def dup_zz_hensel_step(m, f, g, h, s, t, K):
    """
    One step in Hensel lifting in `Z[x]`.

    Given positive integer `m` and `Z[x]` polynomials `f`, `g`, `h`, `s`
    and `t` such that::

        f == g*h (mod m)
        s*g + t*h == 1 (mod m)

        lc(f) is not a zero divisor (mod m)
        lc(h) == 1

        deg(f) == deg(g) + deg(h)
        deg(s) < deg(h)
        deg(t) < deg(g)

    returns polynomials `G`, `H`, `S` and `T`, such that::

        f == G*H (mod m**2)
        S*G + T**H == 1 (mod m**2)

    References
    ==========

    1. [Gathen99]_

    """
    M = m**2

    e = dup_sub_mul(f, g, h, K)
    e = dup_trunc(e, M, K)

    q, r = dup_div(dup_mul(s, e, K), h, K)

    q = dup_trunc(q, M, K)
    r = dup_trunc(r, M, K)

    u = dup_add(dup_mul(t, e, K), dup_mul(q, g, K), K)
    G = dup_trunc(dup_add(g, u, K), M, K)
    H = dup_trunc(dup_add(h, r, K), M, K)

    u = dup_add(dup_mul(s, G, K), dup_mul(t, H, K), K)
    b = dup_trunc(dup_sub(u, [K.one], K), M, K)

    c, d = dup_div(dup_mul(s, b, K), H, K)

    c = dup_trunc(c, M, K)
    d = dup_trunc(d, M, K)

    u = dup_add(dup_mul(t, b, K), dup_mul(c, G, K), K)
    S = dup_trunc(dup_sub(s, d, K), M, K)
    T = dup_trunc(dup_sub(t, u, K), M, K)

    return G, H, S, T

def dup_half_gcdex(f, g, K):
    """
    Half extended Euclidean algorithm in ``F[x]``.

    Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.

    **Examples**

    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.euclidtools import dup_half_gcdex

    >>> f = QQ.map([1, -2, -6, 12, 15])
    >>> g = QQ.map([1, 1, -4, -4])

    >>> dup_half_gcdex(f, g, QQ)
    ([-1/5, 3/5], [1/1, 1/1])

    """
    if not (K.has_Field or not K.is_Exact):
        raise DomainError("can't compute half extended GCD over %s" % K)

    a, b = [K.one], []

    while g:
        q, r = dup_div(f, g, K)
        f, g = g, r
        a, b = b, dup_sub_mul(a, q, b, K)

    a = dup_exquo_ground(a, dup_LC(f, K), K)
    f = dup_monic(f, K)

    return a, f

def test_dup_div():
    f, g, q, r = [5,4,3,2,1], [1,2,3], [5,-6,0], [20,1]

    assert dup_div(f, g, ZZ) == (q, r)
    assert dup_quo(f, g, ZZ) == q
    assert dup_rem(f, g, ZZ) == r

    raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))

    f, g, q, r = [5,4,3,2,1,0], [1,2,0,0,9], [5,-6], [15,2,-44,54]

    assert dup_div(f, g, ZZ) == (q, r)
    assert dup_quo(f, g, ZZ) == q
    assert dup_rem(f, g, ZZ) == r

    raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))

def dup_half_gcdex(f, g, K):
    """
    Half extended Euclidean algorithm in `F[x]`.

    Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x = ring("x", QQ)

    >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
    >>> g = x**3 + x**2 - 4*x - 4

    >>> R.dup_half_gcdex(f, g)
    (-1/5*x + 3/5, x + 1)

    """
    if not K.has_Field:
        raise DomainError("can't compute half extended GCD over %s" % K)

    a, b = [K.one], []

    while g:
        q, r = dup_div(f, g, K)
        f, g = g, r
        a, b = b, dup_sub_mul(a, q, b, K)

    a = dup_quo_ground(a, dup_LC(f, K), K)
    f = dup_monic(f, K)

    return a, f

def _dup_left_decompose(f, h, K):
    """Helper function for :func:`_dup_decompose`."""
    g, i = {}, 0

    while f:
        q, r = dup_div(f, h, K)

        if dup_degree(r) > 0:
            return None
        else:
            g[i] = dup_LC(r, K)
            f, i = q, i + 1

    return dup_from_raw_dict(g, K)

def dup_trial_division(f, factors, K):
    """Determine multiplicities of factors using trial division. """
    result = []

    for factor in factors:
        k = 0

        while True:
            q, r = dup_div(f, factor, K)

            if not r:
                f, k = q, k+1
            else:
                break

        result.append((factor, k))

    return _sort_factors(result)

def dup_zz_factor(f, K):
    """
    Factor (non square-free) polynomials in `Z[x]`.

    Given a univariate polynomial `f` in `Z[x]` computes its complete
    factorization `f_1, ..., f_n` into irreducibles over integers::

                f = content(f) f_1**k_1 ... f_n**k_n

    The factorization is computed by reducing the input polynomial
    into a primitive square-free polynomial and factoring it using
    Zassenhaus algorithm. Trial division is used to recover the
    multiplicities of factors.

    The result is returned as a tuple consisting of::

              (content(f), [(f_1, k_1), ..., (f_n, k_n))

    Consider polynomial `f = 2*x**4 - 2`::

        >>> from sympy.polys.factortools import dup_zz_factor
        >>> from sympy.polys.domains import ZZ

        >>> dup_zz_factor([2, 0, 0, 0, -2], ZZ)
        (2, [([1, -1], 1), ([1, 1], 1), ([1, 0, 1], 1)])

    In result we got the following factorization::

                 f = 2 (x - 1) (x + 1) (x**2 + 1)

    Note that this is a complete factorization over integers,
    however over Gaussian integers we can factor the last term.

    By default, polynomials `x**n - 1` and `x**n + 1` are factored
    using cyclotomic decomposition to speedup computations. To
    disable this behaviour set cyclotomic=False.

    **References**

    1. [Gathen99]_

    """
    cont, g = dup_primitive(f, K)

    n = dup_degree(g)

    if dup_LC(g, K) < 0:
        cont, g = -cont, dup_neg(g, K)

    if n <= 0:
        return cont, []
    elif n == 1:
        return cont, [(g, 1)]

    if query('USE_IRREDUCIBLE_IN_FACTOR'):
        if dup_zz_irreducible_p(g, K):
            return cont, [(g, 1)]

    g = dup_sqf_part(g, K)
    H, factors = None, []

    if query('USE_CYCLOTOMIC_FACTOR'):
        H = dup_zz_cyclotomic_factor(g, K)

    if H is None:
        H = dup_zz_zassenhaus(g, K)

    for h in H:
        k = 0

        while True:
            q, r = dup_div(f, h, K)

            if not r:
                f, k = q, k+1
            else:
                break

        factors.append((h, k))

    return cont, _sort_factors(factors)

def dup_zz_heu_gcd(f, g, K):
    """
    Heuristic polynomial GCD in ``Z[x]``.

    Given univariate polynomials ``f`` and ``g`` in ``Z[x]``, returns
    their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``
    such that::

          h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)

    The algorithm is purely heuristic which means it may fail to compute
    the GCD. This will be signaled by raising an exception. In this case
    you will need to switch to another GCD method.

    The algorithm computes the polynomial GCD by evaluating polynomials
    f and g at certain points and computing (fast) integer GCD of those
    evaluations. The polynomial GCD is recovered from the integer image
    by interpolation.  The final step is to verify if the result is the
    correct GCD. This gives cofactors as a side effect.

    **Examples**

    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.euclidtools import dup_zz_heu_gcd

    >>> f = ZZ.map([1, 0, -1])
    >>> g = ZZ.map([1, -3, 2])

    >>> dup_zz_heu_gcd(f, g, ZZ)
    ([1, -1], [1, 1], [1, -2])

    **References**

    1. [Liao95]_

    """
    result = _dup_rr_trivial_gcd(f, g, K)

    if result is not None:
        return result

    df = dup_degree(f)
    dg = dup_degree(g)

    gcd, f, g = dup_extract(f, g, K)

    if df == 0 or dg == 0:
        return [gcd], f, g

    f_norm = dup_max_norm(f, K)
    g_norm = dup_max_norm(g, K)

    B = 2*min(f_norm, g_norm) + 29

    x = max(min(B, 99*K.sqrt(B)),
            2*min(f_norm // abs(dup_LC(f, K)),
                  g_norm // abs(dup_LC(g, K))) + 2)

    for i in xrange(0, HEU_GCD_MAX):
        ff = dup_eval(f, x, K)
        gg = dup_eval(g, x, K)

        if ff and gg:
            h = K.gcd(ff, gg)

            cff = ff // h
            cfg = gg // h

            h = _dup_zz_gcd_interpolate(h, x, K)
            h = dup_primitive(h, K)[1]

            cff_, r = dup_div(f, h, K)

            if not r:
                cfg_, r = dup_div(g, h, K)

                if not r:
                    h = dup_mul_ground(h, gcd, K)
                    return h, cff_, cfg_

            cff = _dup_zz_gcd_interpolate(cff, x, K)

            h, r = dup_div(f, cff, K)

            if not r:
                cfg_, r = dup_div(g, h, K)

                if not r:
                    h = dup_mul_ground(h, gcd, K)
                    return h, cff, cfg_

            cfg = _dup_zz_gcd_interpolate(cfg, x, K)

            h, r = dup_div(g, cfg, K)

            if not r:
                cff_, r = dup_div(f, h, K)

                if not r:
                    h = dup_mul_ground(h, gcd, K)
#.........这里部分代码省略.........