polyroots.py

"""Algorithms for computing symbolic roots of polynomials. """


import math
from functools import reduce

from sympy.core import S, I, pi
from sympy.core.exprtools import factor_terms
from sympy.core.function import _mexpand
from sympy.core.logic import fuzzy_not
from sympy.core.mul import expand_2arg, Mul
from sympy.core.numbers import Rational, igcd, comp
from sympy.core.power import Pow
from sympy.core.relational import Eq
from sympy.core.sorting import ordered
from sympy.core.symbol import Dummy, Symbol, symbols
from sympy.core.sympify import sympify
from sympy.functions import exp, im, cos, acos, Piecewise
from sympy.functions.elementary.miscellaneous import root, sqrt
from sympy.ntheory import divisors, isprime, nextprime
from sympy.polys.domains import EX
from sympy.polys.polyerrors import (PolynomialError, GeneratorsNeeded,
    DomainError, UnsolvableFactorError)
from sympy.polys.polyquinticconst import PolyQuintic
from sympy.polys.polytools import Poly, cancel, factor, gcd_list, discriminant
from sympy.polys.rationaltools import together
from sympy.polys.specialpolys import cyclotomic_poly
from sympy.utilities import public
from sympy.utilities.misc import filldedent



z = Symbol('z')  # importing from abc cause O to be lost as clashing symbol


def roots_linear(f):
    """Returns a list of roots of a linear polynomial."""
    r = -f.nth(0)/f.nth(1)
    dom = f.get_domain()

    if not dom.is_Numerical:
        if dom.is_Composite:
            r = factor(r)
        else:
            from sympy.simplify.simplify import simplify
            r = simplify(r)

    return [r]


def roots_quadratic(f):
    """Returns a list of roots of a quadratic polynomial. If the domain is ZZ
    then the roots will be sorted with negatives coming before positives.
    The ordering will be the same for any numerical coefficients as long as
    the assumptions tested are correct, otherwise the ordering will not be
    sorted (but will be canonical).
    """

    a, b, c = f.all_coeffs()
    dom = f.get_domain()

    def _sqrt(d):
        # remove squares from square root since both will be represented
        # in the results; a similar thing is happening in roots() but
        # must be duplicated here because not all quadratics are binomials
        co = []
        other = []
        for di in Mul.make_args(d):
            if di.is_Pow and di.exp.is_Integer and di.exp % 2 == 0:
                co.append(Pow(di.base, di.exp//2))
            else:
                other.append(di)
        if co:
            d = Mul(*other)
            co = Mul(*co)
            return co*sqrt(d)
        return sqrt(d)

    def _simplify(expr):
        if dom.is_Composite:
            return factor(expr)
        else:
            from sympy.simplify.simplify import simplify
            return simplify(expr)

    if c is S.Zero:
        r0, r1 = S.Zero, -b/a

        if not dom.is_Numerical:
            r1 = _simplify(r1)
        elif r1.is_negative:
            r0, r1 = r1, r0
    elif b is S.Zero:
        r = -c/a
        if not dom.is_Numerical:
            r = _simplify(r)

        R = _sqrt(r)
        r0 = -R
        r1 = R
    else:
        d = b**2 - 4*a*c
        A = 2*a
        B = -b/A

        if not dom.is_Numerical:
            d = _simplify(d)
            B = _simplify(B)

        D = factor_terms(_sqrt(d)/A)
        r0 = B - D
        r1 = B + D
        if a.is_negative:
            r0, r1 = r1, r0
        elif not dom.is_Numerical:
            r0, r1 = [expand_2arg(i) for i in (r0, r1)]

    return [r0, r1]


def roots_cubic(f, trig=False):
    """Returns a list of roots of a cubic polynomial.

    References
    ==========
    [1] https://en.wikipedia.org/wiki/Cubic_function, General formula for roots,
    (accessed November 17, 2014).
    """
    if trig:
        a, b, c, d = f.all_coeffs()
        p = (3*a*c - b**2)/(3*a**2)
        q = (2*b**3 - 9*a*b*c + 27*a**2*d)/(27*a**3)
        D = 18*a*b*c*d - 4*b**3*d + b**2*c**2 - 4*a*c**3 - 27*a**2*d**2
        if (D > 0) == True:
            rv = []
            for k in range(3):
                rv.append(2*sqrt(-p/3)*cos(acos(q/p*sqrt(-3/p)*Rational(3, 2))/3 - k*pi*Rational(2, 3)))
            return [i - b/3/a for i in rv]

    # a*x**3 + b*x**2 + c*x + d -> x**3 + a*x**2 + b*x + c
    _, a, b, c = f.monic().all_coeffs()

    if c is S.Zero:
        x1, x2 = roots([1, a, b], multiple=True)
        return [x1, S.Zero, x2]

    # x**3 + a*x**2 + b*x + c -> u**3 + p*u + q
    p = b - a**2/3
    q = c - a*b/3 + 2*a**3/27

    pon3 = p/3
    aon3 = a/3

    u1 = None
    if p is S.Zero:
        if q is S.Zero:
            return [-aon3]*3
        u1 = -root(q, 3) if q.is_positive else root(-q, 3)
    elif q is S.Zero:
        y1, y2 = roots([1, 0, p], multiple=True)
        return [tmp - aon3 for tmp in [y1, S.Zero, y2]]
    elif q.is_real and q.is_negative:
        u1 = -root(-q/2 + sqrt(q**2/4 + pon3**3), 3)

    coeff = I*sqrt(3)/2
    if u1 is None:
        u1 = S.One
        u2 = Rational(-1, 2) + coeff
        u3 = Rational(-1, 2) - coeff
        b, c, d = a, b, c  # a, b, c, d = S.One, a, b, c
        D0 = b**2 - 3*c  # b**2 - 3*a*c
        D1 = 2*b**3 - 9*b*c + 27*d  # 2*b**3 - 9*a*b*c + 27*a**2*d
        C = root((D1 + sqrt(D1**2 - 4*D0**3))/2, 3)
        return [-(b + uk*C + D0/C/uk)/3 for uk in [u1, u2, u3]]  # -(b + uk*C + D0/C/uk)/3/a

    u2 = u1*(Rational(-1, 2) + coeff)
    u3 = u1*(Rational(-1, 2) - coeff)

    if p is S.Zero:
        return [u1 - aon3, u2 - aon3, u3 - aon3]

    soln = [
        -u1 + pon3/u1 - aon3,
        -u2 + pon3/u2 - aon3,
        -u3 + pon3/u3 - aon3
    ]

    return soln

def _roots_quartic_euler(p, q, r, a):
    """
    Descartes-Euler solution of the quartic equation

    Parameters
    ==========

    p, q, r: coefficients of ``x**4 + p*x**2 + q*x + r``
    a: shift of the roots

    Notes
    =====

    This is a helper function for ``roots_quartic``.

    Look for solutions of the form ::

      ``x1 = sqrt(R) - sqrt(A + B*sqrt(R))``
      ``x2 = -sqrt(R) - sqrt(A - B*sqrt(R))``
      ``x3 = -sqrt(R) + sqrt(A - B*sqrt(R))``
      ``x4 = sqrt(R) + sqrt(A + B*sqrt(R))``

    To satisfy the quartic equation one must have
    ``p = -2*(R + A); q = -4*B*R; r = (R - A)**2 - B**2*R``
    so that ``R`` must satisfy the Descartes-Euler resolvent equation
    ``64*R**3 + 32*p*R**2 + (4*p**2 - 16*r)*R - q**2 = 0``

    If the resolvent does not have a rational solution, return None;
    in that case it is likely that the Ferrari method gives a simpler
    solution.

    Examples
    ========

    >>> from sympy import S
    >>> from sympy.polys.polyroots import _roots_quartic_euler
    >>> p, q, r = -S(64)/5, -S(512)/125, -S(1024)/3125
    >>> _roots_quartic_euler(p, q, r, S(0))[0]
    -sqrt(32*sqrt(5)/125 + 16/5) + 4*sqrt(5)/5
    """
    # solve the resolvent equation
    x = Dummy('x')
    eq = 64*x**3 + 32*p*x**2 + (4*p**2 - 16*r)*x - q**2
    xsols = list(roots(Poly(eq, x), cubics=False).keys())
    xsols = [sol for sol in xsols if sol.is_rational and sol.is_nonzero]
    if not xsols:
        return None
    R = max(xsols)
    c1 = sqrt(R)
    B = -q*c1/(4*R)
    A = -R - p/2
    c2 = sqrt(A + B)
    c3 = sqrt(A - B)
    return [c1 - c2 - a, -c1 - c3 - a, -c1 + c3 - a, c1 + c2 - a]


def roots_quartic(f):
    r"""
    Returns a list of roots of a quartic polynomial.

    There are many references for solving quartic expressions available [1-5].
    This reviewer has found that many of them require one to select from among
    2 or more possible sets of solutions and that some solutions work when one
    is searching for real roots but do not work when searching for complex roots
    (though this is not always stated clearly). The following routine has been
    tested and found to be correct for 0, 2 or 4 complex roots.

    The quasisymmetric case solution [6] looks for quartics that have the form
    `x**4 + A*x**3 + B*x**2 + C*x + D = 0` where `(C/A)**2 = D`.

    Although no general solution that is always applicable for all
    coefficients is known to this reviewer, certain conditions are tested
    to determine the simplest 4 expressions that can be returned:

      1) `f = c + a*(a**2/8 - b/2) == 0`
      2) `g = d - a*(a*(3*a**2/256 - b/16) + c/4) = 0`
      3) if `f != 0` and `g != 0` and `p = -d + a*c/4 - b**2/12` then
        a) `p == 0`
        b) `p != 0`

    Examples
    ========

        >>> from sympy import Poly
        >>> from sympy.polys.polyroots import roots_quartic

        >>> r = roots_quartic(Poly('x**4-6*x**3+17*x**2-26*x+20'))

        >>> # 4 complex roots: 1+-I*sqrt(3), 2+-I
        >>> sorted(str(tmp.evalf(n=2)) for tmp in r)
        ['1.0 + 1.7*I', '1.0 - 1.7*I', '2.0 + 1.0*I', '2.0 - 1.0*I']

    References
    ==========

    1. http://mathforum.org/dr.math/faq/faq.cubic.equations.html
    2. https://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method
    3. http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html
    4. http://staff.bath.ac.uk/masjhd/JHD-CA.pdf
    5. http://www.albmath.org/files/Math_5713.pdf
    6. http://www.statemaster.com/encyclopedia/Quartic-equation
    7. eqworld.ipmnet.ru/en/solutions/ae/ae0108.pdf
    """
    _, a, b, c, d = f.monic().all_coeffs()

    if not d:
        return [S.Zero] + roots([1, a, b, c], multiple=True)
    elif (c/a)**2 == d:
        x, m = f.gen, c/a

        g = Poly(x**2 + a*x + b - 2*m, x)

        z1, z2 = roots_quadratic(g)

        h1 = Poly(x**2 - z1*x + m, x)
        h2 = Poly(x**2 - z2*x + m, x)

        r1 = roots_quadratic(h1)
        r2 = roots_quadratic(h2)

        return r1 + r2
    else:
        a2 = a**2
        e = b - 3*a2/8
        f = _mexpand(c + a*(a2/8 - b/2))
        aon4 = a/4
        g = _mexpand(d - aon4*(a*(3*a2/64 - b/4) + c))

        if f.is_zero:
            y1, y2 = [sqrt(tmp) for tmp in
                      roots([1, e, g], multiple=True)]
            return [tmp - aon4 for tmp in [-y1, -y2, y1, y2]]
        if g.is_zero:
            y = [S.Zero] + roots([1, 0, e, f], multiple=True)
            return [tmp - aon4 for tmp in y]
        else:
            # Descartes-Euler method, see [7]
            sols = _roots_quartic_euler(e, f, g, aon4)
            if sols:
                return sols
            # Ferrari method, see [1, 2]
            p = -e**2/12 - g
            q = -e**3/108 + e*g/3 - f**2/8
            TH = Rational(1, 3)

            def _ans(y):
                w = sqrt(e + 2*y)
                arg1 = 3*e + 2*y
                arg2 = 2*f/w
                ans = []
                for s in [-1, 1]:
                    root = sqrt(-(arg1 + s*arg2))
                    for t in [-1, 1]:
                        ans.append((s*w - t*root)/2 - aon4)
                return ans

            # whether a Piecewise is returned or not
            # depends on knowing p, so try to put
            # in a simple form
            p = _mexpand(p)


            # p == 0 case
            y1 = e*Rational(-5, 6) - q**TH
            if p.is_zero:
                return _ans(y1)

            # if p != 0 then u below is not 0
            root = sqrt(q**2/4 + p**3/27)
            r = -q/2 + root  # or -q/2 - root
            u = r**TH  # primary root of solve(x**3 - r, x)
            y2 = e*Rational(-5, 6) + u - p/u/3
            if fuzzy_not(p.is_zero):
                return _ans(y2)

            # sort it out once they know the values of the coefficients
            return [Piecewise((a1, Eq(p, 0)), (a2, True))
                    for a1, a2 in zip(_ans(y1), _ans(y2))]


def roots_binomial(f):
    """Returns a list of roots of a binomial polynomial. If the domain is ZZ
    then the roots will be sorted with negatives coming before positives.
    The ordering will be the same for any numerical coefficients as long as
    the assumptions tested are correct, otherwise the ordering will not be
    sorted (but will be canonical).
    """
    n = f.degree()

    a, b = f.nth(n), f.nth(0)
    base = -cancel(b/a)
    alpha = root(base, n)

    if alpha.is_number:
        alpha = alpha.expand(complex=True)

    # define some parameters that will allow us to order the roots.
    # If the domain is ZZ this is guaranteed to return roots sorted
    # with reals before non-real roots and non-real sorted according
    # to real part and imaginary part, e.g. -1, 1, -1 + I, 2 - I
    neg = base.is_negative
    even = n % 2 == 0
    if neg:
        if even == True and (base + 1).is_positive:
            big = True
        else:
            big = False

    # get the indices in the right order so the computed
    # roots will be sorted when the domain is ZZ
    ks = []
    imax = n//2
    if even:
        ks.append(imax)
        imax -= 1
    if not neg:
        ks.append(0)
    for i in range(imax, 0, -1):
        if neg:
            ks.extend([i, -i])
        else:
            ks.extend([-i, i])
    if neg:
        ks.append(0)
        if big:
            for i in range(0, len(ks), 2):
                pair = ks[i: i + 2]
                pair = list(reversed(pair))

    # compute the roots
    roots, d = [], 2*I*pi/n
    for k in ks:
        zeta = exp(k*d).expand(complex=True)
        roots.append((alpha*zeta).expand(power_base=False))

    return roots


def _inv_totient_estimate(m):
    """
    Find ``(L, U)`` such that ``L <= phi^-1(m) <= U``.

    Examples
    ========

    >>> from sympy.polys.polyroots import _inv_totient_estimate

    >>> _inv_totient_estimate(192)
    (192, 840)
    >>> _inv_totient_estimate(400)
    (400, 1750)

    """
    primes = [ d + 1 for d in divisors(m) if isprime(d + 1) ]

    a, b = 1, 1

    for p in primes:
        a *= p
        b *= p - 1

    L = m
    U = int(math.ceil(m*(float(a)/b)))

    P = p = 2
    primes = []

    while P <= U:
        p = nextprime(p)
        primes.append(p)
        P *= p

    P //= p
    b = 1

    for p in primes[:-1]:
        b *= p - 1

    U = int(math.ceil(m*(float(P)/b)))

    return L, U


def roots_cyclotomic(f, factor=False):
    """Compute roots of cyclotomic polynomials. """
    L, U = _inv_totient_estimate(f.degree())

    for n in range(L, U + 1):
        g = cyclotomic_poly(n, f.gen, polys=True)

        if f.expr == g.expr:
            break
    else:  # pragma: no cover
        raise RuntimeError("failed to find index of a cyclotomic polynomial")

    roots = []

    if not factor:
        # get the indices in the right order so the computed
        # roots will be sorted
        h = n//2
        ks = [i for i in range(1, n + 1) if igcd(i, n) == 1]
        ks.sort(key=lambda x: (x, -1) if x <= h else (abs(x - n), 1))
        d = 2*I*pi/n
        for k in reversed(ks):
            roots.append(exp(k*d).expand(complex=True))
    else:
        g = Poly(f, extension=root(-1, n))

        for h, _ in ordered(g.factor_list()[1]):
            roots.append(-h.TC())

    return roots


def roots_quintic(f):
    """
    Calculate exact roots of a solvable irreducible quintic with rational coefficients.
    Return an empty list if the quintic is reducible or not solvable.
    """
    result = []

    coeff_5, coeff_4, p_, q_, r_, s_ = f.all_coeffs()

    if not all(coeff.is_Rational for coeff in (coeff_5, coeff_4, p_, q_, r_, s_)):
        return result

    if coeff_5 != 1:
        f = Poly(f / coeff_5)
        _, coeff_4, p_, q_, r_, s_ = f.all_coeffs()

    # Cancel coeff_4 to form x^5 + px^3 + qx^2 + rx + s
    if coeff_4:
        p = p_ - 2*coeff_4*coeff_4/5
        q = q_ - 3*coeff_4*p_/5 + 4*coeff_4**3/25
        r = r_ - 2*coeff_4*q_/5 + 3*coeff_4**2*p_/25 - 3*coeff_4**4/125
        s = s_ - coeff_4*r_/5 + coeff_4**2*q_/25 - coeff_4**3*p_/125 + 4*coeff_4**5/3125
        x = f.gen
        f = Poly(x**5 + p*x**3 + q*x**2 + r*x + s)
    else:
        p, q, r, s = p_, q_, r_, s_

    quintic = PolyQuintic(f)

    # Eqn standardized. Algo for solving starts here
    if not f.is_irreducible:
        return result
    f20 = quintic.f20
    # Check if f20 has linear factors over domain Z
    if f20.is_irreducible:
        return result
    # Now, we know that f is solvable
    for _factor in f20.factor_list()[1]:
        if _factor[0].is_linear:
            theta = _factor[0].root(0)
            break
    d = discriminant(f)
    delta = sqrt(d)
    # zeta = a fifth root of unity
    zeta1, zeta2, zeta3, zeta4 = quintic.zeta
    T = quintic.T(theta, d)
    tol = S(1e-10)
    alpha = T[1] + T[2]*delta
    alpha_bar = T[1] - T[2]*delta
    beta = T[3] + T[4]*delta
    beta_bar = T[3] - T[4]*delta

    disc = alpha**2 - 4*beta
    disc_bar = alpha_bar**2 - 4*beta_bar

    l0 = quintic.l0(theta)
    Stwo = S(2)
    l1 = _quintic_simplify((-alpha + sqrt(disc)) / Stwo)
    l4 = _quintic_simplify((-alpha - sqrt(disc)) / Stwo)

    l2 = _quintic_simplify((-alpha_bar + sqrt(disc_bar)) / Stwo)
    l3 = _quintic_simplify((-alpha_bar - sqrt(disc_bar)) / Stwo)

    order = quintic.order(theta, d)
    test = (order*delta.n()) - ( (l1.n() - l4.n())*(l2.n() - l3.n()) )
    # Comparing floats
    if not comp(test, 0, tol):
        l2, l3 = l3, l2

    # Now we have correct order of l's
    R1 = l0 + l1*zeta1 + l2*zeta2 + l3*zeta3 + l4*zeta4
    R2 = l0 + l3*zeta1 + l1*zeta2 + l4*zeta3 + l2*zeta4
    R3 = l0 + l2*zeta1 + l4*zeta2 + l1*zeta3 + l3*zeta4
    R4 = l0 + l4*zeta1 + l3*zeta2 + l2*zeta3 + l1*zeta4

    Res = [None, [None]*5, [None]*5, [None]*5, [None]*5]
    Res_n = [None, [None]*5, [None]*5, [None]*5, [None]*5]

    # Simplifying improves performance a lot for exact expressions
    R1 = _quintic_simplify(R1)
    R2 = _quintic_simplify(R2)
    R3 = _quintic_simplify(R3)
    R4 = _quintic_simplify(R4)

    # hard-coded results for [factor(i) for i in _vsolve(x**5 - a - I*b, x)]
    x0 = z**(S(1)/5)
    x1 = sqrt(2)
    x2 = sqrt(5)
    x3 = sqrt(5 - x2)
    x4 = I*x2
    x5 = x4 + I
    x6 = I*x0/4
    x7 = x1*sqrt(x2 + 5)
    sol = [x0, -x6*(x1*x3 - x5), x6*(x1*x3 + x5), -x6*(x4 + x7 - I), x6*(-x4 + x7 + I)]

    R1 = R1.as_real_imag()
    R2 = R2.as_real_imag()
    R3 = R3.as_real_imag()
    R4 = R4.as_real_imag()

    for i, s in enumerate(sol):
        Res[1][i] = _quintic_simplify(s.xreplace({z: R1[0] + I*R1[1]}))
        Res[2][i] = _quintic_simplify(s.xreplace({z: R2[0] + I*R2[1]}))
        Res[3][i] = _quintic_simplify(s.xreplace({z: R3[0] + I*R3[1]}))
        Res[4][i] = _quintic_simplify(s.xreplace({z: R4[0] + I*R4[1]}))

    for i in range(1, 5):
        for j in range(5):
            Res_n[i][j] = Res[i][j].n()
            Res[i][j] = _quintic_simplify(Res[i][j])
    r1 = Res[1][0]
    r1_n = Res_n[1][0]

    for i in range(5):
        if comp(im(r1_n*Res_n[4][i]), 0, tol):
            r4 = Res[4][i]
            break

    # Now we have various Res values. Each will be a list of five
    # values. We have to pick one r value from those five for each Res
    u, v = quintic.uv(theta, d)
    testplus = (u + v*delta*sqrt(5)).n()
    testminus = (u - v*delta*sqrt(5)).n()

    # Evaluated numbers suffixed with _n
    # We will use evaluated numbers for calculation. Much faster.
    r4_n = r4.n()
    r2 = r3 = None

    for i in range(5):
        r2temp_n = Res_n[2][i]
        for j in range(5):
            # Again storing away the exact number and using
            # evaluated numbers in computations
            r3temp_n = Res_n[3][j]
            if (comp((r1_n*r2temp_n**2 + r4_n*r3temp_n**2 - testplus).n(), 0, tol) and
                comp((r3temp_n*r1_n**2 + r2temp_n*r4_n**2 - testminus).n(), 0, tol)):
                r2 = Res[2][i]
                r3 = Res[3][j]
                break
        if r2 is not None:
            break
    else:
        return []  # fall back to normal solve

    # Now, we have r's so we can get roots
    x1 = (r1 + r2 + r3 + r4)/5
    x2 = (r1*zeta4 + r2*zeta3 + r3*zeta2 + r4*zeta1)/5
    x3 = (r1*zeta3 + r2*zeta1 + r3*zeta4 + r4*zeta2)/5
    x4 = (r1*zeta2 + r2*zeta4 + r3*zeta1 + r4*zeta3)/5
    x5 = (r1*zeta1 + r2*zeta2 + r3*zeta3 + r4*zeta4)/5
    result = [x1, x2, x3, x4, x5]

    # Now check if solutions are distinct

    saw = set()
    for r in result:
        r = r.n(2)
        if r in saw:
            # Roots were identical. Abort, return []
            # and fall back to usual solve
            return []
        saw.add(r)

    # Restore to original equation where coeff_4 is nonzero
    if coeff_4:
        result = [x - coeff_4 / 5 for x in result]
    return result


def _quintic_simplify(expr):
    from sympy.simplify.simplify import powsimp
    expr = powsimp(expr)
    expr = cancel(expr)
    return together(expr)


def _integer_basis(poly):
    """Compute coefficient basis for a polynomial over integers.

    Returns the integer ``div`` such that substituting ``x = div*y``
    ``p(x) = m*q(y)`` where the coefficients of ``q`` are smaller
    than those of ``p``.

    For example ``x**5 + 512*x + 1024 = 0``
    with ``div = 4`` becomes ``y**5 + 2*y + 1 = 0``

    Returns the integer ``div`` or ``None`` if there is no possible scaling.

    Examples
    ========

    >>> from sympy.polys import Poly
    >>> from sympy.abc import x
    >>> from sympy.polys.polyroots import _integer_basis
    >>> p = Poly(x**5 + 512*x + 1024, x, domain='ZZ')
    >>> _integer_basis(p)
    4
    """
    monoms, coeffs = list(zip(*poly.terms()))

    monoms, = list(zip(*monoms))
    coeffs = list(map(abs, coeffs))

    if coeffs[0] < coeffs[-1]:
        coeffs = list(reversed(coeffs))
        n = monoms[0]
        monoms = [n - i for i in reversed(monoms)]
    else:
        return None

    monoms = monoms[:-1]
    coeffs = coeffs[:-1]

    # Special case for two-term polynominals
    if len(monoms) == 1:
        r = Pow(coeffs[0], S.One/monoms[0])
        if r.is_Integer:
            return int(r)
        else:
            return None

    divs = reversed(divisors(gcd_list(coeffs))[1:])

    try:
        div = next(divs)
    except StopIteration:
        return None

    while True:
        for monom, coeff in zip(monoms, coeffs):
            if coeff % div**monom != 0:
                try:
                    div = next(divs)
                except StopIteration:
                    return None
                else:
                    break
        else:
            return div


def preprocess_roots(poly):
    """Try to get rid of symbolic coefficients from ``poly``. """
    coeff = S.One

    poly_func = poly.func
    try:
        _, poly = poly.clear_denoms(convert=True)
    except DomainError:
        return coeff, poly

    poly = poly.primitive()[1]
    poly = poly.retract()

    # TODO: This is fragile. Figure out how to make this independent of construct_domain().
    if poly.get_domain().is_Poly and all(c.is_term for c in poly.rep.coeffs()):
        poly = poly.inject()

        strips = list(zip(*poly.monoms()))
        gens = list(poly.gens[1:])

        base, strips = strips[0], strips[1:]

        for gen, strip in zip(list(gens), strips):
            reverse = False

            if strip[0] < strip[-1]:
                strip = reversed(strip)
                reverse = True

            ratio = None

            for a, b in zip(base, strip):
                if not a and not b:
                    continue
                elif not a or not b:
                    break
                elif b % a != 0:
                    break
                else:
                    _ratio = b // a

                    if ratio is None:
                        ratio = _ratio
                    elif ratio != _ratio:
                        break
            else:
                if reverse:
                    ratio = -ratio

                poly = poly.eval(gen, 1)
                coeff *= gen**(-ratio)
                gens.remove(gen)

        if gens:
            poly = poly.eject(*gens)

    if poly.is_univariate and poly.get_domain().is_ZZ:
        basis = _integer_basis(poly)

        if basis is not None:
            n = poly.degree()

            def func(k, coeff):
                return coeff//basis**(n - k[0])

            poly = poly.termwise(func)
            coeff *= basis

    if not isinstance(poly, poly_func):
        poly = poly_func(poly)
    return coeff, poly


@public
def roots(f, *gens,
        auto=True,
        cubics=True,
        trig=False,
        quartics=True,
        quintics=False,
        multiple=False,
        filter=None,
        predicate=None,
        strict=False,
        **flags):
    """
    Computes symbolic roots of a univariate polynomial.

    Given a univariate polynomial f with symbolic coefficients (or
    a list of the polynomial's coefficients), returns a dictionary
    with its roots and their multiplicities.

    Only roots expressible via radicals will be returned.  To get
    a complete set of roots use RootOf class or numerical methods
    instead. By default cubic and quartic formulas are used in
    the algorithm. To disable them because of unreadable output
    set ``cubics=False`` or ``quartics=False`` respectively. If cubic
    roots are real but are expressed in terms of complex numbers
    (casus irreducibilis [1]) the ``trig`` flag can be set to True to
    have the solutions returned in terms of cosine and inverse cosine
    functions.

    To get roots from a specific domain set the ``filter`` flag with
    one of the following specifiers: Z, Q, R, I, C. By default all
    roots are returned (this is equivalent to setting ``filter='C'``).

    By default a dictionary is returned giving a compact result in
    case of multiple roots.  However to get a list containing all
    those roots set the ``multiple`` flag to True; the list will
    have identical roots appearing next to each other in the result.
    (For a given Poly, the all_roots method will give the roots in
    sorted numerical order.)

    If the ``strict`` flag is True, ``UnsolvableFactorError`` will be
    raised if the roots found are known to be incomplete (because
    some roots are not expressible in radicals).

    Examples
    ========

    >>> from sympy import Poly, roots, degree
    >>> from sympy.abc import x, y

    >>> roots(x**2 - 1, x)
    {-1: 1, 1: 1}

    >>> p = Poly(x**2-1, x)
    >>> roots(p)
    {-1: 1, 1: 1}

    >>> p = Poly(x**2-y, x, y)

    >>> roots(Poly(p, x))
    {-sqrt(y): 1, sqrt(y): 1}

    >>> roots(x**2 - y, x)
    {-sqrt(y): 1, sqrt(y): 1}

    >>> roots([1, 0, -1])
    {-1: 1, 1: 1}

    ``roots`` will only return roots expressible in radicals. If
    the given polynomial has some or all of its roots inexpressible in
    radicals, the result of ``roots`` will be incomplete or empty
    respectively.

    Example where result is incomplete:

    >>> roots((x-1)*(x**5-x+1), x)
    {1: 1}

    In this case, the polynomial has an unsolvable quintic factor
    whose roots cannot be expressed by radicals. The polynomial has a
    rational root (due to the factor `(x-1)`), which is returned since
    ``roots`` always finds all rational roots.

    Example where result is empty:

    >>> roots(x**7-3*x**2+1, x)
    {}

    Here, the polynomial has no roots expressible in radicals, so
    ``roots`` returns an empty dictionary.

    The result produced by ``roots`` is complete if and only if the
    sum of the multiplicity of each root is equal to the degree of
    the polynomial. If strict=True, UnsolvableFactorError will be
    raised if the result is incomplete.

    The result can be be checked for completeness as follows:

    >>> f = x**3-2*x**2+1
    >>> sum(roots(f, x).values()) == degree(f, x)
    True
    >>> f = (x-1)*(x**5-x+1)
    >>> sum(roots(f, x).values()) == degree(f, x)
    False


    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Cubic_equation#Trigonometric_and_hyperbolic_solutions

    """
    from sympy.polys.polytools import to_rational_coeffs
    flags = dict(flags)

    if isinstance(f, list):
        if gens:
            raise ValueError('redundant generators given')

        x = Dummy('x')

        poly, i = {}, len(f) - 1

        for coeff in f:
            poly[i], i = sympify(coeff), i - 1

        f = Poly(poly, x, field=True)
    else:
        try:
            F = Poly(f, *gens, **flags)
            if not isinstance(f, Poly) and not F.gen.is_Symbol:
                raise PolynomialError("generator must be a Symbol")
            f = F
        except GeneratorsNeeded:
            if multiple:
                return []
            else:
                return {}
        else:
            n = f.degree()
            if f.length() == 2 and n > 2:
                # check for foo**n in constant if dep is c*gen**m
                con, dep = f.as_expr().as_independent(*f.gens)
                fcon = -(-con).factor()
                if fcon != con:
                    con = fcon
                    bases = []
                    for i in Mul.make_args(con):
                        if i.is_Pow:
                            b, e = i.as_base_exp()
                            if e.is_Integer and b.is_Add:
                                bases.append((b, Dummy(positive=True)))
                    if bases:
                        rv = roots(Poly((dep + con).xreplace(dict(bases)),
                            *f.gens), *F.gens,
                            auto=auto,
                            cubics=cubics,
                            trig=trig,
                            quartics=quartics,
                            quintics=quintics,
                            multiple=multiple,
                            filter=filter,
                            predicate=predicate,
                            **flags)
                        return {factor_terms(k.xreplace(
                            {v: k for k, v in bases})
                        ): v for k, v in rv.items()}

        if f.is_multivariate:
            raise PolynomialError('multivariate polynomials are not supported')

    def _update_dict(result, zeros, currentroot, k):
        if currentroot == S.Zero:
            if S.Zero in zeros:
                zeros[S.Zero] += k
            else:
                zeros[S.Zero] = k
        if currentroot in result:
            result[currentroot] += k
        else:
            result[currentroot] = k

    def _try_decompose(f):
        """Find roots using functional decomposition. """
        factors, roots = f.decompose(), []

        for currentroot in _try_heuristics(factors[0]):
            roots.append(currentroot)

        for currentfactor in factors[1:]:
            previous, roots = list(roots), []

            for currentroot in previous:
                g = currentfactor - Poly(currentroot, f.gen)

                for currentroot in _try_heuristics(g):
                    roots.append(currentroot)

        return roots

    def _try_heuristics(f):
        """Find roots using formulas and some tricks. """
        if f.is_ground:
            return []
        if f.is_monomial:
            return [S.Zero]*f.degree()

        if f.length() == 2:
            if f.degree() == 1:
                return list(map(cancel, roots_linear(f)))
            else:
                return roots_binomial(f)

        result = []

        for i in [-1, 1]:
            if not f.eval(i):
                f = f.quo(Poly(f.gen - i, f.gen))
                result.append(i)
                break

        n = f.degree()

        if n == 1:
            result += list(map(cancel, roots_linear(f)))
        elif n == 2:
            result += list(map(cancel, roots_quadratic(f)))
        elif f.is_cyclotomic:
            result += roots_cyclotomic(f)
        elif n == 3 and cubics:
            result += roots_cubic(f, trig=trig)
        elif n == 4 and quartics:
            result += roots_quartic(f)
        elif n == 5 and quintics:
            result += roots_quintic(f)

        return result

    # Convert the generators to symbols
    dumgens = symbols('x:%d' % len(f.gens), cls=Dummy)
    f = f.per(f.rep, dumgens)

    (k,), f = f.terms_gcd()

    if not k:
        zeros = {}
    else:
        zeros = {S.Zero: k}

    coeff, f = preprocess_roots(f)

    if auto and f.get_domain().is_Ring:
        f = f.to_field()

    # Use EX instead of ZZ_I or QQ_I
    if f.get_domain().is_QQ_I:
        f = f.per(f.rep.convert(EX))

    rescale_x = None
    translate_x = None

    result = {}

    if not f.is_ground:
        dom = f.get_domain()
        if not dom.is_Exact and dom.is_Numerical:
            for r in f.nroots():
                _update_dict(result, zeros, r, 1)
        elif f.degree() == 1:
            _update_dict(result, zeros, roots_linear(f)[0], 1)
        elif f.length() == 2:
            roots_fun = roots_quadratic if f.degree() == 2 else roots_binomial
            for r in roots_fun(f):
                _update_dict(result, zeros, r, 1)
        else:
            _, factors = Poly(f.as_expr()).factor_list()
            if len(factors) == 1 and f.degree() == 2:
                for r in roots_quadratic(f):
                    _update_dict(result, zeros, r, 1)
            else:
                if len(factors) == 1 and factors[0][1] == 1:
                    if f.get_domain().is_EX:
                        res = to_rational_coeffs(f)
                        if res:
                            if res[0] is None:
                                translate_x, f = res[2:]
                            else:
                                rescale_x, f = res[1], res[-1]
                            result = roots(f)
                            if not result:
                                for currentroot in _try_decompose(f):
                                    _update_dict(result, zeros, currentroot, 1)
                        else:
                            for r in _try_heuristics(f):
                                _update_dict(result, zeros, r, 1)
                    else:
                        for currentroot in _try_decompose(f):
                            _update_dict(result, zeros, currentroot, 1)
                else:
                    for currentfactor, k in factors:
                        for r in _try_heuristics(Poly(currentfactor, f.gen, field=True)):
                            _update_dict(result, zeros, r, k)

    if coeff is not S.One:
        _result, result, = result, {}

        for currentroot, k in _result.items():
            result[coeff*currentroot] = k

    if filter not in [None, 'C']:
        handlers = {
            'Z': lambda r: r.is_Integer,
            'Q': lambda r: r.is_Rational,
            'R': lambda r: all(a.is_real for a in r.as_numer_denom()),
            'I': lambda r: r.is_imaginary,
        }

        try:
            query = handlers[filter]
        except KeyError:
            raise ValueError("Invalid filter: %s" % filter)

        for zero in dict(result).keys():
            if not query(zero):
                del result[zero]

    if predicate is not None:
        for zero in dict(result).keys():
            if not predicate(zero):
                del result[zero]
    if rescale_x:
        result1 = {}
        for k, v in result.items():
            result1[k*rescale_x] = v
        result = result1
    if translate_x:
        result1 = {}
        for k, v in result.items():
            result1[k + translate_x] = v
        result = result1

    # adding zero roots after non-trivial roots have been translated
    result.update(zeros)

    if strict and sum(result.values()) < f.degree():
        raise UnsolvableFactorError(filldedent('''
            Strict mode: some factors cannot be solved in radicals, so
            a complete list of solutions cannot be returned. Call
            roots with strict=False to get solutions expressible in
            radicals (if there are any).
            '''))

    if not multiple:
        return result
    else:
        zeros = []

        for zero in ordered(result):
            zeros.extend([zero]*result[zero])

        return zeros


def root_factors(f, *gens, filter=None, **args):
    """
    Returns all factors of a univariate polynomial.

    Examples
    ========

    >>> from sympy.abc import x, y
    >>> from sympy.polys.polyroots import root_factors

    >>> root_factors(x**2 - y, x)
    [x - sqrt(y), x + sqrt(y)]

    """
    args = dict(args)

    F = Poly(f, *gens, **args)

    if not F.is_Poly:
        return [f]

    if F.is_multivariate:
        raise ValueError('multivariate polynomials are not supported')

    x = F.gens[0]

    zeros = roots(F, filter=filter)

    if not zeros:
        factors = [F]
    else:
        factors, N = [], 0

        for r, n in ordered(zeros.items()):
            factors, N = factors + [Poly(x - r, x)]*n, N + n

        if N < F.degree():
            G = reduce(lambda p, q: p*q, factors)
            factors.append(F.quo(G))

    if not isinstance(f, Poly):
        factors = [ f.as_expr() for f in factors ]

    return factors