def _do_test_benchmark_cyclic_4():
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_coloring():
V = range(1, 12 + 1)
E = [(1, 2), (2, 3), (1, 4), (1, 6), (1, 12), (2, 5), (2, 7), (3, 8),
(3, 10), (4, 11), (4, 9), (5, 6), (6, 7), (7, 8), (8, 9), (9, 10),
(10, 11), (11, 12), (5, 12), (5, 9), (6, 10), (7, 11), (8, 12),
(3, 4)]
R, V = xring(["x%d" % v for v in V], QQ, lex)
E = [(V[i - 1], V[j - 1]) for i, j in E]
x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12 = V
I3 = [x**3 - 1 for x in V]
Ig = [x**2 + x * y + y**2 for x, y in E]
I = I3 + Ig
assert groebner(I[:-1], R) == [
x1 + x11 + x12,
x2 - x11,
x3 - x12,
x4 - x12,
x5 + x11 + x12,
x6 - x11,
x7 - x12,
x8 + x11 + x12,
x9 - x11,
x10 + x11 + x12,
x11**2 + x11 * x12 + x12**2,
x12**3 - 1,
]
assert groebner(I, R) == [1]
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 _do_test_benchmark_minimal_polynomial():
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 _do_test_benchmark_katsura_3():
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 _do_test_benchmark_minpoly():
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 + QQ(155, 2067) * z**5 - QQ(355, 689) * z**4 +
QQ(6062, 2067) * z**3 - QQ(3687, 689) * z**2 + QQ(6878, 2067) * z -
QQ(25, 53),
y + QQ(4, 53) * z**5 - QQ(91, 159) * z**4 + QQ(523, 159) * z**3 -
QQ(387, 53) * z**2 + QQ(1043, 159) * z - 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_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 _do_test_benchmark_czichowski():
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 _do_test_groebner():
R, x, y = ring("x,y", QQ, lex)
f = x**2 + 2 * x * y**2
g = x * y + 2 * y**3 - 1
assert groebner([f, g], R) == [x, y**3 - QQ(1, 2)]
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, -QQ(1, 2) * x + 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 + QQ(3, 2) * y**4 - QQ(1, 2) * y**2 + QQ(1, 16),
]
def test_groebner(method):
with config.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)
