Source code for diofant.polys.polyroots

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

import functools
import math

from ..core import (Dummy, Eq, Float, I, Integer, Rational, Symbol, comp,
factor_terms, pi, symbols, sympify)
from ..core.compatibility import ordered
from ..core.mul import expand_2arg
from ..functions import Piecewise, acos, cos, exp, im, root, sqrt
from ..ntheory import divisors, isprime, nextprime
from ..simplify import powsimp, simplify
from .polyerrors import GeneratorsNeeded, PolynomialError
from .polyquinticconst import PolyQuintic
from .polytools import Poly, cancel, discriminant, factor, gcd_list
from .rationaltools import together
from .specialpolys import cyclotomic_poly

__all__ = 'roots',

def roots_linear(f):
"""Returns a list of roots of a linear polynomial."""
r = -f.coeff_monomial(1)/f.coeff_monomial(f.gen)
dom = f.domain

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

return [r]

"""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.domain

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

if c == 0:
r0, r1 = Integer(0), -b/a

if not dom.is_Numerical:
r1 = _simplify(r1)
elif r1.is_negative:
r0, r1 = r1, r0
elif b == 0:
r = -c/a
if not dom.is_Numerical:
r = _simplify(r)

R = sqrt(r).doit()
r0 = -R
r1 = R
else:
d = (b**2 - 4*a*c)/a**2
B = -b/(2*a)

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

D = factor_terms(sqrt(d)/2)
r0 = B - D
r1 = B + D
if 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
==========

* 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.is_positive:
rv = []
for k in range(3):
rv.append(2*sqrt(-p/3)*cos(acos(3*q/2/p*sqrt(-3/p))/3 - k*2*pi/3))
return [i - b/3/a for i in rv]

_, a, b, c = f.monic().all_coeffs()

if c == 0:
x1, x2 = roots([1, a, b], multiple=True)
return [x1, Integer(0), x2]

p = b - a**2/3
q = c - a*b/3 + 2*a**3/27

pon3 = p/3
aon3 = a/3

u1 = None
if p == 0:
if q == 0:
return [-aon3]*3
elif q.is_positive:
u1 = -root(q, 3)
elif q.is_negative:
u1 = root(-q, 3)
elif q.is_extended_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 = Integer(1)
u2 = -Rational(1, 2) + coeff
u3 = -Rational(1, 2) - coeff
a, b, c, d = Integer(1), a, b, c
D0 = b**2 - 3*a*c
D1 = 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/a for uk in [u1, u2, u3]]

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

if p == 0:
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
========

>>> p, q, r = -Rational(64, 5), -Rational(512, 125), -Rational(1024, 3125)
>>> _roots_quartic_euler(p, q, r, Integer(0))
-sqrt(32*sqrt(5)/125 + 16/5) + 4*sqrt(5)/5

"""
# solve the resolvent equation
x = Symbol('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))
xsols = [sol for sol in xsols if sol.is_rational]
if not xsols:
return
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 [1-4].
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 don't 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  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
========

>>> 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(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
==========

* http://mathforum.org/dr.math/faq/faq.cubic.equations.html
* https://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method
* http://staff.bath.ac.uk/masjhd/JHD-CA.pdf
* http://www.albmath.org/files/Math_5713.pdf
* http://www.statemaster.com/encyclopedia/Quartic-equation
* eqworld.ipmnet.ru/en/solutions/ae/ae0108.pdf

"""
_, a, b, c, d = f.monic().all_coeffs()

if not d:
return [Integer(0)] + 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)

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

return r1 + r2
else:
a2 = a**2
e = b - 3*a2/8
f = c + a*(a2/8 - b/2)
g = d - a*(a*(3*a2/256 - b/16) + c/4)
aon4 = a/4

if f == 0:
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 == 0:
y = [Integer(0)] + roots([1, 0, e, f], multiple=True)
return [tmp - aon4 for tmp in y]
else:
# Descartes-Euler method, see 
sols = _roots_quartic_euler(e, f, g, aon4)
if sols:
return sols
# Ferrari method, see [1, 2]
a2 = a**2
e = b - 3*a2/8
f = c + a*(a2/8 - b/2)
g = d - a*(a*(3*a2/256 - b/16) + c/4)
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

# p == 0 case
y1 = -5*e/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 = -5*e/6 + u - p/u/3
if p.is_nonzero:
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.coeff_monomial(f.gen**n), f.coeff_monomial(1)
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 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
========

>>> _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 = 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 = 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 == g:
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 math.gcd(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()):
roots.append(-h.TC())

return roots

def roots_quintic(f):
"""Calulate exact roots of a solvable quintic."""
result = []
coeff_5, coeff_4, p, q, r, s = f.all_coeffs()

# Eqn must be of the form x^5 + px^3 + qx^2 + rx + s
if coeff_4:
return result

if coeff_5 != 1:
l = [p/coeff_5, q/coeff_5, r/coeff_5, s/coeff_5]
if not all(coeff.is_Rational for coeff in l):
return result
f = Poly(f/coeff_5)
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
_factors = f20.factor_list()
assert _factors.is_linear
theta = _factors.root(0)
d = discriminant(f)
delta = sqrt(d)
# zeta = a fifth root of unity
zeta1, zeta2, zeta3, zeta4 = quintic.zeta
T = quintic.T(theta, d)
tol = Float(1e-10)
alpha = T + T*delta
alpha_bar = T - T*delta
beta = T + T*delta
beta_bar = T - T*delta

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

l0 = quintic.l0(theta)

l1 = _quintic_simplify((-alpha + sqrt(disc))/2)
l4 = _quintic_simplify((-alpha - sqrt(disc))/2)

l2 = _quintic_simplify((-alpha_bar + sqrt(disc_bar))/2)
l3 = _quintic_simplify((-alpha_bar - sqrt(disc_bar))/2)

order = quintic.order(theta, d)
test = order*delta - (l1 - l4)*(l2 - l3)
# Comparing floats
if not comp(test.evalf(strict=False), 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]
sol = Symbol('sol')

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

# Solve imported here. Causing problems if imported as 'solve'
# and hence the changed name
from ..solvers import solve as _solve
a, b = symbols('a b', cls=Dummy)
_sol = _solve(sol**5 - a - I*b, sol)
for i in range(5):
_sol[i] = factor(_sol[i][sol])
R1 = R1.as_real_imag()
R2 = R2.as_real_imag()
R3 = R3.as_real_imag()
R4 = R4.as_real_imag()

for i, root in enumerate(_sol):
Res[i] = _quintic_simplify(root.subs({a: R1, b: R1}))
Res[i] = _quintic_simplify(root.subs({a: R2, b: R2}))
Res[i] = _quintic_simplify(root.subs({a: R3, b: R3}))
Res[i] = _quintic_simplify(root.subs({a: R4, b: R4}))

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

for i in range(5):  # pragma: no branch
if comp(im(r1_n*Res_n[i]), 0, tol):
r4 = Res[i]
break

u, v = quintic.uv(theta, d)

# 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)).evalf(strict=False)
testminus = (u - v*delta*sqrt(5)).evalf(strict=False)

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

for i in range(5):  # pragma: no branch
r2temp_n = Res_n[i]
for j in range(5):
# Again storing away the exact number and using
# evaluated numbers in computations
r3temp_n = Res_n[j]

if (comp(r1_n*r2temp_n**2 + r4_n*r3temp_n**2 - testplus, 0, tol) and
comp(r3temp_n*r1_n**2 + r2temp_n*r4_n**2 - testminus, 0, tol)):
r2 = Res[i]
r3 = Res[j]
break
if r2:
break

# 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
return [x1, x2, x3, x4, x5]

def _quintic_simplify(expr):
expr = powsimp(expr)
expr = cancel(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
========

>>> p = Poly(x**5 + 512*x + 1024, x)
>>> _integer_basis(p)
4

"""
if poly.is_zero:
return

monoms, coeffs = list(zip(*poly.terms()))

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

if coeffs < coeffs[-1]:
coeffs = list(reversed(coeffs))
n = monoms
monoms = [n - i for i in reversed(monoms)]
else:
return

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

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

try:
div = next(divs)
except StopIteration:
return

while True:
for monom, coeff in zip(monoms, coeffs):
if coeff % div**monom != 0:
try:
div = next(divs)
except StopIteration:
return
else:
break
else:
return div

def preprocess_roots(poly):
"""Try to get rid of symbolic coefficients from poly."""
coeff = Integer(1)

_, poly = poly.clear_denoms(convert=True)

poly = poly.primitive()
poly = poly.retract()

# TODO: This is fragile. Figure out how to make this independent of construct_domain().
if poly.domain.is_PolynomialRing 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, strips[1:]

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

if strip < 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.domain.is_IntegerRing:
basis = _integer_basis(poly)

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

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

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

return coeff, poly

[docs]def roots(f, *gens, **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) 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.)

Examples
========

>>> 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}

References
==========

* https://en.wikipedia.org/wiki/Cubic_function#Trigonometric_and_hyperbolic_solutions

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

auto = flags.pop('auto', True)
cubics = flags.pop('cubics', True)
trig = flags.pop('trig', False)
quartics = flags.pop('quartics', True)
quintics = flags.pop('quintics', False)
multiple = flags.pop('multiple', False)
filter = flags.pop('filter', None)
predicate = flags.pop('predicate', None)

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)
except GeneratorsNeeded:
if multiple:
return []
else:
return {}

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

def _update_dict(result, root, k):
if root in result:
result[root] += k
else:
result[root] = k

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

for root in _try_heuristics(factors):
roots.append(root)

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

for root in previous:
g = factor - Poly(root, f.gen, extension=False)

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

return roots

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

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

result = []

n = f.degree()

if n == 2:
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

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

if not k:
zeros = {}
else:
zeros = {Integer(0): k}

coeff, f = preprocess_roots(f)

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

rescale_x = None
translate_x = None

result = {}

if not f.is_ground:
if not f.domain.is_Exact:
for r in f.nroots(n=f.domain.dps):
_update_dict(result, r, 1)
elif f.degree() == 1:
result[roots_linear(f)] = 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, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
if len(factors) == 1 and f.degree() == 2:
_update_dict(result, r, 1)
else:
if len(factors) == 1 and factors == 1:
if f.domain.is_SymbolicDomain:
res = to_rational_coeffs(f)
if res:
if res is None:
translate_x, f = res[2:]
else:
rescale_x, f = res, res[-1]
result = roots(f)
else:
for root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for factor, k in factors:
for r in _try_heuristics(Poly(factor, f.gen, field=True)):
_update_dict(result, r, k)

if coeff != 1:
_result, result, = result, {}

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

result.update(zeros)

if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: r.is_extended_real,
'I': lambda r: r.is_imaginary,
}

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

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

if predicate is not None:
for zero in dict(result):
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

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

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

return zeros

def root_factors(f, *gens, **args):
"""
Returns all factors of a univariate polynomial.

Examples
========

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

"""
args = dict(args)
filter = args.pop('filter', None)

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

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

x = F.gens

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 = functools.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