, K)

    >>> s == 1
    True
    >>> f == [[K(1), K(0)], [K(1), K([QQ(-1), QQ(0)]), K(0)]]
    True
    >>> r == [[1, 0, 0], [2, 0, 0, 0], [1, 0, 1, 0, 0]]
    True

    """
    if not u:
        return dup_sqf_norm(f, K)

    if not K.is_Algebraic:
        raise DomainError("ground domain must be algebraic")

    g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
    F = dmp_raise([K.one, -K.unit], u, 0, K)

    s = 0

    while True:
        h, _ = dmp_inject(f, u, K, front=True)
        r = dmp_resultant(g, h, u + 1, K.dom)

        if dmp_sqf_p(r, u, K.dom):
            break
        else:
            f, s = dmp_compose(f, F, u, K), s + 1

    return s, f, r
Beispiel #13
0
def dup_sqf_norm(f, K):
    """
    Square-free norm of ``f`` in ``K[x]``, useful over algebraic domains.

    Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
    is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> from sympy import sqrt

    >>> K = QQ.algebraic_field(sqrt(3))
    >>> R, x = ring("x", K)
    >>> _, X = ring("x", QQ)

    >>> s, f, r = R.dup_sqf_norm(x**2 - 2)

    >>> s == 1
    True
    >>> f == x**2 + K([QQ(-2), QQ(0)])*x + 1
    True
    >>> r == X**4 - 10*X**2 + 1
    True

    """
    if not K.is_Algebraic:
        raise DomainError("ground domain must be algebraic")

    s, g = 0, dmp_raise(K.mod.rep, 1, 0, K.dom)

    while True:
        h, _ = dmp_inject(f, 0, K, front=True)
        r = dmp_resultant(g, h, 1, K.dom)

        if dup_sqf_p(r, K.dom):
            break
        else:
            f, s = dup_shift(f, -K.unit, K), s + 1

    return s, f, r
def dup_sqf_norm(f, K):
    """
    Square-free norm of ``f`` in ``K[x]``, useful over algebraic domains.

    Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
    is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> from sympy import sqrt

    >>> K = QQ.algebraic_field(sqrt(3))
    >>> R, x = ring("x", K)
    >>> _, X = ring("x", QQ)

    >>> s, f, r = R.dup_sqf_norm(x**2 - 2)

    >>> s == 1
    True
    >>> f == x**2 + K([QQ(-2), QQ(0)])*x + 1
    True
    >>> r == X**4 - 10*X**2 + 1
    True

    """
    if not K.is_Algebraic:
        raise DomainError("ground domain must be algebraic")

    s, g = 0, dmp_raise(K.mod.rep, 1, 0, K.dom)

    while True:
        h, _ = dmp_inject(f, 0, K, front=True)
        r = dmp_resultant(g, h, 1, K.dom)

        if dup_sqf_p(r, K.dom):
            break
        else:
            f, s = dup_shift(f, -K.unit, K), s + 1

    return s, f, r
Beispiel #15
0
def test_dmp_inject():
    K = ZZ['x','y']

    assert dmp_inject([], 0, K) == ([[[]]], 2)
    assert dmp_inject([[]], 1, K) == ([[[[]]]], 3)

    assert dmp_inject([K([[1]])], 0, K) == ([[[1]]], 2)
    assert dmp_inject([[K([[1]])]], 1, K) == ([[[[1]]]], 3)

    assert dmp_inject([K([[1]]),K([[2],[3,4]])], 0, K) == ([[[1]],[[2],[3,4]]], 2)

    f = [K([[3],[7,0],[5,0,0]]),K([[2],[]]),K([[]]),K([[1,0,0],[11]])]
    g = [[[3],[7,0],[5,0,0]],[[2],[]],[[]],[[1,0,0],[11]]]

    assert dmp_inject(f, 0, K) == (g, 2)
Beispiel #16
0
def dup_sqf_norm(f, K):
    """
    Square-free norm of ``f`` in ``K[x]``, useful over algebraic domains.

    Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
    is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.

    **Examples**

    >>> from sympy import sqrt
    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.sqfreetools import dup_sqf_norm

    >>> K = QQ.algebraic_field(sqrt(3))

    >>> s, f, r = dup_sqf_norm([K(1), K(0), K(-2)], K)

    >>> s == 1
    True
    >>> f == [K(1), K([QQ(-2), QQ(0)]), K(1)]
    True
    >>> r == [1, 0, -10, 0, 1]
    True

    """
    if not K.is_Algebraic:
        raise DomainError("ground domain must be algebraic")

    s, g = 0, dmp_raise(K.mod.rep, 1, 0, K.dom)

    while True:
        h, _ = dmp_inject(f, 0, K, front=True)
        r = dmp_resultant(g, h, 1, K.dom)

        if dup_sqf_p(r, K.dom):
            break
        else:
            f, s = dup_shift(f, -K.unit, K), s+1

    return s, f, r
Beispiel #17
0
def test_dmp_inject():
    R, x,y = ring("x,y", ZZ)
    K = R.to_domain()

    assert dmp_inject([], 0, K) == ([[[]]], 2)
    assert dmp_inject([[]], 1, K) == ([[[[]]]], 3)

    assert dmp_inject([R(1)], 0, K) == ([[[1]]], 2)
    assert dmp_inject([[R(1)]], 1, K) == ([[[[1]]]], 3)

    assert dmp_inject([R(1), 2*x + 3*y + 4], 0, K) == ([[[1]], [[2], [3, 4]]], 2)

    f = [3*x**2 + 7*x*y + 5*y**2, 2*x, R(0), x*y**2 + 11]
    g = [[[3], [7, 0], [5, 0, 0]], [[2], []], [[]], [[1, 0, 0], [11]]]

    assert dmp_inject(f, 0, K) == (g, 2)
Beispiel #18
0
def dmp_factor_list(f, u, K0):
    """Factor polynomials into irreducibles in `K[X]`. """
    if not u:
        return dup_factor_list(f, K0)

    J, f = dmp_terms_gcd(f, u, K0)

    if not K0.has_CharacteristicZero:  # pragma: no cover
        coeff, factors = dmp_gf_factor(f, u, K0)
    elif K0.is_Algebraic:
        coeff, factors = dmp_ext_factor(f, u, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dmp_convert(f, u, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.has_Field:
            K = K0.get_ring()

            denom, f = dmp_clear_denoms(f, u, K0, K)
            f = dmp_convert(f, u, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            levels, f, v = dmp_exclude(f, u, K)
            coeff, factors = dmp_zz_factor(f, v, K)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_include(f, levels, v, K), k)
        elif K.is_Poly:
            f, v = dmp_inject(f, u, K)

            coeff, factors = dmp_factor_list(f, v, K.dom)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, v, K), k)

            coeff = K.convert(coeff, K.dom)
        else:  # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.has_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_convert(f, u, K, K0), k)

            coeff = K0.convert(coeff, K)
            denom = K0.convert(denom, K)

            coeff = K0.quo(coeff, denom)

        if K0_inexact is not None:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_convert(f, u, K0, K0_inexact), k)

            coeff = K0_inexact.convert(coeff, K0)

    for i, j in enumerate(reversed(J)):
        if not j:
            continue

        term = {(0, ) * (u - i) + (1, ) + (0, ) * i: K0.one}
        factors.insert(0, (dmp_from_dict(term, u, K0), j))

    return coeff, _sort_factors(factors)
Beispiel #19
0
def dmp_factor_list(f, u, K0):
    """Factor polynomials into irreducibles in `K[X]`. """
    if not u:
        return dup_factor_list(f, K0)

    J, f = dmp_terms_gcd(f, u, K0)

    if not K0.has_CharacteristicZero: # pragma: no cover
        coeff, factors = dmp_gf_factor(f, u, K0)
    elif K0.is_Algebraic:
        coeff, factors = dmp_ext_factor(f, u, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dmp_convert(f, u, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.has_Field:
            K = K0.get_ring()

            denom, f = dmp_clear_denoms(f, u, K0, K)
            f = dmp_convert(f, u, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            levels, f, v = dmp_exclude(f, u, K)
            coeff, factors = dmp_zz_factor(f, v, K)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_include(f, levels, v, K), k)
        elif K.is_Poly:
            f, v = dmp_inject(f, u, K)

            coeff, factors = dmp_factor_list(f, v, K.dom)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, v, K), k)

            coeff = K.convert(coeff, K.dom)
        else: # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.has_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_convert(f, u, K, K0), k)

            coeff = K0.convert(coeff, K)
            denom = K0.convert(denom, K)

            coeff = K0.quo(coeff, denom)

        if K0_inexact is not None:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_convert(f, u, K0, K0_inexact), k)

            coeff = K0_inexact.convert(coeff, K0)

    for i, j in enumerate(reversed(J)):
        if not j:
            continue

        term = {(0,)*(u-i) + (1,) + (0,)*i: K0.one}
        factors.insert(0, (dmp_from_dict(term, u, K0), j))

    return coeff, _sort_factors(factors)
Beispiel #20
0
 def inject(f, front=False):
     """Inject ground domain generators into ``f``. """
     F, lev = dmp_inject(f.rep, f.lev, f.dom, front=front)
     return f.__class__(F, f.dom.dom, lev)
Beispiel #21
0
 def inject(f, front=False):
     """Inject ground domain generators into ``f``. """
     F, lev = dmp_inject(f.rep, f.lev, f.dom, front=front)
     return f.__class__(F, f.dom.dom, lev)
Beispiel #22
0
def dmp_factor_list(f, u, K0):
    """Factor multivariate polynomials into irreducibles in `K[X]`. """
    if not u:
        return dup_factor_list(f, K0)

    J, f = dmp_terms_gcd(f, u, K0)
    cont, f = dmp_ground_primitive(f, u, K0)

    if K0.is_FiniteField:  # pragma: no cover
        coeff, factors = dmp_gf_factor(f, u, K0)
    elif K0.is_Algebraic:
        coeff, factors = dmp_ext_factor(f, u, K0)


#     elif K0.is_GaussianRing:
#         coeff, factors = dmp_zz_i_factor(f, u, K0)
#     elif K0.is_GaussianField:
#         coeff, factors = dmp_qq_i_factor(f, u, K0)
    else:
        if not K0.is_Exact:
            K0_inexact, K0 = K0, K0.get_exact()
            f = dmp_convert(f, u, K0_inexact, K0)
        else:
            K0_inexact = None

        if K0.is_Field:
            K = K0.get_ring()

            denom, f = dmp_clear_denoms(f, u, K0, K)
            f = dmp_convert(f, u, K0, K)
        else:
            K = K0

        if K.is_ZZ:
            levels, f, v = dmp_exclude(f, u, K)
            coeff, factors = dmp_zz_factor(f, v, K)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_include(f, levels, v, K), k)
        elif K.is_Poly:
            f, v = dmp_inject(f, u, K)

            coeff, factors = dmp_factor_list(f, v, K.dom)

            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_eject(f, v, K), k)

            coeff = K.convert(coeff, K.dom)
        else:  # pragma: no cover
            raise DomainError('factorization not supported over %s' % K0)

        if K0.is_Field:
            for i, (f, k) in enumerate(factors):
                factors[i] = (dmp_convert(f, u, K, K0), k)

            coeff = K0.convert(coeff, K)
            coeff = K0.quo(coeff, denom)

            if K0_inexact:
                for i, (f, k) in enumerate(factors):
                    max_norm = dmp_max_norm(f, u, K0)
                    f = dmp_quo_ground(f, max_norm, u, K0)
                    f = dmp_convert(f, u, K0, K0_inexact)
                    factors[i] = (f, k)
                    coeff = K0.mul(coeff, K0.pow(max_norm, k))

                coeff = K0_inexact.convert(coeff, K0)
                K0 = K0_inexact

    for i, j in enumerate(reversed(J)):
        if not j:
            continue

        term = {(0, ) * (u - i) + (1, ) + (0, ) * i: K0.one}
        factors.insert(0, (dmp_from_dict(term, u, K0), j))

    return coeff * cont, _sort_factors(factors)