def check(self, f):
# reciprocal version of Dimitrov's first comment in https://mathoverflow.net/questions/214962/criteria-for-irreducibility-using-the-location-of-complex-roots
import mpmath
import sympy
const_coeff = f.TC()
if const_coeff == 0:
return REDUCIBLE, None
# constant coeff needs to be in form +-p^d
primes = sympy.ntheory.factorint(abs(const_coeff))
if len(primes) != 1:
return UNKNOWN, None
# extract p
p = next(iter(primes.keys()))
# linear coefficients must not be divisible by p
a1 = get_coeff(f, 1)
if a1 % p == 0:
return UNKNOWN, None
if self.max_p and abs(const_coeff) >= self.max_p:
return UNKNOWN, None
# check if there are roots inside as well as outside of unit circle
try:
all = mpmath.polyroots(f.all_coeffs(), maxsteps=100)
inside_unit_circle = 0
outside_unit_circle = 0
on_unit_circle = 0
for root in all:
root_size = abs(root)
if root_size == 1:
on_unit_circle += 1
elif root_size < 1:
inside_unit_circle += 1
else:
outside_unit_circle += 1
if (inside_unit_circle + on_unit_circle == 0):
return IRREDUCIBLE, {
"inside/on": inside_unit_circle + on_unit_circle,
"outside": outside_unit_circle,
"p": p
}
except Exception as e:
# could not get complex roots, too bad
pass
return UNKNOWN, None
def check(self, f):
const_coeff = f.TC()
if const_coeff == 0:
return REDUCIBLE, None
lead_coeff = f.LC()
if lead_coeff != 1:
return UNKNOWN, None
a_nm1 = abs(get_coeff(f, f.degree() - 1))
s = 1
for exp, coeff in poly_non_zero_exps(f):
if exp < f.degree() - 1:
s += abs(coeff)
if a_nm1 > s:
return IRREDUCIBLE, None
return UNKNOWN, None
def check(self, f):
# http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/schurtheorem.pdf
# Theorem 1, multiplied to get Z[x]
n = f.degree()
lead_coeff = f.LC()
if abs(lead_coeff) != 1:
return UNKNOWN, None
fact = n
for exp in reversed(range(1, n)):
coeff = get_coeff(f, exp)
if coeff % fact != 0:
return UNKNOWN, None
fact *= exp
# fact now contains n!
const_coeff = f.TC()
if const_coeff != fact:
return UNKNOWN, None
return IRREDUCIBLE, None
def check(self, f):
# see https://www.lehigh.edu/~shw2/preprints/eisenstein.pdf,
# and https://math.stackexchange.com/questions/3589657/prove-polynomial-is-irreducible/3590300#3590300
const_coeff = f.TC()
if const_coeff == 0:
return REDUCIBLE, None
linear_coeff = get_coeff(f, 1)
if linear_coeff == 0:
return UNKNOWN, None
primes = sympy.ntheory.factorint(linear_coeff)
for p in primes:
power = primes[p]
if power == 1:
# good candidate, it must not divide leading term
lead_coeff = f.LC()
if lead_coeff % p == 0:
# bad luck
continue
# it must divide or other terms
for exp, coeff in poly_non_zero_exps(f):
if exp != f.degree():
if coeff % p != 0:
# bad luck...
return UNKNOWN, None
# finally the polynomial must not have rational roots
numerators = sympy.ntheory.factorint(const_coeff)
denominators = sympy.ntheory.factorint(lead_coeff)
for num in numerators:
for denom in denominators:
for sign in (-1, 1):
if f.eval(sympy.Rational(sign * num, denom)) == 0:
# rational root...
return UNKNOWN, None
return IRREDUCIBLE, {"p": p}
return UNKNOWN, None