def test_benchmark_cyclic_4(method):
with 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_benchmark_katsura_4(method):
with 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_sympyissue_23174():
_, x = ring('x', FF(2))
f = (x**16 + x**15 + x**14 + x**13 + x**12 + x**11 + x**10 + x**9 + x**8 +
x**7 + x**6 + x**5 + x**4 + x**3 + x**2 + x + 1)
r = (1, [(x**8 + x**5 + x**4 + x**3 + 1, 1),
(x**8 + x**7 + x**6 + x**4 + x**2 + x + 1, 1)])
with using(gf_factor_method='zassenhaus'):
assert f.factor_list() == r
def test_benchmark_minimal_polynomial(method):
with 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_sympyissue_16620():
_, 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_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 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', FF(5))
f = 3 * x**2 + 2 * x + 4
g = 2 * x**2 + 2 * x + 3
assert f.cofactors(g) == (x + 3, 3 * x + 3, 2 * x + 1)
R, x = ring('x', FF(11))
assert R(0).cofactors(R(0)) == (0, 0, 0)
assert R(2).cofactors(R(0)) == (1, 2, 0)
assert R(0).cofactors(R(2)) == (1, 0, 2)
assert R(2).cofactors(R(2)) == (1, 2, 2)
assert R(0).cofactors(x) == (x, 0, 1)
assert x.cofactors(R(0)) == (x, 1, 0)
assert (3 * x).cofactors(3 * x) == (x, 3, 3)
assert (x**2 + 8 * x + 7).cofactors(x**3 + 7 * x**2 + x + 7) == (x + 7,
x + 1,
x**2 + 1)
R, x = ring('x', ZZ)
for test in (True, False):
for method in ('prs', 'modgcd'):
with using(use_heu_gcd=test, fallback_gcd_zz_method=method):
assert R(0).cofactors(R(0)) == (0, 0, 0)
assert R(0).cofactors(x) == (x, 0, 1)
assert x.cofactors(R(0)) == (x, 1, 0)
assert R(0).cofactors(-x) == (x, 0, -1)
assert (-x).cofactors(R(0)) == (x, -1, 0)
assert (2 * x).cofactors(R(2)) == (2, x, 1)
assert R(2).cofactors(R(0)) == (2, 1, 0)
assert R(-2).cofactors(R(0)) == (2, -1, 0)
assert R(0).cofactors(R(-2)) == (2, 0, -1)
assert R(0).cofactors(2 * x + 4) == (2 * x + 4, 0, 1)
assert (2 * x + 4).cofactors(R(0)) == (2 * x + 4, 1, 0)
assert R(2).cofactors(R(2)) == (2, 1, 1)
assert R(-2).cofactors(R(2)) == (2, -1, 1)
assert R(2).cofactors(R(-2)) == (2, 1, -1)
assert R(-2).cofactors(R(-2)) == (2, -1, -1)
assert (x**2 + 2 * x + 1).cofactors(R(1)) == (1,
x**2 + 2 * x + 1,
1)
assert (x**2 + 2 * x + 1).cofactors(R(2)) == (1,
x**2 + 2 * x + 1,
2)
assert (2 * x**2 + 4 * x + 2).cofactors(R(2)) == (2, x**2 +
2 * x + 1, 1)
assert R(2).cofactors(2 * x**2 + 4 * x + 2) == (2, 1, x**2 +
2 * x + 1)
assert (2 * x**2 + 4 * x + 2).cofactors(x + 1) == (x + 1,
2 * x + 2,
1)
assert (x + 1).cofactors(2 * x**2 + 4 * x + 2) == (x + 1, 1,
2 * x + 2)
assert (x - 31).cofactors(x) == (1, x - 31, x)
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**2 - 1
g = x**2 - 3 * x + 2
assert f.cofactors(g) == (x - 1, x + 1, x - 2)
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)
h = (
365431878023781158602430064717380211405897160759702125019136
* x**21 +
197599133478719444145775798221171663643171734081650688 *
x**14 -
9504116979659010018253915765478924103928886144 * x**7 -
311973482284542371301330321821976049)
cff = (-964661685087874498642420170752 * x**28 +
649736296036977287118848 * x**21 +
658473216967637120 * x**14 - 30463679113 * x**7 - 1)
cfg = (-47268422569305850433478588366848 * x**27 +
30940259392972115602096128 * x**20 +
18261628279718027904 * x**13 - 426497272383 * x**6)
assert f.cofactors(f.diff()) == (h, cff, cfg)
f = 1317378933230047068160 * x + 2945748836994210856960
g = 120352542776360960 * x + 269116466014453760
H, cff, cfg = 120352542776360960 * x + 269116466014453760, 10946, 1
assert f.cofactors(g) == (H, cff, cfg)
with using(heu_gcd_max=0):
assert f.cofactors(g) == (H, cff, cfg)
R, x = ring('x', QQ)
for test in (True, False):
with using(use_heu_gcd=test, fallback_gcd_zz_method='prs'):
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)
assert R(0).cofactors(R(0)) == (0, 0, 0)
f, g = x**2 / 2 + x + QQ(1, 2), x / 2 + QQ(1, 2)
assert f.cofactors(g) == (x + 1, g, QQ(1, 2))
f, g = x**2 - 1, x**2 - 3 * x + 2
assert f.cofactors(g) == (x - 1, x + 1, x - 2)
R, x = ring('x', QQ.algebraic_field(sqrt(2)))
for method in ('modgcd', 'prs'):
with using(gcd_aa_method=method):
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 = ring('x', QQ.algebraic_field(sqrt(2)**(-1) * sqrt(3)))
for method in ('modgcd', 'prs'):
with using(gcd_aa_method=method):
f, g = x + 1, x - 1
assert f.cofactors(g) == (1, f, g)
R, x = ring('x', CC)
f, g = x**2 - 1, x**3 - 3 * x + 2
assert f.cofactors(g) == (x - 1, x + 1, x**2 + x - 2)
R, x, y = ring('x y', ZZ)
for test in (True, False):
for method in ('prs', 'modgcd'):
with using(use_heu_gcd=test, fallback_gcd_zz_method=method):
assert R(0).cofactors(R(0)) == (0, 0, 0)
assert R(2).cofactors(R(0)) == (2, 1, 0)
assert R(-2).cofactors(R(0)) == (2, -1, 0)
assert R(0).cofactors(R(-2)) == (2, 0, -1)
assert R(0).cofactors(2 * x + 4) == (2 * x + 4, 0, 1)
assert (2 * x).cofactors(R(2)) == (2, x, 1)
assert (2 * x + 4).cofactors(R(0)) == (2 * x + 4, 1, 0)
assert R(2).cofactors(R(2)) == (2, 1, 1)
assert R(-2).cofactors(R(2)) == (2, -1, 1)
assert R(2).cofactors(R(-2)) == (2, 1, -1)
assert R(-2).cofactors(R(-2)) == (2, -1, -1)
assert (x**2 + 2 * x + 1).cofactors(R(1)) == (1,
x**2 + 2 * x + 1,
1)
assert (x**2 + 2 * x + 1).cofactors(R(2)) == (1,
x**2 + 2 * x + 1,
2)
assert (2 * x**2 + 4 * x + 2).cofactors(R(2)) == (2, x**2 +
2 * x + 1, 1)
assert R(2).cofactors(2 * x**2 + 4 * x + 2) == (2, 1, x**2 +
2 * x + 1)
f, g = 2 * x**2 + 4 * x + 2, x + 1
assert f.cofactors(g) == (g, 2 * x + 2, 1)
assert g.cofactors(f) == (g, 1, 2 * x + 2)
with using(heu_gcd_max=0):
assert f.cofactors(g) == (g, 2 * x + 2, 1)
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, g = x + 2 * y, x + y
assert f.cofactors(g) == (1, f, g)
f, g = x**2 + 2 * x * y + y**2, x**2 + x * y
assert f.cofactors(g) == (x + y, x + y, x)
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 = 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)
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
)
h = (
-1743436700916730000000000000000000000000000 * x**2 * y -
12525091394814956778444367128798089188206978000000000000000
* x * y -
244093590929967254073813518342605644787785800 * x -
22495499028725631994927113634418779135935898997901327211111875586270479483
* y -
876801128965234839118530545935732755107147297241756982389990
)
assert f.cofactors(g) == (g, h, 1)
R, x, y = ring('x y', QQ)
for test in (True, False):
with using(use_heu_gcd=test, fallback_gcd_zz_method='prs'):
f, g = x**2 / 2 + x + QQ(1, 2), x / 2 + QQ(1, 2)
assert f.cofactors(g) == (x + 1, g, QQ(1, 2))
assert g.cofactors(f) == (x + 1, QQ(1, 2), g)
assert f.gcd(g) == x + 1
with using(fallback_gcd_zz_method='modgcd'):
assert f.gcd(g) == x + 1
assert R(0).cofactors(R(0)) == (0, 0, 0)
assert R(0).cofactors(g) == (x + 1, 0, QQ(1, 2))
f, g = x**2 / 4 + x * y + y**2, x**2 / 2 + x * y
assert f.cofactors(g) == (x + 2 * y, x / 4 + y / 2, x / 2)
f, g = x**2 / 2 + x * y + y**2 / 2, x**2 + x * y
assert f.cofactors(g) == (x + y, x / 2 + y / 2, x)
R, x, y = ring('x y', QQ.algebraic_field(sqrt(2)))
for method in ('modgcd', 'prs'):
with using(gcd_aa_method=method):
f, g = (x + sqrt(2) * y)**2, x + sqrt(2) * y
assert f.cofactors(g) == (g, g, 1)
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)
R, x, y = ring('x y', RR)
for test in (True, False):
with using(use_heu_gcd=test, fallback_gcd_zz_method='prs'):
f, g = 2.1 * x * y**2 - 2.2 * x * y + 2.1 * x, 1.0 * x**3
h = 1.0 * x
assert f.cofactors(g) == (h, 2.1 * y**2 - 2.2 * y + 2.1,
1.0 * x**2)
f, g = 2.1 * x * y**2 - 2.1 * x * y + 2.1 * x, 2.1 * x**3
assert f.cofactors(g) == (h, f // h, g // h)
assert g.cofactors(f) == (h, g // h, f // h)
R, x, y = ring('x y', CC)
f, g = x**2 - y, x**3 - y * x + 2
assert f.cofactors(g) == (1, f, g)
R, x, y, z = ring('x y z', ZZ)
for test in (True, False):
for method in ('prs', 'modgcd'):
with using(use_heu_gcd=test, fallback_gcd_zz_method=method):
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 = ring('x y z', QQ.algebraic_field(sqrt(2), sqrt(3)))
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)
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)
R, x, y, z = ring('x y z', QQ.algebraic_field(I))
for method in ('prs', 'modgcd'):
with using(gcd_aa_method=method):
f, g = R.one, I * z
assert f.cofactors(g) == (1, f, g)
assert g.cofactors(f) == (1, g, f)
R, x, y, z, u = ring('x y z u', ZZ)
for test in (True, False):
for method in ('prs', 'modgcd'):
with using(use_heu_gcd=test, fallback_gcd_zz_method=method):
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 = x + y + z, -x - y - z - u
assert f.cofactors(g) == (1, f, 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
f, g, h = (1199999999999991 * x**17 - y,
2 * y - 19989798798 + x**211,
12 * x * y**7 + x**4 - 1)
for _ in range(10):
assert (f * h).cofactors(g * h) == (h, f, g)
R, x, y, z, u, _ = ring('x y z u v', ZZ)
for test in (True, False):
with using(use_heu_gcd=test, 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
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, _, a, b = ring('x y z u v a b', ZZ)
for test in (True, False):
with using(use_heu_gcd=test, 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, _, a, b, *_ = ring('x y z u v a b c d', ZZ)
for test in (True, False):
with using(use_heu_gcd=test, 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
F, x = field('x', QQ)
R, _ = ring('t', F)
assert R(x).gcd(R(0)) == 1
def test_PolyElement_subresultants():
R, x = ring('x', ZZ)
for check in (True, False):
with using(use_collins_resultant=check):
assert R(0).resultant(R(0)) == 0
assert R(0).resultant(R(0), includePRS=True) == (0, [])
assert R(1).resultant(R(0)) == 0
assert R(1).subresultants(R(0)) == [1]
assert R(0).resultant(R(1)) == 0
assert R(0).resultant(R(1), includePRS=True) == (0, [1])
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 = R(260708)
assert f.subresultants(g) == [f, g, a, b, c, d]
assert f.resultant(g) == d.drop(x)
f = x**2 - 2 * x + 1
g = x**2 - 1
a = 2 * x - 2
assert f.subresultants(g) == [f, g, a]
assert f.resultant(g) == 0
f = x**2 + 1
g = x**2 - 1
a = -2
assert f.subresultants(g) == [f, g, a]
assert f.resultant(g) == 4
assert f.resultant(g,
includePRS=True) == (4,
[x**2 + 1, x**2 - 1, -2])
f = x**2 - 1
g = x**3 - x**2 + 2
assert f.resultant(g) == 0
f = 3 * x**3 - x
g = 5 * x**2 + 1
assert f.resultant(g) == 64
f = x**2 - 2 * x + 7
g = x**3 - x + 5
assert f.resultant(g) == 265
f = x**3 - 6 * x**2 + 11 * x - 6
g = x**3 - 15 * x**2 + 74 * x - 120
assert f.resultant(g) == -8640
f = x**3 - 6 * x**2 + 11 * x - 6
g = x**3 - 10 * x**2 + 29 * x - 20
assert f.resultant(g) == 0
f = x**3 - 1
g = x**3 + 2 * x**2 + 2 * x - 1
assert f.resultant(g) == 16
f = x**8 - 2
g = x - 1
assert f.resultant(g) == -1
# issue sympy/sympy#10666
f = x**3 - 7 * x + 7
g = x
assert f.resultant(g) == -g.resultant(f) == -7
Rt, t = ring('t', ZZ)
_, 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 f.resultant(
g) == 2930944 * t**6 + 2198208 * t**4 + 549552 * t**2 + 45796
assert (x - 1).resultant(x + 1, includePRS=True) == (2, [x - 1, x + 1, 2])
R, x, y = ring('x y', ZZ)
for check in (True, False):
with using(use_collins_resultant=check):
assert R(0).resultant(R(0)) == 0
assert R(0).resultant(R(0), includePRS=True) == (0, [])
assert R(0).resultant(R(1)) == 0
assert R(1).resultant(R(0)) == 0
assert R(1).subresultants(R(0)) == [1]
assert R(0).resultant(R(1), includePRS=True) == (0, [1])
f = x + y + 2
g = 2 * x * y + x + 3
assert f.resultant(g) == (-2 * y**2 - 5 * y + 1).drop(x)
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 = b.drop(x)
rr = (r, [
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 f.subresultants(g) == [f, g, a, b]
assert f.resultant(g) == r
assert f.resultant(g, includePRS=True) == rr
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 = b.drop(x)
assert f.subresultants(g) == [f, g, a]
assert f.resultant(g) == r
assert f.resultant(g, includePRS=True)[0] == r
f = x + y
g = x**2 - x * y + 1
assert f.resultant(g) == (1 + 2 * y**2).drop(x)
g += 1
assert f.resultant(g) == (2 + 2 * y**2).drop(x)
R, x, y = ring('x y', QQ)
for check in (True, False):
with using(use_collins_resultant=check):
assert R(0).resultant(R(0)) == 0
assert R(0).resultant(R(1)) == 0
f = x + y
g = x**2 - x * y + 1
assert f.resultant(g) == (1 + 2 * y**2).drop(x)
f = x / 2 + y + QQ(2, 3)
g = 2 * x * y + x + 3
assert f.resultant(g) == (-2 * y**2 - 7 * y / 3 + QQ(5, 6)).drop(x)
f = 3 * x**2 * y - y**3 - 4
g = x**2 + x * y**3 - 9
assert f.resultant(g) == (-3 * y**10 - 12 * y**7 + y**6 -
54 * y**4 + 8 * y**3 + 729 * y**2 -
216 * y + 16).drop(x)
f = -x**3 + 5
g = 3 * x**2 * y + x**2
assert f.resultant(g) == (675 * y**3 + 675 * y**2 + 225 * y +
25).drop(x)
R, x, y, z, u, v = ring('x y z u v', ZZ)
for check in (True, False):
with using(use_collins_resultant=check):
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 f.resultant(g) == r.drop(x)
R, x, y, z, u, v = ring('x y z u v', QQ)
for check in (True, False):
with using(use_collins_resultant=check):
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 f.resultant(g) == r.drop(x)
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, _ = ring('x y z', QQ)
assert (x**2 + x + 1).is_irreducible
assert (x**2 + 2 * x + 1).is_irreducible is False
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, (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
F4 = FF(2, [1, 1, 1])
R, x = ring('x', F4)
f = x**3 + F4(3) * x + F4(2)
g = (1, [(x + 1, 1), (x**2 + x + F4(2), 1)])
for method in ('berlekamp', '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 = (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
f = x**4 + F8(5) * x**2 + 1
g = (1, [(x + F8(4), 2), (x + F8(7), 2)])
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_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, R(EX(sin(1))).factor_list)
R, x, y = ring('x y', FF(2))
pytest.raises(NotImplementedError, (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__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_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 _ 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_czichowski(method):
# This is very slow (> 2 minutes on 3.4 GHz) without GMPY
with using(groebner=method):
R, x, t = ring('x t', ZZ, lex)
I = [9*x**8 + 36*x**7 - 32*x**6 - 252*x**5 - 78*x**4 + 468*x**3 + 288*x**2 - 108*x + 9,
(-72 - 72*t)*x**7 + (-256 - 252*t)*x**6 + (192 + 192*t)*x**5 + (1280 + 1260*t)*x**4 + (312 + 312*t)*x**3 + (-404*t)*x**2 + (-576 - 576*t)*x + 96 + 108*t]
assert groebner(I, R) == [
3725588592068034903797967297424801242396746870413359539263038139343329273586196480000*x -
160420835591776763325581422211936558925462474417709511019228211783493866564923546661604487873*t**7 -
1406108495478033395547109582678806497509499966197028487131115097902188374051595011248311352864*t**6 -
5241326875850889518164640374668786338033653548841427557880599579174438246266263602956254030352*t**5 -
10758917262823299139373269714910672770004760114329943852726887632013485035262879510837043892416*t**4 -
13119383576444715672578819534846747735372132018341964647712009275306635391456880068261130581248*t**3 -
9491412317016197146080450036267011389660653495578680036574753839055748080962214787557853941760*t**2 -
3767520915562795326943800040277726397326609797172964377014046018280260848046603967211258368000*t -
632314652371226552085897259159210286886724229880266931574701654721512325555116066073245696000,
610733380717522355121*t**8 +
6243748742141230639968*t**7 +
27761407182086143225024*t**6 +
70066148869420956398592*t**5 +
109701225644313784229376*t**4 +
109009005495588442152960*t**3 +
67072101084384786432000*t**2 +
23339979742629593088000*t +
3513592776846090240000,
]
R, x, t = ring('x t', ZZ, grlex)
I = [i.set_ring(R) for i in I]
assert groebner(I, R) == [
16996618586000601590732959134095643086442*t**3*x -
32936701459297092865176560282688198064839*t**3 +
78592411049800639484139414821529525782364*t**2*x -
120753953358671750165454009478961405619916*t**2 +
120988399875140799712152158915653654637280*t*x -
144576390266626470824138354942076045758736*t +
60017634054270480831259316163620768960*x**2 +
61976058033571109604821862786675242894400*x -
56266268491293858791834120380427754600960,
576689018321912327136790519059646508441672750656050290242749*t**4 +
2326673103677477425562248201573604572527893938459296513327336*t**3 +
110743790416688497407826310048520299245819959064297990236000*t**2*x +
3308669114229100853338245486174247752683277925010505284338016*t**2 +
323150205645687941261103426627818874426097912639158572428800*t*x +
1914335199925152083917206349978534224695445819017286960055680*t +
861662882561803377986838989464278045397192862768588480000*x**2 +
235296483281783440197069672204341465480107019878814196672000*x +
361850798943225141738895123621685122544503614946436727532800,
-117584925286448670474763406733005510014188341867*t**3 +
68566565876066068463853874568722190223721653044*t**2*x -
435970731348366266878180788833437896139920683940*t**2 +
196297602447033751918195568051376792491869233408*t*x -
525011527660010557871349062870980202067479780112*t +
517905853447200553360289634770487684447317120*x**3 +
569119014870778921949288951688799397569321920*x**2 +
138877356748142786670127389526667463202210102080*x -
205109210539096046121625447192779783475018619520,
-3725142681462373002731339445216700112264527*t**3 +
583711207282060457652784180668273817487940*t**2*x -
12381382393074485225164741437227437062814908*t**2 +
151081054097783125250959636747516827435040*t*x**2 +
1814103857455163948531448580501928933873280*t*x -
13353115629395094645843682074271212731433648*t +
236415091385250007660606958022544983766080*x**2 +
1390443278862804663728298060085399578417600*x -
4716885828494075789338754454248931750698880,
]
def test_groebner(method):
with using(groebner=method):
R, x, y = ring('x y', QQ, lex)
f = x**2 + 2*x*y**2
g = x*y + 2*y**3 - 1
assert not is_groebner([f, g])
ans = [x, y**3 - QQ(1, 2)]
assert groebner([f, g], R) == ans
assert is_groebner(ans)
assert is_minimal(ans, R)
assert groebner([x, x**2], R) == [x]
assert groebner([x**2, x], R) == [x]
R, x, y = ring('x y', ZZ)
f = x**2*y + y**6 + 1
g = x**2 - 2*x*y
assert not is_groebner([f, g])
ans = [2*x - y**10 - 4*y**7 - y**4 - 4*y,
y**12 + 4*y**9 + 2*y**6 + 4*y**3 + 1]
assert groebner([f, g], R) == ans
assert is_groebner(ans)
assert is_minimal(ans, R) is False
R, y, x = ring('y x', QQ, lex)
f = 2*x**2*y + y**2
g = 2*x**3 + x*y - 1
assert groebner([f, g], R) == [y, x**3 - QQ(1, 2)]
R, x, y, z = ring('x y z', QQ, lex)
f = x - z**2
g = y - z**3
assert groebner([f, g], R) == [f, g]
R, x, y = ring('x y', QQ, grlex)
f = x**3 - 2*x*y
g = x**2*y + x - 2*y**2
assert groebner([f, g], R) == [x**2, x*y, -x/2 + y**2]
R, x, y, z = ring('x y z', QQ, lex)
f = -x**2 + y
g = -x**3 + z
assert groebner([f, g], R) == [x**2 - y, x*y - z, x*z - y**2, y**3 - z**2]
R, x, y, z = ring('x y z', QQ, grlex)
f = -x**2 + y
g = -x**3 + z
assert groebner([f, g], R) == [y**3 - z**2, x**2 - y, x*y - z, x*z - y**2]
R, x, y, z = ring('x y z', QQ, lex)
f = -x**2 + z
g = -x**3 + y
assert groebner([f, g], R) == [x**2 - z, x*y - z**2, x*z - y, y**2 - z**3]
R, x, y, z = ring('x y z', QQ, grlex)
f = -x**2 + z
g = -x**3 + y
assert groebner([f, g], R) == [-y**2 + z**3, x**2 - z, x*y - z**2, x*z - y]
R, x, y, z = ring('x y z', QQ, lex)
f = x - y**2
g = -y**3 + z
assert groebner([f, g], R) == [x - y**2, y**3 - z]
R, x, y, z = ring('x y z', QQ, grlex)
f = x - y**2
g = -y**3 + z
assert groebner([f, g], R) == [x**2 - y*z, x*y - z, -x + y**2]
R, x, y, z = ring('x y z', QQ, lex)
f = x - z**2
g = y - z**3
assert groebner([f, g], R) == [x - z**2, y - z**3]
R, x, y, z = ring('x y z', QQ, grlex)
f = x - z**2
g = y - z**3
assert groebner([f, g], R) == [x**2 - y*z, x*z - y, -x + z**2]
R, x, y, z = ring('x y z', QQ, lex)
f = -y**2 + z
g = x - y**3
assert groebner([f, g], R) == [x - y*z, y**2 - z]
R, x, y, z = ring('x y z', QQ, grlex)
f = -y**2 + z
g = x - y**3
assert groebner([f, g], R) == [-x**2 + z**3, x*y - z**2, y**2 - z, -x + y*z]
R, x, y, z = ring('x y z', QQ, lex)
f = y - z**2
g = x - z**3
assert groebner([f, g], R) == [x - z**3, y - z**2]
R, x, y, z = ring('x y z', QQ, grlex)
f = y - z**2
g = x - z**3
assert groebner([f, g], R) == [-x**2 + y**3, x*z - y**2, -x + y*z, -y + z**2]
R, x, y, z = ring('x y z', QQ, lex)
f = 4*x**2*y**2 + 4*x*y + 1
g = x**2 + y**2 - 1
assert groebner([f, g], R) == [
x - 4*y**7 + 8*y**5 - 7*y**3 + 3*y,
y**8 - 2*y**6 + 3*y**4/2 - y**2/2 + QQ(1, 16),
]
b = [y**2 + x*y + x**2, y + x, y, x**2, x]
assert is_groebner(b)
assert not is_minimal(b, R)