1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
|
restart
load "Elimination.m2"
load "GTZ.m2"
R = ZZ/32003[a,b,c,d,e,f]
I = ideal(
a^2*c*d*f^2,
b^2*c*d*f^2,
a^2*b*d*f^2,
b^3*d*f^2,
a^3*d*f^2,
a*b^2*d*f^2,
a^2*c*d*e,
b^2*c*d*e,
a^2*b*d*e,
b^3*d*e,
a^3*d*e,
a*b^2*d*e,
a^2*c*d^2,
b^2*c*d^2,
a^2*b*d^2,
b^3*d^2,
a^3*d^2,
a*b^2*d^2,
a^2*c^2*f^2,
b^2*c^2*f^2,
a^2*b*c*f^2,
b^3*c*f^2,
a^3*c*f^2,
a*b^2*c*f^2,
a^2*b^2*f^2,
b^4*f^2,
a^3*b*f^2,
a*b^3*f^2,
a^4*f^2)
GTZ1 I
R = ZZ/32003[b,s,t,u,v,w,x,y,z]
I = ideal(
b*v+s*u,
b*w+t*u,
s*w+t*v,
b*y+s*x,
b*z+t*x,
s*z+t*y,
u*y+v*x,
u*z+w*x,
v*z+w*y)
GTZ1 I
R = ZZ/32003[x,y,z]
I = ideal(
x*y^2*z^2-x*y^2*z+x*y*z^2-x*y*z,
x*y^3*z+x*y^2*z,
x*y^4-x*y^2,
x^2*y*z^2-x^2*y*z,
x^2*y^3-x^2*y^2,
x^4*z^3-x^4*z^2+2*x^3*z^3-2*x^3*z^2+x^2*z^3-x^2*z^2,
x^2*y^2*z,
x^4*y*z+x^3*y*z,
2*x^4*y^2+6*x^3*y^2+6*x^2*y^2+x*y^3+x*y^2,
x^5*z+x^4*z^2+x^4*z+2*x^3*z^2-x^3*z+x^2*z^2-x^2*z,
x^6*y+3*x^5*y+3*x^4*y+x^3*y)
GTZ1 I
R = ZZ/32003[b,s,t,u,v,w,x,y,z]
I = ideal(
s*u-b*v,
t*v-s*w,
v*x-u*y,
w*y-v*z)
GTZ1 I -- becomes non-binomial...
R = ZZ/32003[a,b,c,d,e,f,g,h]
I = ideal(
h+f+e-d-a,
2*f*b+2*e*c+2*d*a-2*a^2-a-1,
3*f*b^2+3*e*c^2-3*d*a^2-d+3*a^3+3*a^2+4*a,
6*g*e*b-6*d*a^2-3*d*a-d+6*a^3+6*a^2+4*a,
4*f*b^3+4*e*c^3+4*d*a^3+4*d*a-4*a^4-6*a^3-10*a^2-a-1,
8*g*e*c*b+8*d*a^3+4*d*a^2+4*d*a-8*a^4-12*a^3-14*a^2-3*a-1,
12*g*e*b^2+12*d*a^3+12*d*a^2+8*d*a-12*a^4-18*a^3-14*a^2-a-1,
-24*d*a^3-24*d*a^2-8*d*a+24*a^4+36*a^3+26*a^2+7*a+1)
codim I
I1 = eliminate(I, {d,e,f,h});
degree I
GTZ0 I
independentSets I
flatt(I,b*c*g)
-----------------------------------------------------
needsPackage "Markov"
G = makeGraph {{},{1},{1},{1},{2,3,4}}
R = markovRing(2,2,2,2,2)
F = marginMap(1,R)
I = F markovIdeal(R, localMarkovStmts G)
transpose gens I
time codim I -- takes a while to get 14
degree I -- 336
debug PrimaryDecomposition
time GTZ0 I;
J0 = oo_0;
J1 = ooo_1;
codim J1
degree J0
degree J1
time GTZ0 J1;
J2 = oo_1;
time GTZ0 J2;
J3 = oo_1;
codim J3
degree J3
time GTZ0 J3;
-----------------------------------------------------
restart
errorDepth = 0
debug PrimaryDecomposition
R = ZZ/32003[a,b,c,d,f,g,h,k,l,s,t,u,v,w,x,y,z, MonomialSize=>8]
I = ideal(
-a*b-a*d+2*a*h,
a*d-b*d-c*f-2*a*h+2*b*h+2*c*k,
a*b-a*d-2*b*h+2*d*h-2*c*k+2*f*k+2*g*l,
a*c-2*c*s-a*t+2*b*t,
a*c-c*s-2*a*t+b*t,
-d-3*s+4*u,
-f-3*t+4*v,
-g+4*w,
-a+2*x,
-b^2-c^2+2*b*x+2*c*y,
-d^2-f^2-g^2+2*d*x+2*f*y+2*g*z)
time minimalPrimes I -- 7.06 sec, or 5.53 sec
time primaryDecomposition(I, Strategy=>ShimoyamaYokoyama)
debug PrimaryDecomposition
time GTZ1 I;
J = minPres I
(J1,g) = saturation(J,x)
J1
J2 = trim(J + ideal g)
J2 = trim substitute(J, x=>0)
intersect(J1,J2) == J
C1 = time primaryDecomposition(J1, Strategy=>ShimoyamaYokoyama);
C2 = time primaryDecomposition(J2, Strategy=>ShimoyamaYokoyama);
time minimalPrimes J -- this is much worse than 'minimalPrimes I'
transpose gens J
time I = trim I
L = ideal apply(sort apply(flatten entries gens I, f -> (size f, first degree f, f)), g -> g#2)
time minimalPrimes L
gbTrace = 3
time (L1, M1) = GTZ0 I;
time gens gb I;
independentSets I
leadTerm(1,gens gb I)
|
