def test_dmp_from_to_dict():
    assert dmp_from_dict({}, 1, ZZ) == [[]]
    assert dmp_to_dict([[]], 1) == {}

    f = [[3],[],[],[2],[],[],[],[],[8]]
    g = {(8,0): 3, (5,0): 2, (0,0): 8}

    assert dmp_from_dict(g, 1, ZZ) == f
    assert dmp_to_dict(f, 1) == g

def test_dmp_from_to_dict():
    assert dmp_from_dict({}, 1, ZZ) == [[]]
    assert dmp_to_dict([[]], 1) == {}

    assert dmp_to_dict([], 0, ZZ, zero=True) == {(0,): ZZ(0)}
    assert dmp_to_dict([[]], 1, ZZ, zero=True) == {(0,0): ZZ(0)}

    f = [[3],[],[],[2],[],[],[],[],[8]]
    g = {(8,0): 3, (5,0): 2, (0,0): 8}

    assert dmp_from_dict(g, 1, ZZ) == f
    assert dmp_to_dict(f, 1) == g

    def to_sympy_dict(f, zero=False):
        """Convert `f` to a dict representation with SymPy coefficients. """
        rep = dmp_to_dict(f.rep, f.lev, f.dom, zero=zero)

        for k, v in rep.iteritems():
            rep[k] = f.dom.to_sympy(v)

        return rep

    def to_sympy_dict(f):
        """Convert ``f`` to a dict representation with SymPy coefficients. """
        rep = dmp_to_dict(f.rep, 0, f.dom)

        for k, v in rep.iteritems():
            rep[k] = f.dom.to_sympy(v)

        return rep

def dmp_lift(f, u, K):
    """
    Convert algebraic coefficients to integers in ``K[X]``.

    Examples
    ========

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

    >>> K = QQ.algebraic_field(I)
    >>> R, x = ring("x", K)

    >>> f = x**2 + K([QQ(1), QQ(0)])*x + K([QQ(2), QQ(0)])

    >>> R.dmp_lift(f)
    x**8 + 2*x**6 + 9*x**4 - 8*x**2 + 16

    """
    if not K.is_Algebraic:
        raise DomainError(
            'computation can be done only in an algebraic domain')

    F, monoms, polys = dmp_to_dict(f, u), [], []

    for monom, coeff in F.items():
        if not coeff.is_ground:
            monoms.append(monom)

    perms = variations([-1, 1], len(monoms), repetition=True)

    for perm in perms:
        G = dict(F)

        for sign, monom in zip(perm, monoms):
            if sign == -1:
                G[monom] = -G[monom]

        polys.append(dmp_from_dict(G, u, K))

    return dmp_convert(dmp_expand(polys, u, K), u, K, K.dom)

def dmp_lift(f, u, K):
    """
    Convert algebraic coefficients to integers in ``K[X]``.

    Examples
    ========

    >>> from sympy import I
    >>> from sympy.polys.domains import QQ
    >>> from sympy.polys.densetools import dmp_lift

    >>> K = QQ.algebraic_field(I)
    >>> f = [K(1), K([QQ(1), QQ(0)]), K([QQ(2), QQ(0)])]

    >>> dmp_lift(f, 0, K)
    [1/1, 0/1, 2/1, 0/1, 9/1, 0/1, -8/1, 0/1, 16/1]

    """
    if not K.is_Algebraic:
        raise DomainError(
            'computation can be done only in an algebraic domain')

    F, monoms, polys = dmp_to_dict(f, u), [], []

    for monom, coeff in F.iteritems():
        if not coeff.is_ground:
            monoms.append(monom)

    perms = variations([-1, 1], len(monoms), repetition=True)

    for perm in perms:
        G = dict(F)

        for sign, monom in zip(perm, monoms):
            if sign == -1:
                G[monom] = -G[monom]

        polys.append(dmp_from_dict(G, u, K))

    return dmp_convert(dmp_expand(polys, u, K), u, K, K.dom)

 def is_quadratic(f):
     """Returns `True` if `f` is quadratic in all its variables. """
     return all([ sum(monom) <= 2 for monom in dmp_to_dict(f.rep, f.lev, f.dom).keys() ])

 def is_linear(f):
     """Returns `True` if `f` is linear in all its variables. """
     return all([ sum(monom) <= 1 for monom in dmp_to_dict(f.rep, f.lev, f.dom).keys() ])

 def to_dict(f, zero=False):
     """Convert `f` to a dict representation with native coefficients. """
     return dmp_to_dict(f.rep, f.lev, f.dom, zero=zero)