def dmp_fateman_poly_F_1(n, K):
    """Fateman's GCD benchmark: trivial GCD """
    u = [K(1), K(0)]

    for i in xrange(0, n):
        u = [dmp_one(i, K), u]

    v = [K(1), K(0), K(0)]

    for i in xrange(0, n):
        v = [dmp_one(i, K), dmp_zero(i), v]

    m = n - 1

    U = dmp_add_term(u, dmp_ground(K(1), m), 0, n, K)
    V = dmp_add_term(u, dmp_ground(K(2), m), 0, n, K)

    f = [[-K(3), K(0)], [], [K(1), K(0), -K(1)]]

    W = dmp_add_term(v, dmp_ground(K(1), m), 0, n, K)
    Y = dmp_raise(f, m, 1, K)

    F = dmp_mul(U, V, n, K)
    G = dmp_mul(W, Y, n, K)

    H = dmp_one(n, K)

    return F, G, H

def dmp_compose(f, g, u, K):
    """
    Evaluate functional composition ``f(g)`` in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_compose(x*y + 2*x + y, y)
    y**2 + 3*y

    """
    if not u:
        return dup_compose(f, g, K)

    if dmp_zero_p(f, u):
        return f

    h = [f[0]]

    for c in f[1:]:
        h = dmp_mul(h, g, u, K)
        h = dmp_add_term(h, c, 0, u, K)

    return h

def dmp_compose(f, g, u, K):
    """
    Evaluate functional composition ``f(g)`` in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.densetools import dmp_compose

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

    >>> dmp_compose(f, g, 1, ZZ)
    [[1, 3, 0]]

    """
    if not u:
        return dup_compose(f, g, K)

    if dmp_zero_p(f, u):
        return f

    h = [f[0]]

    for c in f[1:]:
        h = dmp_mul(h, g, u, K)
        h = dmp_add_term(h, c, 0, u, K)

    return h

def dmp_fateman_poly_F_2(n, K):
    """Fateman's GCD benchmark: linearly dense quartic inputs """
    u = [K(1), K(0)]

    for i in xrange(0, n - 1):
        u = [dmp_one(i, K), u]

    m = n - 1

    v = dmp_add_term(u, dmp_ground(K(2), m - 1), 0, n, K)

    f = dmp_sqr([dmp_one(m, K), dmp_neg(v, m, K)], n, K)
    g = dmp_sqr([dmp_one(m, K), v], n, K)

    v = dmp_add_term(u, dmp_one(m - 1, K), 0, n, K)

    h = dmp_sqr([dmp_one(m, K), v], n, K)

    return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h

def dup_real_imag(f, K):
    """
    Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.

    Examples
    ========

    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.densetools import dup_real_imag

    >>> dup_real_imag([ZZ(1), ZZ(1), ZZ(1), ZZ(1)], ZZ)
    ([[1], [1], [-3, 0, 1], [-1, 0, 1]], [[3, 0], [2, 0], [-1, 0, 1, 0]])

    """
    if not K.is_ZZ and not K.is_QQ:
        raise DomainError(
            "computing real and imaginary parts is not supported over %s" % K)

    f1 = dmp_zero(1)
    f2 = dmp_zero(1)

    if not f:
        return f1, f2

    g = [[[K.one, K.zero]], [[K.one], []]]
    h = dmp_ground(f[0], 2)

    for c in f[1:]:
        h = dmp_mul(h, g, 2, K)
        h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)

    H = dup_to_raw_dict(h)

    for k, h in H.iteritems():
        m = k % 4

        if not m:
            f1 = dmp_add(f1, h, 1, K)
        elif m == 1:
            f2 = dmp_add(f2, h, 1, K)
        elif m == 2:
            f1 = dmp_sub(f1, h, 1, K)
        else:
            f2 = dmp_sub(f2, h, 1, K)

    return f1, f2

def dup_real_imag(f, K):
    """
    Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.

    Examples
    ========

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

    >>> R.dup_real_imag(x**3 + x**2 + x + 1)
    (x**3 + x**2 - 3*x*y**2 + x - y**2 + 1, 3*x**2*y + 2*x*y - y**3 + y)

    """
    if not K.is_ZZ and not K.is_QQ:
        raise DomainError("computing real and imaginary parts is not supported over %s" % K)

    f1 = dmp_zero(1)
    f2 = dmp_zero(1)

    if not f:
        return f1, f2

    g = [[[K.one, K.zero]], [[K.one], []]]
    h = dmp_ground(f[0], 2)

    for c in f[1:]:
        h = dmp_mul(h, g, 2, K)
        h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)

    H = dup_to_raw_dict(h)

    for k, h in H.items():
        m = k % 4

        if not m:
            f1 = dmp_add(f1, h, 1, K)
        elif m == 1:
            f2 = dmp_add(f2, h, 1, K)
        elif m == 2:
            f1 = dmp_sub(f1, h, 1, K)
        else:
            f2 = dmp_sub(f2, h, 1, K)

    return f1, f2

def dmp_fateman_poly_F_3(n, K):
    """Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """
    u = dup_from_raw_dict({n+1: K.one}, K)

    for i in xrange(0, n-1):
        u = dmp_add_term([u], dmp_one(i, K), n+1, i+1, K)

    v = dmp_add_term(u, dmp_ground(K(2), n-2), 0, n, K)

    f = dmp_sqr(dmp_add_term([dmp_neg(v, n-1, K)], dmp_one(n-1, K), n+1, n, K), n, K)
    g = dmp_sqr(dmp_add_term([v], dmp_one(n-1, K), n+1, n, K), n, K)

    v = dmp_add_term(u, dmp_one(n-2, K), 0, n-1, K)

    h = dmp_sqr(dmp_add_term([v], dmp_one(n-1, K), n+1, n, K), n, K)

    return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h

def test_dmp_add_term():
    assert dmp_add_term([ZZ(1),ZZ(1),ZZ(1)], ZZ(1), 2, 0, ZZ) == \
           dup_add_term([ZZ(1),ZZ(1),ZZ(1)], ZZ(1), 2, ZZ)
    assert dmp_add_term(f_0, [[]], 3, 2, ZZ) == f_0
    assert dmp_add_term(F_0, [[]], 3, 2, QQ) == F_0