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_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_10996():
R, x, y, z = ring('x,y,z', ZZ)
f = 12 * x**6 * y**7 * z**3 - 3 * x**4 * y**9 * z**3 + 12 * x**3 * y**5 * z**4
g = (-48 * x**7 * y**8 * z**3 + 12 * x**5 * y**10 * z**3 -
48 * x**5 * y**7 * z**2 + 36 * x**4 * y**7 * z -
48 * x**4 * y**6 * z**4 + 12 * x**3 * y**9 * z**2 - 48 * x**3 * y**4 -
9 * x**2 * y**9 * z - 48 * x**2 * y**5 * z**3 + 12 * x * y**6 +
36 * x * y**5 * z**2 - 48 * y**2 * z)
H, cff, cfg = heugcd(f, g)
assert H == 12 * x**3 * y**4 - 3 * x * y**6 + 12 * y**2 * z
assert H * cff == f and H * cfg == g
def test_heugcd_multivariate_integers():
R, x, y = ring("x,y", ZZ)
f, g = 2 * x**2 + 4 * x + 2, x + 1
assert heugcd(f, g) == (x + 1, 2 * x + 2, 1)
f, g = x + 1, 2 * x**2 + 4 * x + 2
assert heugcd(f, g) == (x + 1, 1, 2 * x + 2)
R, x, y, z, u = ring("x,y,z,u", ZZ)
f, g = u**2 + 2 * u + 1, 2 * u + 2
assert heugcd(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 heugcd(f, g) == (h, cff, cfg)
assert heugcd(g, f) == (h, cfg, cff)
R, x, y, z = ring("x,y,z", ZZ)
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = heugcd(f, 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)
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = heugcd(f, 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)
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = heugcd(f, 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)
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = heugcd(f, g)
assert H == h and H * cff == f and H * cfg == g
R, x, y, z = ring("x,y,z", ZZ)
f, g, h = R.fateman_poly_F_2()
H, cff, cfg = heugcd(f, g)
assert H == h and H * cff == f and H * cfg == g
f, g, h = R.fateman_poly_F_3()
H, cff, cfg = heugcd(f, g)
assert H == h and H * cff == f and H * cfg == g
R, x, y, z, t = ring("x,y,z,t", ZZ)
f, g, h = R.fateman_poly_F_3()
H, cff, cfg = heugcd(f, g)
assert H == h and H * cff == f and H * cfg == g
def test_heugcd_multivariate_integers():
R, x, y = ring("x,y", ZZ)
f, g = 2*x**2 + 4*x + 2, x + 1
assert heugcd(f, g) == (x + 1, 2*x + 2, 1)
f, g = x + 1, 2*x**2 + 4*x + 2
assert heugcd(f, g) == (x + 1, 1, 2*x + 2)
R, x, y, z, u = ring("x,y,z,u", ZZ)
f, g = u**2 + 2*u + 1, 2*u + 2
assert heugcd(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 heugcd(f, g) == (h, cff, cfg)
assert heugcd(g, f) == (h, cfg, cff)
R, x, y, z = ring("x,y,z", ZZ)
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = heugcd(f, 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)
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = heugcd(f, 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)
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = heugcd(f, 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)
f, g, h = R.fateman_poly_F_1()
H, cff, cfg = heugcd(f, g)
assert H == h and H*cff == f and H*cfg == g
R, x, y, z = ring("x,y,z", ZZ)
f, g, h = R.fateman_poly_F_2()
H, cff, cfg = heugcd(f, g)
assert H == h and H*cff == f and H*cfg == g
f, g, h = R.fateman_poly_F_3()
H, cff, cfg = heugcd(f, g)
assert H == h and H*cff == f and H*cfg == g
R, x, y, z, t = ring("x,y,z,t", ZZ)
f, g, h = R.fateman_poly_F_3()
H, cff, cfg = heugcd(f, g)
assert H == h and H*cff == f and H*cfg == g
def test_dmp_gcd():
R, x = ring('x', ZZ)
for test in (True, False):
with using(use_heu_gcd=test, fallback_gcd_zz_method='prs'):
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 + 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 = 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()) == g
f = 1317378933230047068160 * x + 2945748836994210856960
g = 120352542776360960 * x + 269116466014453760
assert f.cofactors(g) == (120352542776360960 * x +
269116466014453760, 10946, 1)
with using(heu_gcd_max=0):
pytest.raises(HeuristicGCDFailed, lambda: heugcd(f, g))
f, g = x**2 - 1, x**2 - 3 * x + 2
for test in (True, False):
for method in ('prs', 'modgcd'):
with using(use_heu_gcd=test, fallback_gcd_zz_method=method):
assert f.gcd(g) == x - 1
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))
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', CC)
f, g = x**2 - 1, x**3 - 3 * x + 2
assert f.cofactors(g) == (1, f, g)
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 = ring('x,y', ZZ)
for test in (True, False):
with using(use_heu_gcd=test, fallback_gcd_zz_method='prs'):
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 + 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):
pytest.raises(HeuristicGCDFailed, lambda: heugcd(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 = (
-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)
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', 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
R, x, y, z = ring('x,y,z', ZZ)
for test in (True, False):
with using(use_heu_gcd=test, fallback_gcd_zz_method='prs'):
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(2, ZZ))
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
f, g, h = map(R.from_dense, dmp_fateman_poly_F_2(2, ZZ))
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(2, ZZ))
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
R, x, y, z, u = ring('x,y,z,u', ZZ)
for test in (True, False):
with using(use_heu_gcd=test, fallback_gcd_zz_method='prs'):
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)
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 = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(4, ZZ))
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)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(6, ZZ))
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)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(8, ZZ))
H, cff, cfg = f.cofactors(g)
assert H == h and H * cff == f and H * cfg == g
F, x = field('x', QQ)
R, t = ring('t', F)
assert R(x).gcd(R(0)) == 1