def test_heugcd_univariate_integers():
R, x = ring("x", ZZ)
f = x**4 + 8 * x**3 + 21 * x**2 + 22 * x + 8
g = x**3 + 6 * x**2 + 11 * x + 6
h = x**2 + 3 * x + 2
cff = x**2 + 5 * x + 4
cfg = x + 3
assert heugcd(f, g) == (h, cff, cfg)
f = x**4 - 4
g = x**4 + 4 * x**2 + 4
h = x**2 + 2
cff = x**2 - 2
cfg = x**2 + 2
assert heugcd(f, g) == (h, cff, cfg)
f = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5
g = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21
h = 1
cff = f
cfg = g
assert heugcd(f, g) == (h, cff, cfg)
f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \
+ 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \
+ 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \
+ 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \
- 12278371209708240950316872681744825481125965781519138077173235712*x**21 \
+ 289127344604779611146960547954288113529690984687482920704*x**14 \
+ 19007977035740498977629742919480623972236450681*x**7 \
+ 311973482284542371301330321821976049
g = 365431878023781158602430064717380211405897160759702125019136*x**21 \
+ 197599133478719444145775798221171663643171734081650688*x**14 \
- 9504116979659010018253915765478924103928886144*x**7 \
- 311973482284542371301330321821976049
# TODO: assert heugcd(f, f.diff(x))[0] == g
f = 1317378933230047068160 * x + 2945748836994210856960
g = 120352542776360960 * x + 269116466014453760
h = 120352542776360960 * x + 269116466014453760
cff = 10946
cfg = 1
assert heugcd(f, g) == (h, cff, cfg)
with using(heu_gcd_max=0):
pytest.raises(HeuristicGCDFailed, lambda: heugcd(f, g))
def test_benchmark_cyclic_4(method):
with config.using(groebner=method):
R, a, b, c, d = ring('a b c d', ZZ, lex)
I = [
a + b + c + d, a * b + a * d + b * c + b * d,
a * b * c + a * b * d + a * c * d + b * c * d, a * b * c * d - 1
]
assert groebner(I, R) == [
4 * a + 3 * d**9 - 4 * d**5 - 3 * d,
4 * b + 4 * c - 3 * d**9 + 4 * d**5 + 7 * d,
4 * c**2 + 3 * d**10 - 4 * d**6 - 3 * d**2,
4 * c * d**4 + 4 * c - d**9 + 4 * d**5 + 5 * d,
d**12 - d**8 - d**4 + 1
]
R, a, b, c, d = ring('a b c d', ZZ, grlex)
I = [i.set_ring(R) for i in I]
assert groebner(I, R) == [
3 * b * c - c**2 + d**6 - 3 * d**2,
-b + 3 * c**2 * d**3 - c - d**5 - 4 * d,
-b + 3 * c * d**4 + 2 * c + 2 * d**5 + 2 * d,
c**4 + 2 * c**2 * d**2 - d**4 - 2, c**3 * d + c * d**3 + d**4 + 1,
b * c**2 - c**3 - c**2 * d - 2 * c * d**2 - d**3, b**2 - c**2,
b * d + c**2 + c * d + d**2, a + b + c + d
]
def test_heugcd_univariate_integers():
R, x = ring("x", ZZ)
f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8
g = x**3 + 6*x**2 + 11*x + 6
h = x**2 + 3*x + 2
cff = x**2 + 5*x + 4
cfg = x + 3
assert heugcd(f, g) == (h, cff, cfg)
f = x**4 - 4
g = x**4 + 4*x**2 + 4
h = x**2 + 2
cff = x**2 - 2
cfg = x**2 + 2
assert heugcd(f, g) == (h, cff, cfg)
f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
h = 1
cff = f
cfg = g
assert heugcd(f, g) == (h, cff, cfg)
f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \
+ 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \
+ 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \
+ 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \
- 12278371209708240950316872681744825481125965781519138077173235712*x**21 \
+ 289127344604779611146960547954288113529690984687482920704*x**14 \
+ 19007977035740498977629742919480623972236450681*x**7 \
+ 311973482284542371301330321821976049
g = 365431878023781158602430064717380211405897160759702125019136*x**21 \
+ 197599133478719444145775798221171663643171734081650688*x**14 \
- 9504116979659010018253915765478924103928886144*x**7 \
- 311973482284542371301330321821976049
# TODO: assert heugcd(f, f.diff(x))[0] == g
f = 1317378933230047068160*x + 2945748836994210856960
g = 120352542776360960*x + 269116466014453760
h = 120352542776360960*x + 269116466014453760
cff = 10946
cfg = 1
assert heugcd(f, g) == (h, cff, cfg)
with using(heu_gcd_max=0):
pytest.raises(HeuristicGCDFailed, lambda: heugcd(f, g))
def test_sympyissue_16620():
R, x = ring('x', FF(2))
f = x**17 + 1
g = (1, [(x + 1, 1), (x**8 + x**5 + x**4 + x**3 + 1, 1),
(x**8 + x**7 + x**6 + x**4 + x**2 + x + 1, 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = x**31 + 1
g = (1, [(x + 1, 1), (x**5 + x**2 + 1, 1), (x**5 + x**3 + 1, 1),
(x**5 + x**3 + x**2 + x + 1, 1), (x**5 + x**4 + x**2 + x + 1, 1),
(x**5 + x**4 + x**3 + x + 1, 1),
(x**5 + x**4 + x**3 + x**2 + 1, 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
def test_PolyElement_gcd():
R, x, y = ring("x,y", QQ)
f = x**2/2 + x + QQ(1, 2)
g = x/2 + QQ(1, 2)
assert f.gcd(g) == x + 1
with using(use_heu_gcd=False):
assert f.gcd(g) == x + 1
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
assert f.gcd(g) == x + 1
R, x, y = ring("x,y", QQ.algebraic_field(sqrt(2)))
f, g = (x + sqrt(2)*y)**2, x + sqrt(2)*y
assert f.gcd(g) == g
with using(gcd_aa_method='modgcd'):
assert f.gcd(g) == g
def test_modgcd_algebraic_field_random():
A = QQ.algebraic_field(sqrt(2), sqrt(3))
R, x, y, z = ring('x, y, z', A)
with using(gcd_aa_method='modgcd'):
h = x**2 * y**3 + 1111 * sqrt(6) / 12 * z
a, b = 11 * y**3 + 2, (y + x - 1) * h
assert (h * a).cofactors(h * b) == (h, a, b)
a, b = 12 * y + 2 * x - 1, (y + x - 1) * h
assert (h * a).cofactors(h * b) == (h, a, b)
def test_gf_irreducible_p():
assert gf_irred_p_ben_or([7], 11, ZZ) is True
assert gf_irred_p_ben_or([7, 3], 11, ZZ) is True
assert gf_irred_p_ben_or([7, 3, 1], 11, ZZ) is False
assert gf_irred_p_rabin([7], 11, ZZ) is True
assert gf_irred_p_rabin([7, 3], 11, ZZ) is True
assert gf_irred_p_rabin([7, 3, 1], 11, ZZ) is False
assert gf_irred_p_ben_or([2, 3, 4, 5, 6], 13, ZZ) is False
assert gf_irred_p_ben_or([2, 3, 4, 5, 8], 13, ZZ) is True
with config.using(gf_irred_method='ben-or'):
assert gf_irreducible_p([7], 11, ZZ) is True
assert gf_irreducible_p([7, 3], 11, ZZ) is True
assert gf_irreducible_p([7, 3, 1], 11, ZZ) is False
with config.using(gf_irred_method='rabin'):
assert gf_irreducible_p([7], 11, ZZ) is True
assert gf_irreducible_p([7, 3], 11, ZZ) is True
assert gf_irreducible_p([7, 3, 1], 11, ZZ) is False
with config.using(gf_irred_method='other'):
pytest.raises(KeyError, lambda: gf_irreducible_p([7], 11, ZZ))
f = [1, 9, 9, 13, 16, 15, 6, 7, 7, 7, 10]
g = [1, 7, 16, 7, 15, 13, 13, 11, 16, 10, 9]
h = gf_mul(f, g, 17, ZZ)
assert gf_irred_p_ben_or(f, 17, ZZ) is True
assert gf_irred_p_ben_or(g, 17, ZZ) is True
assert gf_irred_p_ben_or(h, 17, ZZ) is False
assert gf_irred_p_rabin(f, 17, ZZ) is True
assert gf_irred_p_rabin(g, 17, ZZ) is True
assert gf_irred_p_rabin(h, 17, ZZ) is False
def test_modgcd_bivariate_integers():
R, x, y = ring('x,y', ZZ)
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
f, g = R.zero, R.zero
assert f.cofactors(g) == (0, 0, 0)
f, g = 2 * x, R(2)
assert f.cofactors(g) == (2, x, 1)
f, g = x + 2 * y, x + y
assert f.cofactors(g) == (1, f, g)
f, g = x**2 + 2 * x * y + y**2, x**3 + y**3
assert f.cofactors(g) == (x + y, x + y, x**2 - x * y + y**2)
f, g = x * y**2 + 2 * x * y + x, x * y**3 + x
assert f.cofactors(g) == (x * y + x, y + 1, y**2 - y + 1)
f, g = x**2 * y**2 + x**2 * y + 1, x * y**2 + x * y + 1
assert f.cofactors(g) == (1, f, g)
f = 2 * x * y**2 + 4 * x * y + 2 * x + y**2 + 2 * y + 1
g = 2 * x * y**3 + 2 * x + y**3 + 1
assert f.cofactors(g) == (2 * x * y + 2 * x + y + 1, y + 1,
y**2 - y + 1)
f, g = 2 * x**2 + 4 * x + 2, x + 1
assert f.cofactors(g) == (x + 1, 2 * x + 2, 1)
f, g = x + 1, 2 * x**2 + 4 * x + 2
assert f.cofactors(g) == (x + 1, 1, 2 * x + 2)
f = 2 * x**2 + 4 * x * y - 2 * x - 4 * y
g = x**2 + x - 2
assert f.cofactors(g) == (x - 1, 2 * x + 4 * y, x + 2)
f = 2 * x**2 + 2 * x * y - 3 * x - 3 * y
g = 4 * x * y - 2 * x + 4 * y**2 - 2 * y
assert f.cofactors(g) == (x + y, 2 * x - 3, 4 * y - 2)
def test_benchmark_minimal_polynomial(method):
with config.using(groebner=method):
R, x, y, z = ring('x y z', QQ, lex)
F = [x**3 + x + 1, y**2 + y + 1, (x + y) * z - (x**2 + y)]
G = [
x + 155 * z**5 / 2067 - 355 * z**4 / 689 + 6062 * z**3 / 2067 -
3687 * z**2 / 689 + 6878 * z / 2067 - QQ(25, 53),
y + 4 * z**5 / 53 - 91 * z**4 / 159 + 523 * z**3 / 159 -
387 * z**2 / 53 + 1043 * z / 159 - QQ(308, 159),
z**6 - 7 * z**5 + 41 * z**4 - 82 * z**3 + 89 * z**2 - 46 * z + 13
]
assert groebner(F, R) == G
def test_dup_gf_irreducible_p():
R, x = ring('x', FF(11))
f = R.to_dense(R(7))
g = R.to_dense(7 * x + 3)
h = R.to_dense(7 * x**2 + 3 * x + 1)
for method in ('ben-or', 'rabin'):
with using(gf_irred_method=method):
assert dup_gf_irreducible_p(f, R.domain) is True
assert dup_gf_irreducible_p(g, R.domain) is True
assert dup_gf_irreducible_p(h, R.domain) is False
with using(gf_irred_method='other'):
pytest.raises(KeyError, lambda: dup_gf_irreducible_p(f, R.domain))
R, x = ring('x', FF(13))
f = R.to_dense(2 * x**4 + 3 * x**3 + 4 * x**2 + 5 * x + 6)
g = R.to_dense(2 * x**4 + 3 * x**3 + 4 * x**2 + 5 * x + 8)
with using(gf_irred_method='ben-or'):
assert dup_gf_irreducible_p(f, R.domain) is False
assert dup_gf_irreducible_p(g, R.domain) is True
R, x = ring('x', FF(17))
f = (x**10 + 9 * x**9 + 9 * x**8 + 13 * x**7 + 16 * x**6 + 15 * x**5 +
6 * x**4 + 7 * x**3 + 7 * x**2 + 7 * x + 10)
g = (x**10 + 7 * x**9 + 16 * x**8 + 7 * x**7 + 15 * x**6 + 13 * x**5 +
13 * x**4 + 11 * x**3 + 16 * x**2 + 10 * x + 9)
h = f * g
f, g, h = map(R.to_dense, (f, g, h))
for method in ('ben-or', 'rabin'):
with using(gf_irred_method=method):
assert dup_gf_irreducible_p(f, R.domain) is True
assert dup_gf_irreducible_p(g, R.domain) is True
assert dup_gf_irreducible_p(h, R.domain) is False
F9 = FF(3, [1, 2, 2])
R, x = ring('x', F9)
f = R.to_dense(x**3 + F9(8) * x**2 + F9(8) * x + F9(4))
for method in ('ben-or', 'rabin'):
with using(gf_irred_method=method):
assert dup_gf_irreducible_p(f, R.domain) is False
F27 = FF(3, [1, 2, 0, 1])
R, x = ring('x', F27)
f = R.to_dense(x**3 + F27(8) * x**2 + F27(19) * x + F27(24))
for method in ('ben-or', 'rabin'):
with using(gf_irred_method=method):
assert dup_gf_irreducible_p(f, R.domain) is True
def test_benchmark_katsura_4(method):
with config.using(groebner=method):
R, x0, x1, x2, x3 = ring('x:4', ZZ, lex)
I = [
x0 + 2 * x1 + 2 * x2 + 2 * x3 - 1,
x0**2 + 2 * x1**2 + 2 * x2**2 + 2 * x3**2 - x0,
2 * x0 * x1 + 2 * x1 * x2 + 2 * x2 * x3 - x1,
x1**2 + 2 * x0 * x2 + 2 * x1 * x3 - x2
]
assert groebner(I, R) == [
5913075 * x0 - 159690237696 * x3**7 + 31246269696 * x3**6 +
27439610544 * x3**5 - 6475723368 * x3**4 - 838935856 * x3**3 +
275119624 * x3**2 + 4884038 * x3 - 5913075,
1971025 * x1 - 97197721632 * x3**7 + 73975630752 * x3**6 -
12121915032 * x3**5 - 2760941496 * x3**4 + 814792828 * x3**3 -
1678512 * x3**2 - 9158924 * x3,
5913075 * x2 + 371438283744 * x3**7 - 237550027104 * x3**6 +
22645939824 * x3**5 + 11520686172 * x3**4 - 2024910556 * x3**3 -
132524276 * x3**2 + 30947828 * x3,
128304 * x3**8 - 93312 * x3**7 + 15552 * x3**6 + 3144 * x3**5 -
1120 * x3**4 + 36 * x3**3 + 15 * x3**2 - x3,
]
R, x0, x1, x2, x3 = ring('x:4', ZZ, grlex)
I = [i.set_ring(R) for i in I]
assert groebner(I, R) == [
393 * x1 - 4662 * x2**2 + 4462 * x2 * x3 - 59 * x2 +
224532 * x3**4 - 91224 * x3**3 - 678 * x3**2 + 2046 * x3,
-x1 + 196 * x2**3 - 21 * x2**2 + 60 * x2 * x3 - 18 * x2 -
168 * x3**3 + 83 * x3**2 - 9 * x3,
-6 * x1 + 1134 * x2**2 * x3 - 189 * x2**2 - 466 * x2 * x3 +
32 * x2 - 630 * x3**3 + 57 * x3**2 + 51 * x3,
33 * x1 + 63 * x2**2 + 2268 * x2 * x3**2 - 188 * x2 * x3 +
34 * x2 + 2520 * x3**3 - 849 * x3**2 + 3 * x3,
7 * x1**2 - x1 - 7 * x2**2 - 24 * x2 * x3 + 3 * x2 - 15 * x3**2 +
5 * x3,
14 * x1 * x2 - x1 + 14 * x2**2 + 18 * x2 * x3 - 4 * x2 +
6 * x3**2 - 2 * x3,
14 * x1 * x3 - x1 + 7 * x2**2 + 32 * x2 * x3 - 4 * x2 +
27 * x3**2 - 9 * x3,
x0 + 2 * x1 + 2 * x2 + 2 * x3 - 1,
]
def test_dmp_ext_factor(method):
with using(aa_factor_method=method):
R, x = ring('x', QQ.algebraic_field(I))
assert R(0).factor_list() == (0, [])
assert (x + 1).factor_list() == (1, [(x + 1, 1)])
assert (2 * x + 2).factor_list() == (2, [(x + 1, 1)])
assert (7 * x**4 + 1).factor_list() == (7, [(x**4 + QQ(1, 7), 1)])
assert (x**4 + 1).factor_list() == (1, [(x**2 - I, 1), (x**2 + I, 1)])
assert (4 * x**2 + 9).factor_list() == (4, [(x - 3 * I / 2, 1),
(x + 3 * I / 2, 1)])
assert (4 * x**4 + 8 * x**3 + 77 * x**2 + 18 * x +
153).factor_list() == (4, [(x - 3 * I / 2, 1),
(x + 1 + 4 * I, 1),
(x + 1 - 4 * I, 1),
(x + 3 * I / 2, 1)])
assert (x**2 + 1).factor_list() == (1, [(x - I, 1), (x + I, 1)])
R, x = ring('x', QQ.algebraic_field(sqrt(2)))
assert (x**4 + 1).factor_list() == (1, [(x**2 - sqrt(2) * x + 1, 1),
(x**2 + sqrt(2) * x + 1, 1)])
f = x**2 + 2 * sqrt(2) * x + 2
assert f.factor_list() == (1, [(x + sqrt(2), 2)])
assert (f**3).factor_list() == (1, [(x + sqrt(2), 6)])
f *= 2
assert f.factor_list() == (2, [(x + sqrt(2), 2)])
assert (f**3).factor_list() == (8, [(x + sqrt(2), 6)])
R, x, y = ring('x y', QQ.algebraic_field(sqrt(2)))
assert R(0).factor_list() == (0, [])
assert (x + 1).factor_list() == (1, [(x + 1, 1)])
assert (2 * x + 2).factor_list() == (2, [(x + 1, 1)])
assert (x**2 - 2 * y**2).factor_list() == (1, [(x - sqrt(2) * y, 1),
(x + sqrt(2) * y, 1)])
assert (2 * x**2 - 4 * y**2).factor_list() == (2,
[(x - sqrt(2) * y, 1),
(x + sqrt(2) * y, 1)])
def test_benchmark_katsura_3(method):
with config.using(groebner=method):
R, x0, x1, x2 = ring('x:3', ZZ, lex)
I = [
x0 + 2 * x1 + 2 * x2 - 1, x0**2 + 2 * x1**2 + 2 * x2**2 - x0,
2 * x0 * x1 + 2 * x1 * x2 - x1
]
assert groebner(I, R) == [
-7 + 7 * x0 + 8 * x2 + 158 * x2**2 - 420 * x2**3,
7 * x1 + 3 * x2 - 79 * x2**2 + 210 * x2**3,
x2 + x2**2 - 40 * x2**3 + 84 * x2**4,
]
R, x0, x1, x2 = ring('x:3', ZZ, grlex)
I = [i.set_ring(R) for i in I]
assert groebner(I, R) == [
7 * x1 + 3 * x2 - 79 * x2**2 + 210 * x2**3,
-x1 + x2 - 3 * x2**2 + 5 * x1**2,
-x1 - 4 * x2 + 10 * x1 * x2 + 12 * x2**2,
-1 + x0 + 2 * x1 + 2 * x2,
]
def test_dmp_gcd():
R, x = ring("x", ZZ)
f, g = 0, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (0, 0, 0)
f, g = 2, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 0)
f, g = -2, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 0)
f, g = 0, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 0, -1)
f, g = 0, 2*x + 4
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 0, 1)
f, g = 2*x + 4, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 1, 0)
f, g = 2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 1)
f, g = -2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 1)
f, g = 2, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, -1)
f, g = -2, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, -1)
f, g = x**2 + 2*x + 1, 1
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1)
f, g = x**2 + 2*x + 1, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2)
f, g = 2*x**2 + 4*x + 2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1)
f, g = 2, 2*x**2 + 4*x + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1)
f, g = 2*x**2 + 4*x + 2, x + 1
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1)
f, g = x + 1, 2*x**2 + 4*x + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2)
f, g = x - 31, x
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, f, g)
f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8
g = x**3 + 6*x**2 + 11*x + 6
h = x**2 + 3*x + 2
cff = x**2 + 5*x + 4
cfg = x + 3
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
f = x**4 - 4
g = x**4 + 4*x**2 + 4
h = x**2 + 2
cff = x**2 - 2
cfg = x**2 + 2
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
h = 1
cff = f
cfg = g
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
R, x = ring("x", QQ)
f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
h = 1
cff = f
cfg = g
assert R.dmp_qq_heu_gcd(f, g) == (h, cff, cfg)
assert R.dup_ff_prs_gcd(f, g) == (h, cff, cfg)
assert R.dup_ff_prs_gcd(R.zero, R.zero) == ([], [], [])
R, x = ring("x", ZZ)
f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \
+ 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \
+ 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \
+ 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \
- 12278371209708240950316872681744825481125965781519138077173235712*x**21 \
+ 289127344604779611146960547954288113529690984687482920704*x**14 \
+ 19007977035740498977629742919480623972236450681*x**7 \
+ 311973482284542371301330321821976049
g = 365431878023781158602430064717380211405897160759702125019136*x**21 \
+ 197599133478719444145775798221171663643171734081650688*x**14 \
- 9504116979659010018253915765478924103928886144*x**7 \
- 311973482284542371301330321821976049
assert R.dmp_zz_heu_gcd(f, R.dmp_diff_in(f, 1, 0))[0] == g
assert R.dup_rr_prs_gcd(f, R.dmp_diff_in(f, 1, 0))[0] == g
R, x = ring("x", QQ)
f = x**2/2 + x + QQ(1, 2)
g = x/2 + QQ(1, 2)
h = x + 1
assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dup_ff_prs_gcd(f, g) == (h, g, QQ(1, 2))
R, x = ring("x", ZZ)
f = 1317378933230047068160*x + 2945748836994210856960
g = 120352542776360960*x + 269116466014453760
h = 120352542776360960*x + 269116466014453760
cff = 10946
cfg = 1
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
with using(heu_gcd_max=0):
pytest.raises(HeuristicGCDFailed, lambda: R.dmp_zz_heu_gcd(f, g))
R, x = ring("x", CC)
f, g = (x**2 - 1, x**3 - 3*x + 2)
assert R.dmp_inner_gcd(f, g) == (1, f, g)
R, x, y = ring("x,y", CC)
f, g = (x**2 - y, x**3 - y*x + 2)
assert R.dmp_inner_gcd(f, g) == (1, f, g)
R, x, y = ring("x,y", ZZ)
f, g = 0, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (0, 0, 0)
f, g = 2, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 0)
f, g = -2, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 0)
f, g = 0, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 0, -1)
f, g = 0, 2*x + 4
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 0, 1)
f, g = 2*x + 4, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 1, 0)
f, g = 2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 1)
f, g = -2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 1)
f, g = 2, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, -1)
f, g = -2, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, -1)
f, g = x**2 + 2*x + 1, 1
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1)
f, g = x**2 + 2*x + 1, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2)
with using(use_simplify_gcd=0):
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2)
f, g = 2*x**2 + 4*x + 2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1)
f, g = 2, 2*x**2 + 4*x + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1)
f, g = 2*x**2 + 4*x + 2, x + 1
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1)
f, g = x + 1, 2*x**2 + 4*x + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2)
with using(heu_gcd_max=0):
pytest.raises(HeuristicGCDFailed, lambda: R.dmp_zz_heu_gcd(f, g))
f = x**2 + 2*x*y + y**2
g = x**2 + x*y
assert R.dmp_rr_prs_gcd(f, g) == (x + y, x + y, x)
R, x, y, z, u = ring("x,y,z,u", ZZ)
f, g = u**2 + 2*u + 1, 2*u + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (u + 1, u + 1, 2)
f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1
h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
assert R.dmp_rr_prs_gcd(f, g) == (h, cff, cfg)
assert R.dmp_zz_heu_gcd(g, f) == (h, cfg, cff)
assert R.dmp_rr_prs_gcd(g, f) == (h, cfg, cff)
R, x, y, z = ring("x,y,z", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(2, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(4, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(6, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(8, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z = ring("x,y,z", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_2(2, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(2, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(4, ZZ))
H, cff, cfg = R.dmp_inner_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y = ring("x,y", QQ)
f = x**2/2 + x + QQ(1, 2)
g = x/2 + QQ(1, 2)
h = x + 1
assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dmp_ff_prs_gcd(f, g) == (h, g, QQ(1, 2))
with using(use_simplify_gcd=0):
assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dmp_ff_prs_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dmp_ff_prs_gcd(R.zero, R.zero) == (0, 0, 0)
assert R.dmp_qq_heu_gcd(R.zero, R.zero) == (0, 0, 0)
assert R.dmp_ff_prs_gcd(R.zero, g) == (x + 1, R.zero, QQ(1, 2))
assert R.dmp_qq_heu_gcd(R.zero, g) == (x + 1, R.zero, QQ(1, 2))
R, x, y = ring("x,y", RR)
f = 2.1*x*y**2 - 2.2*x*y + 2.1*x
g = 1.0*x**3
assert R.dmp_ff_prs_gcd(f, g) == \
(1.0*x, 2.1*y**2 - 2.2*y + 2.1, 1.0*x**2)
R, x, y = ring("x,y", ZZ)
f = (-17434367009167300000000000000000000000000000000000000000000000000000000*x**4*y -
250501827896299135568887342575961783764139560000000000000000000000000000000000000000000*x**3*y -
2440935909299672540738135183426056447877858000000000000000000000000000000*x**3 -
1349729941723537919695626818065131519270095220127010623905326719279566297660000000000000000000000000000*x**2*y -
26304033868956978374552886858060487282904504027042515077682955951658838800000000000000000*x**2 -
3232215785736369696036755035364398565076440134133908303058376297547504030528179314849416971379040931276000000000000000*x*y -
94485916261760032526508027937078714464844205539023800247528621905831259414691631156161537919255129011800*x -
2902585888465621357542575571971656665554321652262249362701116665830760628936600958940851960635161420991047110815678789984677193092993*y -
113133324167442997472440652189550843502029192913459268196939183295294085146407870078840385860571627108778756267503630290)
g = (10000000000000000000000000000*x**2 + 71841388839807267676152024786000000000000000*x +
129029628760809605749020969023932901278290735413660734705971)
assert (R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) ==
(g,
-1743436700916730000000000000000000000000000*x**2*y -
12525091394814956778444367128798089188206978000000000000000*x*y -
244093590929967254073813518342605644787785800*x -
22495499028725631994927113634418779135935898997901327211111875586270479483*y -
876801128965234839118530545935732755107147297241756982389990, 1))
R, x = ring("x", ZZ)
f, g = x**2 - 1, x**2 - 3*x + 2
assert R.dmp_gcd(f, g) == x - 1
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
R.dmp_gcd(f, g) == x - 1
R, x = ring("x", QQ)
f, g = x**2/2 + x + QQ(1, 2), x/2 + QQ(1, 2)
assert R.dmp_gcd(f, g) == x + 1
with using(use_heu_gcd=False):
R.dmp_gcd(f, g) == x + 1
R, x, y = ring("x,y", QQ.algebraic_field(sqrt(2)))
f, g = (x + sqrt(2)*y)**2, x + sqrt(2)*y
assert R.dmp_gcd(f, g) == g
with using(gcd_aa_method='modgcd'):
assert R.dmp_gcd(f, g) == g
def test_dmp_subresultants():
R, x = ring("x", ZZ)
assert R.dmp_resultant(0, 0) == 0
assert R.dmp_resultant(1, 0) == 0
assert R.dmp_resultant(0, 1) == 0
f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
a = 15*x**4 - 3*x**2 + 9
b = 65*x**2 + 125*x - 245
c = 9326*x - 12300
d = 260708
assert R.dmp_subresultants(f, g) == [f, g, a, b, c, d]
assert R.dmp_resultant(f, g) == R.dmp_LC(d)
f = x**2 - 2*x + 1
g = x**2 - 1
a = 2*x - 2
assert R.dmp_subresultants(f, g) == [f, g, a]
assert R.dmp_resultant(f, g) == 0
f = x**2 + 1
g = x**2 - 1
a = -2
assert R.dmp_subresultants(f, g) == [f, g, a]
assert R.dmp_resultant(f, g) == 4
assert R.dmp_resultant(f, g, includePRS=True) == (4, [x**2 + 1, x**2 - 1, -2])
f = x**2 - 1
g = x**3 - x**2 + 2
assert R.dmp_resultant(f, g) == 0
f = 3*x**3 - x
g = 5*x**2 + 1
assert R.dmp_resultant(f, g) == 64
f = x**2 - 2*x + 7
g = x**3 - x + 5
assert R.dmp_resultant(f, g) == 265
f = x**3 - 6*x**2 + 11*x - 6
g = x**3 - 15*x**2 + 74*x - 120
assert R.dmp_resultant(f, g) == -8640
f = x**3 - 6*x**2 + 11*x - 6
g = x**3 - 10*x**2 + 29*x - 20
assert R.dmp_resultant(f, g) == 0
f = x**3 - 1
g = x**3 + 2*x**2 + 2*x - 1
assert R.dmp_resultant(f, g) == 16
f = x**8 - 2
g = x - 1
assert R.dmp_resultant(f, g) == -1
assert R.dup_inner_subresultants(0, 0) == ([], [])
assert R.dup_inner_subresultants(0, 1) == ([1], [1])
R, x, y = ring("x,y", ZZ)
assert R.dmp_resultant(0, 0) == 0
assert R.dmp_prs_resultant(0, 0)[0] == 0
assert R.dmp_zz_collins_resultant(0, 0) == 0
assert R.dmp_qq_collins_resultant(0, 0) == 0
assert R.dmp_resultant(1, 0) == 0
assert R.dmp_resultant(1, 0) == 0
assert R.dmp_resultant(1, 0) == 0
assert R.dmp_resultant(0, 1) == 0
assert R.dmp_prs_resultant(0, 1)[0] == 0
assert R.dmp_zz_collins_resultant(0, 1) == 0
assert R.dmp_qq_collins_resultant(0, 1) == 0
assert R.dmp_inner_subresultants(0, 0) == ([], [])
assert R.dmp_inner_subresultants(0, 1) == ([R.one], [[1]])
f = 3*x**2*y - y**3 - 4
g = x**2 + x*y**3 - 9
a = 3*x*y**4 + y**3 - 27*y + 4
b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16
r = R.dmp_LC(b)
Y = R.drop(x).y
rr = (-3*Y**10 - 12*Y**7 + Y**6 - 54*Y**4 + 8*Y**3 + 729*Y**2 - 216*Y + 16,
[3*x**2*y - y**3 - 4, x**2 + x*y**3 - 9, 3*x*y**4 + y**3 - 27*y + 4,
-3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16])
assert R.dmp_subresultants(f, g) == [f, g, a, b]
assert R.dmp_resultant(f, g) == r
assert R.dmp_resultant(f, g, includePRS=True) == rr
assert R.dmp_prs_resultant(f, g)[0] == r
assert R.dmp_zz_collins_resultant(f, g) == r
assert R.dmp_qq_collins_resultant(f, g) == r
f = -x**3 + 5
g = 3*x**2*y + x**2
a = 45*y**2 + 30*y + 5
b = 675*y**3 + 675*y**2 + 225*y + 25
r = R.dmp_LC(b)
assert R.dmp_subresultants(f, g) == [f, g, a]
assert R.dmp_resultant(f, g) == r
assert R.dmp_prs_resultant(f, g)[0] == r
assert R.dmp_zz_collins_resultant(f, g) == r
assert R.dmp_qq_collins_resultant(f, g) == r
R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
f = 6*x**2 - 3*x*y - 2*x*z + y*z
g = x**2 - x*u - x*v + u*v
r = y**2*z**2 - 3*y**2*z*u - 3*y**2*z*v + 9*y**2*u*v - 2*y*z**2*u \
- 2*y*z**2*v + 6*y*z*u**2 + 12*y*z*u*v + 6*y*z*v**2 - 18*y*u**2*v \
- 18*y*u*v**2 + 4*z**2*u*v - 12*z*u**2*v - 12*z*u*v**2 + 36*u**2*v**2
assert R.dmp_zz_collins_resultant(f, g) == r.drop(x)
R, x, y, z, u, v = ring("x,y,z,u,v", QQ)
f = x**2 - x*y/2 - x*z/3 + y*z/6
g = x**2 - x*u - x*v + u*v
r = y**2*z**2/36 - y**2*z*u/12 - y**2*z*v/12 + y**2*u*v/4 \
- y*z**2*u/18 - y*z**2*v/18 + y*z*u**2/6 + y*z*u*v/3 \
+ y*z*v**2/6 - y*u**2*v/2 - y*u*v**2/2 + z**2*u*v/9 \
- z*u**2*v/3 - z*u*v**2/3 + u**2*v**2
assert R.dmp_qq_collins_resultant(f, g) == r.drop(x)
Rt, t = ring("t", ZZ)
Rx, x = ring("x", Rt)
f = x**6 - 5*x**4 + 5*x**2 + 4
g = -6*t*x**5 + x**4 + 20*t*x**3 - 3*x**2 - 10*t*x + 6
assert Rx.dmp_resultant(f, g) == 2930944*t**6 + 2198208*t**4 + 549552*t**2 + 45796
assert Rx.dmp_prs_resultant(x - 1, x + 1) == (2, [x - 1, x + 1, 2])
R, x, y = ring("x,y", ZZ)
f = x + y
g = x**2 - x*y + 1
assert R.dmp_resultant(f, g) == (1 + 2*y**2).drop(x)
g += 1
with using(use_collins_resultant=True):
assert R.dmp_resultant(f, g) == (2 + 2*y**2).drop(x)
R, x, y = ring("x,y", QQ)
f = x + y
g = x**2 - x*y + 1
with using(use_collins_resultant=True):
assert R.dmp_resultant(f, g) == (1 + 2*y**2).drop(x)
R, x, y = ring("x y", ZZ)
f = x + y + 2
g = 2*x*y + x + 3
assert R.dmp_zz_collins_resultant(f, g) == (-2*y**2 - 5*y + 1).drop(x)
def test_modgcd_algebraic_field():
A = QQ.algebraic_field(sqrt(2))
R, x = ring('x', A)
with using(gcd_aa_method='modgcd'):
f, g = 2 * x, R(2)
assert f.cofactors(g) == (1, f, g)
f, g = 2 * x, R(sqrt(2))
assert f.cofactors(g) == (1, f, g)
f, g = 2 * x + 2, 6 * x**2 - 6
assert f.cofactors(g) == (x + 1, 2, 6 * x - 6)
R, x, y = ring('x, y', A)
with using(gcd_aa_method='modgcd'):
f, g = x + sqrt(2) * y, x + y
assert f.cofactors(g) == (1, f, g)
f, g = x * y + sqrt(2) * y**2, sqrt(2) * y
assert f.cofactors(g) == (y, x + sqrt(2) * y, sqrt(2))
f, g = x**2 + 2 * sqrt(2) * x * y + 2 * y**2, x + sqrt(2) * y
assert f.cofactors(g) == (g, g, 1)
A = QQ.algebraic_field(sqrt(2), sqrt(3))
R, x, y, z = ring('x, y, z', A)
with using(gcd_aa_method='modgcd'):
h = x**2 * y**7 + sqrt(6) / 21 * z
f, g = h * (27 * y**3 + 1), h * (y + x)
assert f.cofactors(g) == (h, 27 * y**3 + 1, x + y)
h = x**13 * y**3 + x**10 / 2 + 1 / sqrt(2)
f, g = h * (x + 1), h * sqrt(2) / sqrt(3)
assert f.cofactors(g) == (h, x + 1, sqrt(2) / sqrt(3))
h = x**4 * y**9 + sqrt(6) / 22 * z
f, g = h * (21 * y**3 + 1), h * (y + x)
assert f.cofactors(g) == (x**4 * y**9 + sqrt(6) / 22 * z,
21 * y**3 + 1, x + y)
h = x**4 * y**3 + sqrt(6) / 22 * z
f, g = h * (11 * y**3 + 1), h * (y + x)
assert f.cofactors(g) == (x**4 * y**3 + sqrt(6) / 22 * z,
11 * y**3 + 1, x + y)
A = QQ.algebraic_field(sqrt(2)**(-1) * sqrt(3))
R, x = ring('x', A)
with using(gcd_aa_method='modgcd'):
f, g = x + 1, x - 1
assert f.cofactors(g) == (1, f, g)
def test_modgcd_multivariate_integers():
R, x, y = ring('x,y', ZZ)
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
f, g = R.zero, R.zero
assert f.cofactors(g) == (0, 0, 0)
f, g = 2 * x**2 + 4 * x + 2, x + 1
assert f.cofactors(g) == (x + 1, 2 * x + 2, 1)
f, g = x + 1, 2 * x**2 + 4 * x + 2
assert f.cofactors(g) == (x + 1, 1, 2 * x + 2)
f = 2 * x**2 + 2 * x * y - 3 * x - 3 * y
g = 4 * x * y - 2 * x + 4 * y**2 - 2 * y
assert f.cofactors(g) == (x + y, 2 * x - 3, 4 * y - 2)
f, g = x * y**2 + 2 * x * y + x, x * y**3 + x
assert f.cofactors(g) == (x * y + x, y + 1, y**2 - y + 1)
f, g = x**2 * y**2 + x**2 * y + 1, x * y**2 + x * y + 1
assert f.cofactors(g) == (1, f, g)
f = x**4 + 8 * x**3 + 21 * x**2 + 22 * x + 8
g = x**3 + 6 * x**2 + 11 * x + 6
assert f.cofactors(g) == (x**2 + 3 * x + 2, x**2 + 5 * x + 4, x + 3)
R, x, y, z, u = ring('x,y,z,u', ZZ)
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
f, g = x + y + z, -x - y - z - u
assert f.cofactors(g) == (1, f, g)
f, g = u**2 + 2 * u + 1, 2 * u + 2
assert f.cofactors(g) == (u + 1, u + 1, 2)
f, g = z**2 * u**2 + 2 * z**2 * u + z**2 + z * u + z, u**2 + 2 * u + 1
h, cff, cfg = u + 1, z**2 * u + z**2 + z, u + 1
assert f.cofactors(g) == (h, cff, cfg)
assert g.cofactors(f) == (h, cfg, cff)
f, g, h = (1199999999999991 * x**17 - y, 2 * y - 19989798798 + x**211,
12 * x * y**7 + x**4 - 1)
for i in range(10):
assert (f * h).cofactors(g * h) == (h, f, g)
R, x, y, z = ring('x,y,z', ZZ)
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
f, g = x - y * z, x - y * z
assert f.cofactors(g) == (x - y * z, 1, 1)
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
f, g, h = R.fateman_poly_F_2()
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
f, g, h = R.fateman_poly_F_3()
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
R, x, y, z, u, v = ring('x,y,z,u,v', ZZ)
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
R, x, y, z, u, v, a, b = ring('x,y,z,u,v,a,b', ZZ)
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
R, x, y, z, u, v, a, b, c, d = ring('x,y,z,u,v,a,b,c,d', ZZ)
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
R, x, y, z, t = ring('x,y,z,t', ZZ)
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
f, g, h = R.fateman_poly_F_3()
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
def test_benchmark_cyclic_4_f5b():
with config.using(groebner='f5b'):
_do_test_benchmark_cyclic_4()
def test_groebner_buchberger():
with config.using(groebner='buchberger'):
_do_test_groebner()
def test__zz_wang():
R, x, y, z = ring('x y z', ZZ)
UV, _x = ring('x', ZZ)
p = ZZ(nextprime(R._zz_mignotte_bound(w_1)))
assert p == 6291469
t_1, k_1, e_1 = y, 1, ZZ(-14)
t_2, k_2, e_2 = z, 2, ZZ(3)
t_3, k_3, e_3 = y + z, 2, ZZ(-11)
t_4, k_4, e_4 = y - z, 1, ZZ(-17)
T = [t_1, t_2, t_3, t_4]
K = [k_1, k_2, k_3, k_4]
E = [e_1, e_2, e_3, e_4]
T = list(zip([t.drop(x) for t in T], K))
A = [ZZ(-14), ZZ(3)]
S = w_1.eval([(y, A[0]), (z, A[1])])
cs, s = S.primitive()
assert cs == 1 and s == S == (1036728 * _x**6 + 915552 * _x**5 +
55748 * _x**4 + 105621 * _x**3 -
17304 * _x**2 - 26841 * _x - 644)
assert R._zz_wang_non_divisors(E, cs, ZZ(4)) == [7, 3, 11, 17]
assert s.is_squarefree and s.degree() == w_1.degree()
_, H = UV._zz_factor_sqf(s)
h_1 = 187 * _x**2 - 23
h_2 = 44 * _x**2 + 42 * _x + 1
h_3 = 126 * _x**2 - 9 * _x + 28
LC = [lc.drop(x) for lc in [y**2 - z**2, -4 * y - 4 * z, -y * z**2]]
factors = R._zz_wang_hensel_lifting(w_1, H, LC, A, p)
assert H == [h_1, h_2, h_3]
assert R._zz_wang_lead_coeffs(w_1, T, cs, E, H, A) == (w_1, H, LC)
assert functools.reduce(operator.mul, factors) == w_1
# coverage tests
f = x**6 + 5 * x**4 * y - 5 * x**2 * y**2 - y**3
assert R._zz_wang(f, mod=4,
seed=1) == [x**2 - y, x**4 + 6 * x**2 * y + y**2]
# This tests a bug in the Wang algorithm that occured only with a very
# specific set of random numbers; issue sympy/sympy#6355.
random_sequence = [
-1, -1, 0, 0, 0, 0, -1, -1, 0, -1, 3, -1, 3, 3, 3, 3, -1, 3
]
R, x, y, z = ring('x y z', ZZ)
f = 2 * x**2 + y * z - y - z**2 + z
assert R._zz_wang(f, seed=random_sequence) == [f]
with using(eez_restart_if_needed=False):
pytest.raises(ExtraneousFactors,
lambda: R._zz_wang(f, seed=random_sequence))
def test_PolyElement_is_irreducible():
R, x = ring('x', FF(5))
f = (x**10 + 4 * x**9 + 2 * x**8 + 2 * x**7 + 3 * x**6 + 2 * x**5 +
4 * x**4 + x**3 + 4 * x**2 + 4)
g = 3 * x**2 + 2 * x + 4
for method in ('ben-or', 'rabin'):
with using(gf_irred_method=method):
assert f.is_irreducible is True
assert g.is_irreducible is False
R, x = ring('x', FF(11))
f = R(7)
g = 7 * x + 3
h = 7 * x**2 + 3 * x + 1
for method in ('ben-or', 'rabin'):
with using(gf_irred_method=method):
assert f.is_irreducible is True
assert g.is_irreducible is True
assert h.is_irreducible is False
with using(gf_irred_method='other'):
pytest.raises(KeyError, lambda: f.is_irreducible)
R, x = ring('x', FF(13))
f = 2 * x**4 + 3 * x**3 + 4 * x**2 + 5 * x + 6
g = 2 * x**4 + 3 * x**3 + 4 * x**2 + 5 * x + 8
with using(gf_irred_method='ben-or'):
assert f.is_irreducible is False
assert g.is_irreducible is True
R, x = ring('x', FF(17))
f = (x**10 + 9 * x**9 + 9 * x**8 + 13 * x**7 + 16 * x**6 + 15 * x**5 +
6 * x**4 + 7 * x**3 + 7 * x**2 + 7 * x + 10)
g = (x**10 + 7 * x**9 + 16 * x**8 + 7 * x**7 + 15 * x**6 + 13 * x**5 +
13 * x**4 + 11 * x**3 + 16 * x**2 + 10 * x + 9)
h = f * g
for method in ('ben-or', 'rabin'):
with using(gf_irred_method=method):
assert f.is_irreducible is True
assert g.is_irreducible is True
assert h.is_irreducible is False
F9 = FF(3, [2, 2, 1])
R, x = ring('x', F9)
f = x**3 + F9(8) * x**2 + F9(8) * x + F9(4)
for method in ('ben-or', 'rabin'):
with using(gf_irred_method=method):
assert f.is_irreducible is False
F27 = FF(3, [1, 0, 2, 1])
R, x = ring('x', F27)
f = x**3 + F27(8) * x**2 + F27(19) * x + F27(24)
for method in ('ben-or', 'rabin'):
with using(gf_irred_method=method):
assert f.is_irreducible is True
R, x = ring('x', ZZ)
assert x.is_irreducible is True
assert (x**2 + x + 1).is_irreducible is True
assert (x**2 + 2 * x + 1).is_irreducible is False
assert (x**2 - 1).is_irreducible is False
f = 3 * x**4 + 2 * x**3 + 6 * x**2 + 8 * x
assert (f + 7).is_irreducible is True
assert (f + 4).is_irreducible is True
assert (f + 10).is_irreducible is True
assert (f + 14).is_irreducible is True
R, x, y = ring('x y', ZZ)
assert R(2).is_irreducible is True
assert (x**2 + x + 1).is_irreducible is True
assert (x**2 + 2 * x + 1).is_irreducible is False
assert ((x - 2 * y) * (x + y)).is_irreducible is False
assert (x**2 + y**2).is_irreducible is True
R, x, y, z = ring('x y z', QQ)
assert (x**2 + x + 1).is_irreducible
assert (x**2 + 2 * x + 1).is_irreducible is False
def test_factor_list():
R, x = ring('x', FF(2))
assert (x**2 + 1).factor_list() == (1, [(x + 1, 2)])
R, x = ring('x', ZZ)
assert R(0).factor_list() == (0, [])
assert R(7).factor_list() == (7, [])
assert (x**2 + 2 * x + 1).factor_list() == (1, [(x + 1, 2)])
# issue sympy/sympy#8037
assert (6 * x**2 - 5 * x - 6).factor_list() == (1, [(2 * x - 3, 1),
(3 * x + 2, 1)])
R, x = ring('x', QQ)
assert R(0).factor_list() == (0, [])
assert R(QQ(1, 7)).factor_list() == (QQ(1, 7), [])
assert (x**2 / 2 + x + QQ(1, 2)).factor_list() == (QQ(1, 2), [(x + 1, 2)])
R, x = ring('x', QQ.algebraic_field(I))
f = x**4 + 2 * x**2
assert f.factor_list() == (1, [(x, 2), (x**2 + 2, 1)])
R, x = ring('x', RR)
assert (1.0 * x**2 + 2.0 * x + 1.0).factor_list() == (1.0, [(1.0 * x + 1.0,
2)])
assert (2.0 * x**2 + 4.0 * x + 2.0).factor_list() == (2.0, [(1.0 * x + 1.0,
2)])
f = 6.7225336055071 * x**2 - 10.6463972754741 * x - 0.33469524022264
assert f.factor_list() == (1.0, [(f, 1)])
# issue diofant/diofant#238
f = 0.1 * x**2 + 1.1 * x + 1.0
assert f.factor_list() == (10.0, [(0.1 * x + 0.1, 1), (0.1 * x + 1.0, 1)])
f = 0.25 + 1.0 * x + 1.0 * x**2
assert f.factor_list() == (4.0, [(0.25 + 0.5 * x, 2)])
Rt, t = ring('t', ZZ)
R, x = ring('x', Rt)
assert R(0).factor_list() == (0, [])
assert R(7).factor_list() == (7, [])
assert (4 * t * x**2 + 4 * t**2 * x).factor_list() == (4 * t, [(x, 1),
(x + t, 1)])
Rt, t = ring('t', QQ)
R, x = ring('x', Rt)
assert R(0).factor_list() == (0, [])
assert R(QQ(1, 7)).factor_list() == (QQ(1, 7), [])
assert (t * x**2 / 2 + t**2 * x / 2).factor_list() == (t / 2, [(x, 1),
(x + t, 1)])
R, x = ring('x', EX)
pytest.raises(DomainError, lambda: R(EX(sin(1))).factor_list())
R, x, y = ring('x y', FF(2))
pytest.raises(NotImplementedError, lambda: (x**2 + y**2).factor_list())
R, x, y = ring('x y', ZZ)
assert R(0).factor_list() == (0, [])
assert R(7).factor_list() == (7, [])
assert (x**2 + 2 * x + 1).factor_list() == (1, [(x + 1, 2)])
assert (4 * x**2 * y + 4 * x * y**2).factor_list() == (4, [(y, 1), (x, 1),
(x + y, 1)])
R, x, y = ring('x y', QQ)
assert R(0).factor_list() == (0, [])
assert R(QQ(1, 7)).factor_list() == (QQ(1, 7), [])
assert (x**2 / 2 + x + QQ(1, 2)).factor_list() == (QQ(1, 2), [(x + 1, 2)])
assert (x**2 * y / 2 + x * y**2 / 2).factor_list() == (QQ(1,
2), [(y, 1),
(x, 1),
(x + y, 1)])
R, x, y = ring('x y', QQ.algebraic_field(I))
f, r = x**2 + y**2, (1, [(x - I * y, 1), (x + I * y, 1)])
for method in ('trager', 'modular'):
with using(aa_factor_method=method):
assert f.factor_list() == r
R, x, y = ring('x y', RR)
f = 2.0 * x**2 - 8.0 * y**2
assert f.factor_list() == (2.0, [(1.0 * x - 2.0 * y, 1),
(1.0 * x + 2.0 * y, 1)])
f = 6.7225336055071 * x**2 * y**2 - 10.6463972754741 * x * y - 0.33469524022264
assert f.factor_list() == (1.0, [(f, 1)])
Rt, t = ring('t', ZZ)
R, x, y = ring('x y', Rt)
assert R(0).factor_list() == (0, [])
assert R(7).factor_list() == (7, [])
assert (4 * t * x**2 + 4 * t**2 * x).factor_list() == (4 * t, [(x, 1),
(x + t, 1)])
Rt, t = ring('t', QQ)
R, x, y = ring('x y', Rt)
assert R(0).factor_list() == (0, [])
assert R(QQ(1, 7)).factor_list() == (QQ(1, 7), [])
assert (t * x**2 / 2 + t**2 * x / 2).factor_list() == (t / 2, [(x, 1),
(x + t, 1)])
R, x, y = ring('x y', EX)
pytest.raises(DomainError, lambda: R(EX(sin(1))).factor_list())
# issue diofant/diofant#238
R, x, y, z = ring('x y z', RR)
f = x * y + x * z + 0.1 * y + 0.1 * z
assert f.factor_list() == (10.0, [(x + 0.1, 1), (0.1 * y + 0.1 * z, 1)])
f = 0.25 * x**2 + 1.0 * x * y * z + 1.0 * y**2 * z**2
assert f.factor_list() == (4.0, [(0.25 * x + 0.5 * y * z, 2)])
R, *X = ring('x:200', ZZ)
f, g = X[0]**2 + 2 * X[0] + 1, X[0] + 1
assert f.factor_list() == (1, [(g, 2)])
f, g = X[-1]**2 + 2 * X[-1] + 1, X[-1] + 1
assert f.factor_list() == (1, [(g, 2)])
def test_dup_zz_factor():
R, x = ring("x", ZZ)
assert R.dup_zz_factor(0) == (0, [])
assert R.dup_zz_factor(7) == (7, [])
assert R.dup_zz_factor(-7) == (-7, [])
assert R.dup_zz_factor_sqf(0) == (0, [])
assert R.dup_zz_factor_sqf(7) == (7, [])
assert R.dup_zz_factor_sqf(-7) == (-7, [])
assert R.dup_zz_factor(2*x + 4) == (2, [(x + 2, 1)])
assert R.dup_zz_factor_sqf(2*x + 4) == (2, [x + 2])
f = x**4 + x + 1
for i in range(20):
assert R.dup_zz_factor(f) == (1, [(f, 1)])
assert R.dup_zz_factor(x**2 + 2*x + 2) == (1, [(x**2 + 2*x + 2, 1)])
with config.using(use_irreducible_in_factor=True):
assert R.dup_zz_factor(x**2 + 2*x + 2) == (1, [(x**2 + 2*x + 2, 1)])
assert R.dup_zz_factor(18*x**2 + 12*x + 2) == (2, [(3*x + 1, 2)])
with config.using(use_irreducible_in_factor=True):
assert R.dup_zz_factor(18*x**2 + 12*x + 2) == (2, [(3*x + 1, 2)])
assert R.dup_zz_factor(-9*x**2 + 1) == \
(-1, [(3*x - 1, 1),
(3*x + 1, 1)])
with config.using(use_irreducible_in_factor=True):
assert R.dup_zz_factor_sqf(3*x**4 + 2*x**3 +
6*x**2 + 8*x + 10) == (1, [3*x**4 + 2*x**3 +
6*x**2 + 8*x + 10])
assert R.dup_zz_factor_sqf(-9*x**2 + 1) == (-1, [3*x - 1, 3*x + 1])
with config.using(use_irreducible_in_factor=True):
assert R.dup_zz_factor_sqf(-9*x**2 + 1) == (-1, [3*x - 1, 3*x + 1])
with config.using(use_cyclotomic_factor=False):
assert R.dup_zz_factor_sqf(-9*x**2 + 1) == (-1, [3*x - 1, 3*x + 1])
assert R.dup_zz_factor(x**3 - 6*x**2 + 11*x - 6) == \
(1, [(x - 3, 1),
(x - 2, 1),
(x - 1, 1)])
assert R.dup_zz_factor_sqf(x**3 - 6*x**2 + 11*x - 6) == \
(1, [x - 3,
x - 2,
x - 1])
assert R.dup_zz_factor(3*x**3 + 10*x**2 + 13*x + 10) == \
(1, [(x + 2, 1),
(3*x**2 + 4*x + 5, 1)])
assert R.dup_zz_factor_sqf(3*x**3 + 10*x**2 + 13*x + 10) == \
(1, [x + 2,
3*x**2 + 4*x + 5])
assert R.dup_zz_factor(-x**6 + x**2) == \
(-1, [(x - 1, 1),
(x + 1, 1),
(x, 2),
(x**2 + 1, 1)])
f = 1080*x**8 + 5184*x**7 + 2099*x**6 + 744*x**5 + 2736*x**4 - 648*x**3 + 129*x**2 - 324
assert R.dup_zz_factor(f) == \
(1, [(5*x**4 + 24*x**3 + 9*x**2 + 12, 1),
(216*x**4 + 31*x**2 - 27, 1)])
f = -29802322387695312500000000000000000000*x**25 \
+ 2980232238769531250000000000000000*x**20 \
+ 1743435859680175781250000000000*x**15 \
+ 114142894744873046875000000*x**10 \
- 210106372833251953125*x**5 \
+ 95367431640625
assert R.dup_zz_factor(f) == \
(-95367431640625, [(5*x - 1, 1),
(100*x**2 + 10*x - 1, 2),
(625*x**4 + 125*x**3 + 25*x**2 + 5*x + 1, 1),
(10000*x**4 - 3000*x**3 + 400*x**2 - 20*x + 1, 2),
(10000*x**4 + 2000*x**3 + 400*x**2 + 30*x + 1, 2)])
f = x**10 - 1
config.setup('USE_CYCLOTOMIC_FACTOR', True)
F_0 = R.dup_zz_factor(f)
config.setup('USE_CYCLOTOMIC_FACTOR', False)
F_1 = R.dup_zz_factor(f)
assert F_0 == F_1 == \
(1, [(x - 1, 1),
(x + 1, 1),
(x**4 - x**3 + x**2 - x + 1, 1),
(x**4 + x**3 + x**2 + x + 1, 1)])
config.setup('USE_CYCLOTOMIC_FACTOR')
f = x**10 + 1
config.setup('USE_CYCLOTOMIC_FACTOR', True)
F_0 = R.dup_zz_factor(f)
config.setup('USE_CYCLOTOMIC_FACTOR', False)
F_1 = R.dup_zz_factor(f)
assert F_0 == F_1 == \
(1, [(x**2 + 1, 1),
(x**8 - x**6 + x**4 - x**2 + 1, 1)])
config.setup('USE_CYCLOTOMIC_FACTOR')
def test_benchmark_minpoly_f5b():
with config.using(groebner='f5b'):
_do_test_benchmark_minpoly()
def test_benchmark_minpoly_buchberger():
with config.using(groebner='buchberger'):
_do_test_benchmark_minpoly()
def test_groebner_f5b():
with config.using(groebner='f5b'):
_do_test_groebner()
def test_modgcd_univariate_integers():
R, x = ring('x', ZZ)
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
f, g = R.zero, R.zero
assert f.cofactors(g) == (0, 0, 0)
f, g = R.zero, x
assert f.cofactors(g) == (x, 0, 1)
assert g.cofactors(f) == (x, 1, 0)
f, g = R.zero, -x
assert f.cofactors(g) == (x, 0, -1)
assert g.cofactors(f) == (x, -1, 0)
f, g = 2 * x, R(2)
assert f.cofactors(g) == (2, x, 1)
f, g = 2 * x + 2, 6 * x**2 - 6
assert f.cofactors(g) == (2 * x + 2, 1, 3 * x - 3)
f, g = [1000000000000 * x + 998549000000] * 2
assert f.cofactors(g) == (f, 1, 1)
f, g = 999530000000 * x + 1000000000000, 999530000000 * x + 999999000000
assert f.cofactors(g) == (1000000, 999530 * x + 1000000,
999530 * x + 999999)
f = x**4 + 8 * x**3 + 21 * x**2 + 22 * x + 8
g = x**3 + 6 * x**2 + 11 * x + 6
assert f.cofactors(g) == (x**2 + 3 * x + 2, x**2 + 5 * x + 4, x + 3)
f = x**4 - 4
g = x**4 + 4 * x**2 + 4
assert f.cofactors(g) == (x**2 + 2, x**2 - 2, x**2 + 2)
f = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5
g = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21
assert f.cofactors(g) == (1, f, g)
f = (
-352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272
* x**49 +
46818041807522713962450042363465092040687472354933295397472942006618953623327997952
* x**42 +
378182690892293941192071663536490788434899030680411695933646320291525827756032
* x**35 +
112806468807371824947796775491032386836656074179286744191026149539708928
* x**28 -
12278371209708240950316872681744825481125965781519138077173235712 *
x**21 +
289127344604779611146960547954288113529690984687482920704 * x**14 +
19007977035740498977629742919480623972236450681 * x**7 +
311973482284542371301330321821976049)
g = (365431878023781158602430064717380211405897160759702125019136 *
x**21 +
197599133478719444145775798221171663643171734081650688 * x**14 -
9504116979659010018253915765478924103928886144 * x**7 -
311973482284542371301330321821976049)
assert f.gcd(f.diff(x)) == g
f = 1317378933230047068160 * x + 2945748836994210856960
g = 120352542776360960 * x + 269116466014453760
assert f.cofactors(g) == (120352542776360960 * x + 269116466014453760,
10946, 1)
def test_dup_zz_factor():
R, x = ring('x', ZZ)
assert R(0).factor_list() == (0, [])
assert R(7).factor_list() == (7, [])
assert R(-7).factor_list() == (-7, [])
assert R._zz_factor_sqf(R(0)) == (0, [])
assert R._zz_factor_sqf(R(7)) == (7, [])
assert R._zz_factor_sqf(R(-7)) == (-7, [])
assert (2 * x + 4).factor_list() == (2, [(x + 2, 1)])
assert R._zz_factor_sqf(2 * x + 4) == (2, [x + 2])
f = x**4 + x + 1
for i in range(20):
assert f.factor_list() == (1, [(f, 1)])
f = x**5 - x**3 - x**2 + 1
assert f.factor_list() == (1, [(x + 1, 1), (x - 1, 2), (x**2 + x + 1, 1)])
for test in (True, False):
with using(use_irreducible_in_factor=test):
assert (x**2 + 2 * x + 2).factor_list() == (1, [(x**2 + 2 * x + 2,
1)])
assert (18 * x**2 + 12 * x + 2).factor_list() == (2, [(3 * x + 1,
2)])
f = -9 * x**2 + 1
assert R._zz_factor_sqf(f) == (-1, [3 * x - 1, 3 * x + 1])
assert f.factor_list() == (-1, [(3 * x - 1, 1), (3 * x + 1, 1)])
assert R._zz_factor_sqf(3 * x**4 + 2 * x**3 + 6 * x**2 + 8 * x +
10) == (1, [
3 * x**4 + 2 * x**3 + 6 * x**2 +
8 * x + 10
])
with using(use_cyclotomic_factor=False):
assert R._zz_factor_sqf(-9 * x**2 + 1) == (-1, [3 * x - 1, 3 * x + 1])
assert (x**3 - 6 * x**2 + 11 * x - 6).factor_list() == (1, [(x - 3, 1),
(x - 2, 1),
(x - 1, 1)])
assert R._zz_factor_sqf(x**3 - 6 * x**2 + 11 * x -
6) == (1, [x - 3, x - 2, x - 1])
assert (3 * x**3 + 10 * x**2 + 13 * x + 10).factor_list() == (1, [
(x + 2, 1), (3 * x**2 + 4 * x + 5, 1)
])
assert R._zz_factor_sqf(3 * x**3 + 10 * x**2 + 13 * x +
10) == (1, [x + 2, 3 * x**2 + 4 * x + 5])
assert (-x**6 + x**2).factor_list() == (-1, [(x, 2), (x - 1, 1),
(x + 1, 1), (x**2 + 1, 1)])
f = (1080 * x**8 + 5184 * x**7 + 2099 * x**6 + 744 * x**5 + 2736 * x**4 -
648 * x**3 + 129 * x**2 - 324)
assert f.factor_list() == (1, [(216 * x**4 + 31 * x**2 - 27, 1),
(5 * x**4 + 24 * x**3 + 9 * x**2 + 12, 1)])
f = (-29802322387695312500000000000000000000 * x**25 +
2980232238769531250000000000000000 * x**20 +
1743435859680175781250000000000 * x**15 +
114142894744873046875000000 * x**10 - 210106372833251953125 * x**5 +
+95367431640625)
assert (f.factor_list() == (-95367431640625, [
(5 * x - 1, 1), (100 * x**2 + 10 * x - 1, 2),
(625 * x**4 + 125 * x**3 + 25 * x**2 + 5 * x + 1, 1),
(10000 * x**4 - 3000 * x**3 + 400 * x**2 - 20 * x + 1, 2),
(10000 * x**4 + 2000 * x**3 + 400 * x**2 + 30 * x + 1, 2)
]))
f = x**10 - 1
for test in (True, False):
with using(use_cyclotomic_factor=test):
f = x**10 - 1
assert f.factor_list() == (1, [(x - 1, 1), (x + 1, 1),
(x**4 - x**3 + x**2 - x + 1, 1),
(x**4 + x**3 + x**2 + x + 1, 1)])
f = x**10 + 1
assert f.factor_list() == (1, [(x**2 + 1, 1),
(x**8 - x**6 + x**4 - x**2 + 1, 1)])
def test_benchmark_cyclic_4_buchberger():
with config.using(groebner='buchberger'):
_do_test_benchmark_cyclic_4()
def test_benchmark_kastura_4_f5b():
with config.using(groebner='f5b'):
_do_test_benchmark_katsura_4()
def test_dmp_gcd():
R, x, y = ring("x,y", ZZ)
f, g = 0, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (0, 0, 0)
f, g = 2, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 0)
f, g = -2, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 0)
f, g = 0, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 0, -1)
f, g = 0, 2 * x + 4
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f,
g) == (2 * x + 4, 0, 1)
f, g = 2 * x + 4, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f,
g) == (2 * x + 4, 1, 0)
f, g = 2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 1)
f, g = -2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 1)
f, g = 2, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, -1)
f, g = -2, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, -1)
f, g = x**2 + 2 * x + 1, 1
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(
f, g) == (1, x**2 + 2 * x + 1, 1)
f, g = x**2 + 2 * x + 1, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(
f, g) == (1, x**2 + 2 * x + 1, 2)
with using(use_simplify_gcd=0):
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(
f, g) == (1, x**2 + 2 * x + 1, 2)
f, g = 2 * x**2 + 4 * x + 2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(
f, g) == (2, x**2 + 2 * x + 1, 1)
f, g = 2, 2 * x**2 + 4 * x + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(
f, g) == (2, 1, x**2 + 2 * x + 1)
f, g = 2 * x**2 + 4 * x + 2, x + 1
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(
f, g) == (x + 1, 2 * x + 2, 1)
f, g = x + 1, 2 * x**2 + 4 * x + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(
f, g) == (x + 1, 1, 2 * x + 2)
with using(heu_gcd_max=0):
pytest.raises(HeuristicGCDFailed, lambda: R.dmp_zz_heu_gcd(f, g))
R, x, y, z, u = ring("x,y,z,u", ZZ)
f, g = u**2 + 2 * u + 1, 2 * u + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f,
g) == (u + 1, u + 1, 2)
f, g = z**2 * u**2 + 2 * z**2 * u + z**2 + z * u + z, u**2 + 2 * u + 1
h, cff, cfg = u + 1, z**2 * u + z**2 + z, u + 1
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
assert R.dmp_rr_prs_gcd(f, g) == (h, cff, cfg)
assert R.dmp_zz_heu_gcd(g, f) == (h, cfg, cff)
assert R.dmp_rr_prs_gcd(g, f) == (h, cfg, cff)
R, x, y, z = ring("x,y,z", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(2, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(4, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(6, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(8, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z = ring("x,y,z", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_2(2, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(2, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(4, ZZ))
H, cff, cfg = R.dmp_inner_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y = ring("x,y", QQ)
f = x**2 / 2 + x + QQ(1, 2)
g = x / 2 + QQ(1, 2)
h = x + 1
assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dmp_ff_prs_gcd(f, g) == (h, g, QQ(1, 2))
with using(use_simplify_gcd=0):
assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dmp_ff_prs_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dmp_ff_prs_gcd(R.zero, R.zero) == (0, 0, 0)
assert R.dmp_qq_heu_gcd(R.zero, R.zero) == (0, 0, 0)
assert R.dmp_ff_prs_gcd(R.zero, g) == (x + 1, R.zero, QQ(1, 2))
assert R.dmp_qq_heu_gcd(R.zero, g) == (x + 1, R.zero, QQ(1, 2))
R, x, y = ring("x,y", RR)
f = 2.1 * x * y**2 - 2.2 * x * y + 2.1 * x
g = 1.0 * x**3
assert R.dmp_ff_prs_gcd(f, g) == \
(1.0*x, 2.1*y**2 - 2.2*y + 2.1, 1.0*x**2)
R, x, y = ring("x,y", ZZ)
f = (
-17434367009167300000000000000000000000000000000000000000000000000000000
* x**4 * y -
250501827896299135568887342575961783764139560000000000000000000000000000000000000000000
* x**3 * y -
2440935909299672540738135183426056447877858000000000000000000000000000000
* x**3 -
1349729941723537919695626818065131519270095220127010623905326719279566297660000000000000000000000000000
* x**2 * y -
26304033868956978374552886858060487282904504027042515077682955951658838800000000000000000
* x**2 -
3232215785736369696036755035364398565076440134133908303058376297547504030528179314849416971379040931276000000000000000
* x * y -
94485916261760032526508027937078714464844205539023800247528621905831259414691631156161537919255129011800
* x -
2902585888465621357542575571971656665554321652262249362701116665830760628936600958940851960635161420991047110815678789984677193092993
* y -
113133324167442997472440652189550843502029192913459268196939183295294085146407870078840385860571627108778756267503630290
)
g = (10000000000000000000000000000 * x**2 +
71841388839807267676152024786000000000000000 * x +
129029628760809605749020969023932901278290735413660734705971)
assert (R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (
g, -1743436700916730000000000000000000000000000 * x**2 * y -
12525091394814956778444367128798089188206978000000000000000 * x * y -
244093590929967254073813518342605644787785800 * x -
22495499028725631994927113634418779135935898997901327211111875586270479483
* y - 876801128965234839118530545935732755107147297241756982389990, 1))
def test_gf_factor():
R, x = ring('x', FF(2))
f = x**4 + x
g = (1, [(x, 1), (x + 1, 1), (x**2 + x + 1, 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = x**18 + x**17 + x**16 + x**14 + x**12 + x**11 + x**8 + x**5 + x**3 + 1
g = (1, [(x + 1, 4), (x**4 + x**3 + 1, 1),
(x**10 + x**8 + x**7 + x**5 + 1, 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = x**63 + 1
g = (1, [(x + 1, 1), (x**2 + x + 1, 1),
(x**3 + x + 1, 1), (x**6 + x + 1, 1), (x**3 + x**2 + 1, 1),
(x**6 + x**3 + 1, 1), (x**6 + x**5 + 1, 1),
(x**6 + x**4 + x**2 + x + 1, 1), (x**6 + x**5 + x**2 + x + 1, 1),
(x**6 + x**4 + x**3 + x + 1, 1), (x**6 + x**5 + x**4 + x + 1, 1),
(x**6 + x**5 + x**3 + x**2 + 1, 1),
(x**6 + x**5 + x**4 + x**2 + 1, 1)])
for method in ('zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = (x**28 + x**27 + x**26 + x**25 + x**24 + x**20 + x**19 + x**17 +
x**16 + x**15 + x**14 + x**13 + x**12 + x**11 + x**9 + x**8 + x**5 +
x**4 + x**2 + x)
g = (1, [(x, 1), (x + 1, 2), (x**5 + x**4 + x**3 + x + 1, 1),
(x**10 + x**9 + x**8 + x**7 + 1, 1),
(x**10 + x**9 + x**8 + x**5 + x**4 + x**2 + 1, 1)])
for method in ('zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
R, x = ring('x', FF(3))
f = x**6 - x**5 + x**4 + x**3 - x
g = (1, [(x, 1), (x + 1, 1), (x**2 + 1, 1), (x**2 + x + 2, 1)])
for method in ('zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = x**4 + x**3 + x + 2
g = (1, [(x**2 + 1, 1), (x**2 + x + 2, 1)])
for method in ('zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
R, x = ring('x', FF(11))
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert R(0).factor_list() == (0, [])
assert R(1).factor_list() == (1, [])
assert x.factor_list() == (1, [(x, 1)])
assert (x + 1).factor_list() == (1, [(x + 1, 1)])
assert (2 * x + 3).factor_list() == (2, [(x + 7, 1)])
assert (5 * x**3 + 2 * x**2 + 7 * x + 2).factor_list() == (5, [(x + 2, 1),
(x + 8, 2)])
f = x**6 + 8 * x**5 + x**4 + 8 * x**3 + 10 * x**2 + 8 * x + 1
g = (1, [(x + 1, 1), (x**2 + 5 * x + 3, 1),
(x**3 + 2 * x**2 + 3 * x + 4, 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = x**3 + 5 * x**2 + 8 * x + 4
g = (1, [(x + 1, 1), (x + 2, 2)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = x**9 + x**8 + 10 * x**7 + x**6 + 10 * x**4 + 10 * x**3 + 10 * x**2
g = (1, [(x, 2), (x**2 + 9 * x + 5, 1),
(x**5 + 3 * x**4 + 8 * x**2 + 5 * x + 2, 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = x**32 + 1
g = (1, [(x**16 + 3 * x**8 + 10, 1), (x**16 + 8 * x**8 + 10, 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = 8 * x**32 + 5
g = (8, [(x + 3, 1), (x + 8, 1), (x**2 + 9, 1), (x**2 + 2 * x + 2, 1),
(x**2 + 9 * x + 2, 1), (x**8 + x**4 + 6, 1),
(x**8 + 10 * x**4 + 6, 1), (x**4 + 5 * x**2 + 7, 1),
(x**4 + 6 * x**2 + 7, 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = 8 * x**63 + 5
g = (8,
[(x + 7, 1), (x**6 + 9 * x**3 + 4, 1), (x**2 + 4 * x + 5, 1),
(x**3 + 6 * x**2 + 8 * x + 2, 1), (x**3 + 9 * x**2 + 9 * x + 2, 1),
(x**6 + 2 * x**5 + 6 * x**4 + 8 * x**2 + 4 * x + 4, 1),
(x**6 + 2 * x**5 + 8 * x**3 + 4 * x**2 + 6 * x + 4, 1),
(x**6 + 5 * x**5 + 6 * x**4 + 8 * x**2 + 6 * x + 4, 1),
(x**6 + 2 * x**5 + 3 * x**4 + 8 * x**3 + 6 * x + 4, 1),
(x**6 + 10 * x**5 + 4 * x**4 + 7 * x**3 + 10 * x**2 + 7 * x + 4, 1),
(x**6 + 3 * x**5 + 3 * x**4 + x**3 + 6 * x**2 + 8 * x + 4, 1),
(x**6 + 6 * x**5 + 2 * x**4 + 7 * x**3 + 9 * x**2 + 8 * x + 4, 1),
(x**6 + 10 * x**5 + 10 * x**4 + x**3 + 4 * x**2 + 9 * x + 4, 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
f = x**15 - 1
g = (1, [(x + 2, 1), (x + 6, 1), (x + 7, 1), (x + 8, 1), (x + 10, 1),
(x**2 + x + 1, 1), (x**2 + 5 * x + 3, 1), (x**2 + 9 * x + 4, 1),
(x**2 + 4 * x + 5, 1), (x**2 + 3 * x + 9, 1)])
for method in ('zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
with using(gf_factor_method='other'):
pytest.raises(KeyError, lambda: (x + 1).factor_list())
R, x = ring('x', FF(13))
f = x**8 + x**6 + 10 * x**4 + 10 * x**3 + 8 * x**2 + 2 * x + 8
g = (1, [(x + 3, 1), (x**3 + 8 * x**2 + 4 * x + 12, 1),
(x**4 + 2 * x**3 + 3 * x**2 + 4 * x + 6, 1)])
with using(gf_factor_method='berlekamp'):
assert f.factor_list() == g
R, x = ring('x', FF(809))
f = (x**10 + 2 * x**9 + 5 * x**8 + 26 * x**7 + 677 * x**6 + 436 * x**5 +
791 * x**4 + 325 * x**3 + 456 * x**2 + 24 * x + 577)
g = (1, [(x + 701, 1),
(x**9 + 110 * x**8 + 559 * x**7 + 532 * x**6 + 694 * x**5 +
151 * x**4 + 110 * x**3 + 70 * x**2 + 735 * x + 122, 1)])
for method in ('zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
# Gathen polynomials: x**n + x + 1 (mod p > 2**n * pi)
R, x = ring('x', FF(nextprime(2**15 * pi)))
f = x**15 + x + 1
g = (1, [
(x**2 + 22730 * x + 68144, 1),
(x**4 + 81553 * x**3 + 77449 * x**2 + 86810 * x + 4724, 1),
(x**4 + 86276 * x**3 + 56779 * x**2 + 14859 * x + 31575, 1),
(x**5 + 15347 * x**4 + 95022 * x**3 + 84569 * x**2 + 94508 * x + 92335,
1)
])
for method in ('zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
# Shoup polynomials: f = a_0 x**n + a_1 x**(n-1) + ... + a_n
# (mod p > 2**(n-2) * pi), where a_n = a_{n-1}**2 + 1, a_0 = 1
R, x = ring('x', FF(nextprime(2**4 * pi)))
f = x**6 + 2 * x**5 + 5 * x**4 + 26 * x**3 + 41 * x**2 + 39 * x + 38
g = (1, [(x**2 + 44 * x + 26, 1),
(x**4 + 11 * x**3 + 25 * x**2 + 18 * x + 30, 1)])
for method in ('zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
F8 = FF(2, [1, 1, 0, 1])
R, x = ring('x', F8)
f = x**10 + x**9 + F8(2) * x**8 + F8(2) * x**7 + F8(5) * x**6 + F8(
3) * x**5
g = (F8(1), [(x, 5), (x + F8(3), 1), (x + F8(6), 1),
(x**3 + F8(4) * x**2 + x + F8(3), 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
F9 = FF(3, [2, 2, 1])
R, x = ring('x', F9)
f = x**5 + F9(2) * x**4 + F9(6) * x**3 + F9(8) * x**2 + F9(5) * x + F9(4)
g = (1, [(x + F9(8), 1), (x**2 + 2 * x + F9(4), 1),
(x**2 + F9(4) * x + F9(4), 1)])
for method in ('berlekamp', 'zassenhaus', 'shoup'):
with using(gf_factor_method=method):
assert f.factor_list() == g
def test_benchmark_czichowski_buchberger():
with config.using(groebner='buchberger'):
_do_test_benchmark_czichowski()
def test_benchmark_kastura_4_buchberger():
with config.using(groebner='buchberger'):
_do_test_benchmark_katsura_4()
def test_benchmark_czichowski_f5b():
with config.using(groebner='f5b'):
_do_test_benchmark_czichowski()
