def primitive_element(extension, x=None, *, ex=False, polys=False):
"""Construct a common number field for all extensions. """
if not extension:
raise ValueError("can't compute primitive element for empty extension")
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not ex:
gen, coeffs = extension[0], [1]
g = minimal_polynomial(gen, x, polys=True)
for ext in extension[1:]:
_, factors = factor_list(g, extension=ext)
g = _choose_factor(factors, x, gen)
s, _, g = g.sqf_norm()
gen += s * ext
coeffs.append(s)
if not polys:
return g.as_expr(), coeffs
else:
return cls(g), coeffs
gen, coeffs = extension[0], [1]
f = minimal_polynomial(gen, x, polys=True)
K = QQ.algebraic_field((f, gen)) # incrementally constructed field
reps = [K.unit] # representations of extension elements in K
for ext in extension[1:]:
p = minimal_polynomial(ext, x, polys=True)
L = QQ.algebraic_field((p, ext))
_, factors = factor_list(f, domain=L)
f = _choose_factor(factors, x, gen)
s, g, f = f.sqf_norm()
gen += s * ext
coeffs.append(s)
K = QQ.algebraic_field((f, gen))
h = _switch_domain(g, K)
erep = _linsolve(h.gcd(p)) # ext as element of K
ogen = K.unit - s * erep # old gen as element of K
reps = [dup_eval(_.rep, ogen, K) for _ in reps] + [erep]
H = [_.rep for _ in reps]
if not polys:
return f.as_expr(), coeffs, H
else:
return f, coeffs, H
def primitive_element(extension, x=None, *, ex=False, polys=False):
r"""
Find a single generator for a number field given by several generators.
Explanation
===========
The basic problem is this: Given several algebraic numbers
$\alpha_1, \alpha_2, \ldots, \alpha_n$, find a single algebraic number
$\theta$ such that
$\mathbb{Q}(\alpha_1, \alpha_2, \ldots, \alpha_n) = \mathbb{Q}(\theta)$.
This function actually guarantees that $\theta$ will be a linear
combination of the $\alpha_i$, with non-negative integer coefficients.
Furthermore, if desired, this function will tell you how to express each
$\alpha_i$ as a $\mathbb{Q}$-linear combination of the powers of $\theta$.
Examples
========
>>> from sympy import primitive_element, sqrt, S, minpoly, simplify
>>> from sympy.abc import x
>>> f, lincomb, reps = primitive_element([sqrt(2), sqrt(3)], x, ex=True)
Then ``lincomb`` tells us the primitive element as a linear combination of
the given generators ``sqrt(2)`` and ``sqrt(3)``.
>>> print(lincomb)
[1, 1]
This means the primtiive element is $\sqrt{2} + \sqrt{3}$.
Meanwhile ``f`` is the minimal polynomial for this primitive element.
>>> print(f)
x**4 - 10*x**2 + 1
>>> print(minpoly(sqrt(2) + sqrt(3), x))
x**4 - 10*x**2 + 1
Finally, ``reps`` (which was returned only because we set keyword arg
``ex=True``) tells us how to recover each of the generators $\sqrt{2}$ and
$\sqrt{3}$ as $\mathbb{Q}$-linear combinations of the powers of the
primitive element $\sqrt{2} + \sqrt{3}$.
>>> print([S(r) for r in reps[0]])
[1/2, 0, -9/2, 0]
>>> theta = sqrt(2) + sqrt(3)
>>> print(simplify(theta**3/2 - 9*theta/2))
sqrt(2)
>>> print([S(r) for r in reps[1]])
[-1/2, 0, 11/2, 0]
>>> print(simplify(-theta**3/2 + 11*theta/2))
sqrt(3)
Parameters
==========
extension : list of :py:class:`~.Expr`
Each expression must represent an algebraic number $\alpha_i$.
x : :py:class:`~.Symbol`, optional (default=None)
The desired symbol to appear in the computed minimal polynomial for the
primitive element $\theta$. If ``None``, we use a dummy symbol.
ex : boolean, optional (default=False)
If and only if ``True``, compute the representation of each $\alpha_i$
as a $\mathbb{Q}$-linear combination over the powers of $\theta$.
polys : boolean, optional (default=False)
If ``True``, return the minimal polynomial as a :py:class:`~.Poly`.
Otherwise return it as an :py:class:`~.Expr`.
Returns
=======
Pair (f, coeffs) or triple (f, coeffs, reps), where:
``f`` is the minimal polynomial for the primitive element.
``coeffs`` gives the primitive element as a linear combination of the
given generators.
``reps`` is present if and only if argument ``ex=True`` was passed,
and is a list of lists of rational numbers. Each list gives the
coefficients of falling powers of the primitive element, to recover
one of the original, given generators.
"""
if not extension:
raise ValueError("Cannot compute primitive element for empty extension")
extension = [_sympify(ext) for ext in extension]
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not ex:
gen, coeffs = extension[0], [1]
g = minimal_polynomial(gen, x, polys=True)
for ext in extension[1:]:
if ext.is_Rational:
coeffs.append(0)
continue
_, factors = factor_list(g, extension=ext)
g = _choose_factor(factors, x, gen)
s, _, g = g.sqf_norm()
gen += s*ext
coeffs.append(s)
if not polys:
return g.as_expr(), coeffs
else:
return cls(g), coeffs
gen, coeffs = extension[0], [1]
f = minimal_polynomial(gen, x, polys=True)
K = QQ.algebraic_field((f, gen)) # incrementally constructed field
reps = [K.unit] # representations of extension elements in K
for ext in extension[1:]:
if ext.is_Rational:
coeffs.append(0) # rational ext is not included in the expression of a primitive element
reps.append(K.convert(ext)) # but it is included in reps
continue
p = minimal_polynomial(ext, x, polys=True)
L = QQ.algebraic_field((p, ext))
_, factors = factor_list(f, domain=L)
f = _choose_factor(factors, x, gen)
s, g, f = f.sqf_norm()
gen += s*ext
coeffs.append(s)
K = QQ.algebraic_field((f, gen))
h = _switch_domain(g, K)
erep = _linsolve(h.gcd(p)) # ext as element of K
ogen = K.unit - s*erep # old gen as element of K
reps = [dup_eval(_.rep, ogen, K) for _ in reps] + [erep]
if K.ext.root.is_Rational: # all extensions are rational
H = [K.convert(_).rep for _ in extension]
coeffs = [0]*len(extension)
f = cls(x, domain=QQ)
else:
H = [_.rep for _ in reps]
if not polys:
return f.as_expr(), coeffs, H
else:
return f, coeffs, H
def test_dup_eval():
assert dup_eval([], 7, ZZ) == 0
assert dup_eval([1, 2], 0, ZZ) == 2
assert dup_eval([1, 2, 3], 7, ZZ) == 66
def dmp_zz_modular_resultant(f, g, p, u, K):
"""
Compute resultant of ``f`` and ``g`` modulo a prime ``p``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_zz_modular_resultant
>>> f = ZZ.map([[1], [1, 2]])
>>> g = ZZ.map([[2, 1], [3]])
>>> dmp_zz_modular_resultant(f, g, ZZ(5), 1, ZZ)
[-2, 0, 1]
"""
if not u:
return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)
v = u - 1
n = dmp_degree(f, u)
m = dmp_degree(g, u)
N = dmp_degree_in(f, 1, u)
M = dmp_degree_in(g, 1, u)
B = n*M + m*N
D, a = [K.one], -K.one
r = dmp_zero(v)
while dup_degree(D) <= B:
while True:
a += K.one
if a == p:
raise HomomorphismFailed('no luck')
F = dmp_eval_in(f, gf_int(a, p), 1, u, K)
if dmp_degree(F, v) == n:
G = dmp_eval_in(g, gf_int(a, p), 1, u, K)
if dmp_degree(G, v) == m:
break
R = dmp_zz_modular_resultant(F, G, p, v, K)
e = dmp_eval(r, a, v, K)
if not v:
R = dup_strip([R])
e = dup_strip([e])
else:
R = [R]
e = [e]
d = K.invert(dup_eval(D, a, K), p)
d = dup_mul_ground(D, d, K)
d = dmp_raise(d, v, 0, K)
c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
r = dmp_add(r, c, v, K)
r = dmp_ground_trunc(r, p, v, K)
D = dup_mul(D, [K.one, -a], K)
D = dup_trunc(D, p, K)
return r
def dup_zz_heu_gcd(f, g, K):
"""
Heuristic polynomial GCD in ``Z[x]``.
Given univariate polynomials ``f`` and ``g`` in ``Z[x]``, returns
their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``
such that::
h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)
The algorithm is purely heuristic which means it may fail to compute
the GCD. This will be signaled by raising an exception. In this case
you will need to switch to another GCD method.
The algorithm computes the polynomial GCD by evaluating polynomials
f and g at certain points and computing (fast) integer GCD of those
evaluations. The polynomial GCD is recovered from the integer image
by interpolation. The final step is to verify if the result is the
correct GCD. This gives cofactors as a side effect.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_zz_heu_gcd
>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, -3, 2])
>>> dup_zz_heu_gcd(f, g, ZZ)
([1, -1], [1, 1], [1, -2])
**References**
1. [Liao95]_
"""
result = _dup_rr_trivial_gcd(f, g, K)
if result is not None:
return result
df = dup_degree(f)
dg = dup_degree(g)
gcd, f, g = dup_extract(f, g, K)
if df == 0 or dg == 0:
return [gcd], f, g
f_norm = dup_max_norm(f, K)
g_norm = dup_max_norm(g, K)
B = 2*min(f_norm, g_norm) + 29
x = max(min(B, 99*K.sqrt(B)),
2*min(f_norm // abs(dup_LC(f, K)),
g_norm // abs(dup_LC(g, K))) + 2)
for i in xrange(0, HEU_GCD_MAX):
ff = dup_eval(f, x, K)
gg = dup_eval(g, x, K)
if ff and gg:
h = K.gcd(ff, gg)
cff = ff // h
cfg = gg // h
h = _dup_zz_gcd_interpolate(h, x, K)
h = dup_primitive(h, K)[1]
cff_, r = dup_div(f, h, K)
if not r:
cfg_, r = dup_div(g, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff_, cfg_
cff = _dup_zz_gcd_interpolate(cff, x, K)
h, r = dup_div(f, cff, K)
if not r:
cfg_, r = dup_div(g, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff, cfg_
cfg = _dup_zz_gcd_interpolate(cfg, x, K)
h, r = dup_div(g, cfg, K)
if not r:
cff_, r = dup_div(f, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff, cfg
x = 73794*x * K.sqrt(K.sqrt(x)) // 27011
raise HeuristicGCDFailed('no luck')
def test_dup_eval():
assert dup_eval([], 7, ZZ) == 0
assert dup_eval([1, 2], 0, ZZ) == 2
assert dup_eval([1, 2, 3], 7, ZZ) == 66
def dmp_zz_modular_resultant(f, g, p, u, K):
"""
Compute resultant of `f` and `g` modulo a prime `p`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x + y + 2
>>> g = 2*x*y + x + 3
>>> R.dmp_zz_modular_resultant(f, g, 5)
-2*y**2 + 1
"""
if not u:
return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)
v = u - 1
n = dmp_degree(f, u)
m = dmp_degree(g, u)
N = dmp_degree_in(f, 1, u)
M = dmp_degree_in(g, 1, u)
B = n*M + m*N
D, a = [K.one], -K.one
r = dmp_zero(v)
while dup_degree(D) <= B:
while True:
a += K.one
if a == p:
raise HomomorphismFailed('no luck')
F = dmp_eval_in(f, gf_int(a, p), 1, u, K)
if dmp_degree(F, v) == n:
G = dmp_eval_in(g, gf_int(a, p), 1, u, K)
if dmp_degree(G, v) == m:
break
R = dmp_zz_modular_resultant(F, G, p, v, K)
e = dmp_eval(r, a, v, K)
if not v:
R = dup_strip([R])
e = dup_strip([e])
else:
R = [R]
e = [e]
d = K.invert(dup_eval(D, a, K), p)
d = dup_mul_ground(D, d, K)
d = dmp_raise(d, v, 0, K)
c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
r = dmp_add(r, c, v, K)
r = dmp_ground_trunc(r, p, v, K)
D = dup_mul(D, [K.one, -a], K)
D = dup_trunc(D, p, K)
return r
def dmp_zz_modular_resultant(f, g, p, u, K):
"""
Compute resultant of `f` and `g` modulo a prime `p`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x + y + 2
>>> g = 2*x*y + x + 3
>>> R.dmp_zz_modular_resultant(f, g, 5)
-2*y**2 + 1
"""
if not u:
return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)
v = u - 1
n = dmp_degree(f, u)
m = dmp_degree(g, u)
N = dmp_degree_in(f, 1, u)
M = dmp_degree_in(g, 1, u)
B = n*M + m*N
D, a = [K.one], -K.one
r = dmp_zero(v)
while dup_degree(D) <= B:
while True:
a += K.one
if a == p:
raise HomomorphismFailed('no luck')
F = dmp_eval_in(f, gf_int(a, p), 1, u, K)
if dmp_degree(F, v) == n:
G = dmp_eval_in(g, gf_int(a, p), 1, u, K)
if dmp_degree(G, v) == m:
break
R = dmp_zz_modular_resultant(F, G, p, v, K)
e = dmp_eval(r, a, v, K)
if not v:
R = dup_strip([R])
e = dup_strip([e])
else:
R = [R]
e = [e]
d = K.invert(dup_eval(D, a, K), p)
d = dup_mul_ground(D, d, K)
d = dmp_raise(d, v, 0, K)
c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
r = dmp_add(r, c, v, K)
r = dmp_ground_trunc(r, p, v, K)
D = dup_mul(D, [K.one, -a], K)
D = dup_trunc(D, p, K)
return r
def dup_zz_heu_gcd(f, g, K):
"""
Heuristic polynomial GCD in `Z[x]`.
Given univariate polynomials `f` and `g` in `Z[x]`, returns
their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``
such that::
h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)
The algorithm is purely heuristic which means it may fail to compute
the GCD. This will be signaled by raising an exception. In this case
you will need to switch to another GCD method.
The algorithm computes the polynomial GCD by evaluating polynomials
f and g at certain points and computing (fast) integer GCD of those
evaluations. The polynomial GCD is recovered from the integer image
by interpolation. The final step is to verify if the result is the
correct GCD. This gives cofactors as a side effect.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_zz_heu_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
References
==========
1. [Liao95]_
"""
result = _dup_rr_trivial_gcd(f, g, K)
if result is not None:
return result
df = dup_degree(f)
dg = dup_degree(g)
gcd, f, g = dup_extract(f, g, K)
if df == 0 or dg == 0:
return [gcd], f, g
f_norm = dup_max_norm(f, K)
g_norm = dup_max_norm(g, K)
B = K(2*min(f_norm, g_norm) + 29)
x = max(min(B, 99*K.sqrt(B)),
2*min(f_norm // abs(dup_LC(f, K)),
g_norm // abs(dup_LC(g, K))) + 2)
for i in xrange(0, HEU_GCD_MAX):
ff = dup_eval(f, x, K)
gg = dup_eval(g, x, K)
if ff and gg:
h = K.gcd(ff, gg)
cff = ff // h
cfg = gg // h
h = _dup_zz_gcd_interpolate(h, x, K)
h = dup_primitive(h, K)[1]
cff_, r = dup_div(f, h, K)
if not r:
cfg_, r = dup_div(g, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff_, cfg_
cff = _dup_zz_gcd_interpolate(cff, x, K)
h, r = dup_div(f, cff, K)
if not r:
cfg_, r = dup_div(g, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff, cfg_
cfg = _dup_zz_gcd_interpolate(cfg, x, K)
h, r = dup_div(g, cfg, K)
if not r:
cff_, r = dup_div(f, h, K)
if not r:
h = dup_mul_ground(h, gcd, K)
return h, cff_, cfg
x = 73794*x * K.sqrt(K.sqrt(x)) // 27011
raise HeuristicGCDFailed('no luck')
def dmp_zz_modular_resultant(f, g, p, u, K):
"""
Compute resultant of `f` and `g` modulo a prime `p`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_zz_modular_resultant
>>> f = ZZ.map([[1], [1, 2]])
>>> g = ZZ.map([[2, 1], [3]])
>>> dmp_zz_modular_resultant(f, g, ZZ(5), 1, ZZ)
[-2, 0, 1]
"""
if not u:
return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)
v = u - 1
n = dmp_degree(f, u)
m = dmp_degree(g, u)
N = dmp_degree_in(f, 1, u)
M = dmp_degree_in(g, 1, u)
B = n * M + m * N
D, a = [K.one], -K.one
r = dmp_zero(v)
while dup_degree(D) <= B:
while True:
a += K.one
if a == p:
raise HomomorphismFailed('no luck')
F = dmp_eval_in(f, gf_int(a, p), 1, u, K)
if dmp_degree(F, v) == n:
G = dmp_eval_in(g, gf_int(a, p), 1, u, K)
if dmp_degree(G, v) == m:
break
R = dmp_zz_modular_resultant(F, G, p, v, K)
e = dmp_eval(r, a, v, K)
if not v:
R = dup_strip([R])
e = dup_strip([e])
else:
R = [R]
e = [e]
d = K.invert(dup_eval(D, a, K), p)
d = dup_mul_ground(D, d, K)
d = dmp_raise(d, v, 0, K)
c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
r = dmp_add(r, c, v, K)
r = dmp_ground_trunc(r, p, v, K)
D = dup_mul(D, [K.one, -a], K)
D = dup_trunc(D, p, K)
return r