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
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
|
TEST /// -- embedding cubic surface (with 3 singular points) in P^3 via 5 sections of O(2)
setRandomSeed 0
elapsedTime d = dim ker map(QQ[x,y,z,w]/ideal(x^3 - y*z*w), QQ[a_0..a_4], {x*w + 2*x*y, x*w-3*y^2, z^2, x^2 + y^2 + z^2 - w^2, 3*y*w - 2*x^2})
R = CC[x,y,z,w]
I = ideal(x^3 - y*z*w)
F = {x*w + 2*x*y, x*w-3*y^2, z^2, x^2 + y^2 + z^2 - w^2, 3*y*w - 2*x^2}
assert(numericalImageDim(F, I) == d)
-- Cf. also: non-homogeneous ideal (x^5 - y*z*w) (~35 seconds for GB computation), kernel over finite fields
///
TEST /// -- twisted cubic
setRandomSeed 0
R = CC[s,t]
F = basis(3,R)
J = monomialCurveIdeal(QQ[a_0..a_3], {1,2,3})
assert(all(1..5, d -> (numericalHilbertFunction(F,ideal 0_R,d)).hilbertFunctionValue == numcols super basis(d,J)))
assert(all(1..5, d -> (numericalHilbertFunction(F,ideal 0_R,d,UseSLP=>true)).hilbertFunctionValue == numcols super basis(d,J)))
W = pseudoWitnessSet(F, ideal 0_R);
assert(W.degree == 3)
assert(isOnImage(W, first numericalImageSample(F,ideal 0_R)) == true)
assert(isOnImage(W, point random(CC^1,CC^(numcols F))) == false)
///
TEST /// -- Rational quartic curve in P^3
setRandomSeed 0
R = CC[s,t]
F = flatten entries basis(4, R) - set{s^2*t^2}
I = ideal 0_R
S = QQ[a_0..a_3]
I3 = super basis(3, ker map(QQ[s,t], S, {s^4,s^3*t,s*t^3,t^4}))
T = numericalHilbertFunction(F, I, 3);
M = extractImageEquations(T, AttemptZZ => true)
assert(image M == image (map(ring M, S, gens ring M))(I3))
elapsedTime PW = pseudoWitnessSet(F,I)
assert(PW.degree == 4)
///
TEST /// -- Grassmannian Gr(3, 5) = G(P^2,P^4)
setRandomSeed 0
(k, n) = (3,5)
R = CC[x_(1,1)..x_(k,n)]
I = ideal 0_R
F = (minors(k, genericMatrix(R, k, n)))_*
assert(numericalImageDim(F, I) == 1 + k*(n-k))
T = numericalHilbertFunction(F, I, 2)
J = super basis(2, Grassmannian(k-1,n-1))
assert(T.hilbertFunctionValue == numcols J)
I2 = image extractImageEquations(T, AttemptZZ => true)
assert(image (map(ring I2, ring J, gens ring I2))(J) == I2)
time W = pseudoWitnessSet(F, I, Repeats => 2)
assert(W.degree == 5)
-- (n, m) = (5, 10)
-- pointList = numericalImageSample(F, I, n);
-- assert(all(pointList, q -> (tally apply(m, i -> isOnImage(W, q)))#true / m >= 8/10)) -- too slow for test
///
-* disabled due to issue 2230
TEST /// -- random canonical curve of genus 4, under random projection to P^2 by cubics
setRandomSeed 0
R = CC[x_0..x_3]
I = ideal(random(2,R),random(3,R))
F = random(R^1,R^{3:-3})
assert(numericalImageDegree(F,I) == 18)
assert((numericalHilbertFunction(F,I,18)).hilbertFunctionValue == 1)
///
*-
TEST /// -- Segre + Veronese
setRandomSeed 0
-- Veronese surface P^2 in P^5
(d, n) = (2, 2)
R = CC[x_0..x_n]
F = basis(d, R)
PW = pseudoWitnessSet(F, ideal 0_R)
assert(PW.degree == 4)
assert((pseudoWitnessSet(PW.map, PW.sourceEquations, PW.witnessPointPairs_{1}, PW.imageSlice)).degree == 4)
assert((pseudoWitnessSet(PW.map, PW.sourceEquations, PW.witnessPointPairs_{0,2}, PW.imageSlice)).degree == 4)
I2 = ideal extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
S = QQ[y_0..y_(binomial(d+n,d)-1)]
RQ = QQ[x_0..x_n]
J = ker map(RQ, S, basis(d, RQ))
assert((map(ring I2, S, gens ring I2))(J) == I2)
-- Segre P^2 x P^3
(n1, n2) = (2, 4)
R = CC[s_0..s_(n1), t_0..t_(n2)]
F = (ideal(s_0..s_(n1))*ideal(t_0..t_(n2)))_*
I2 = ideal extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
RQ = QQ[s_0..s_(n1), t_0..t_(n2)]
S = QQ[y_0..y_((n1+1)*(n2+1)-1)]
J = ker map(RQ, S, (ideal(s_0..s_(n1))*ideal(t_0..t_(n2)))_*)
assert((map(ring I2, S, gens ring I2))(J) == I2)
///
TEST /// -- Iterated Veronese
(d1,d2) = (2,3)
R = CC[x_0..x_(d1)]
I = ideal(x_0*x_2 - x_1^2)
W = first components numericalIrreducibleDecomposition I
I.cache.WitnessSet = W;
S = CC[y_0..y_(binomial(d1+d2,d2)-1)]
F = map(R,S,basis(d2,R))
eps = 1e-10
p1 = first numericalSourceSample(I)
assert(clean(eps, sub(gens I, matrix p1)) == 0)
p2 = first numericalSourceSample(I, W)
assert(clean(eps, sub(gens I, matrix p2)) == 0)
p3 = first numericalSourceSample(I, p1)
assert(clean(eps, sub(gens I, matrix p3)) == 0)
P = numericalSourceSample(I, W, 10)
assert(all(P/matrix, p -> clean(eps, sub(gens I, p)) == 0))
q1 = numericalImageSample(F, I)
S = numericalImageSample(F,I,55);
T = numericalHilbertFunction(F,I,S,2)
assert(T.hilbertFunctionValue == 42)
assert(isWellDefined T)
PW = pseudoWitnessSet(F, I, p1)
assert(isWellDefined PW)
assert(PW.degree == 6)
assert(numericalImageDegree(F, I) == 6)
assert((pseudoWitnessSet(F, I, {PW.witnessPointPairs#0}, PW.imageSlice)).degree == 6)
assert((pseudoWitnessSet(F, I, PW.witnessPointPairs, PW.imageSlice)).degree == 6)
///
TEST /// -- Orthogonal group O(n)
setRandomSeed 0
n = 4
R = CC[x_0..x_(n^2-1)]
A = genericMatrix(R,n,n)
I = ideal(A*transpose A - id_(R^n));
F = vars R
p = point id_((coefficientRing R)^n)
assert(numericalImageDim(F,I,p) == binomial(n,2))
degSOn = 2^(n-1)*det matrix table(floor(n/2), floor(n/2), (i,j) -> binomial(2*n - 2*i - 2*j - 4, n - 2*i - 2))
elapsedTime PW = pseudoWitnessSet(F,I,p, Repeats=>2, Threshold=>3, MaxThreads=>allowableThreads) -- ~ 40 seconds
assert(numericalImageDegree PW == degSOn)
///
TEST /// -- Twisted cubic projections
-- no-check-flag #2183
R = CC[x_0..x_3]
I = monomialCurveIdeal(R, {1,2,3})
F1 = random(R^1, R^{3:-1})
p = numericalSourceSample I
imagePts = numericalImageSample(F1, I, 10);
assert(numericalImageDim(F1, I, p#0) == 2)
assert((numericalHilbertFunction(F1, I, imagePts, 2)).hilbertFunctionValue == 0)
assert((numericalHilbertFunction(F1, I, imagePts, 3)).hilbertFunctionValue == 1)
F2 = (gens R)_{0,2,3}
T = numericalHilbertFunction(F2, I, 3)
nodalCubic = ideal extractImageEquations(T, AttemptZZ => true)
S = ring nodalCubic
assert(nodalCubic == ideal(S_1^3 - S_0*S_2^2))
F3 = (gens R)_{0,1,2}
assert((numericalHilbertFunction(F3, I, 2)).hilbertFunctionValue == 1)
assert((pseudoWitnessSet(F2, I, p#0)).degree == 3)
///
TEST /// -- 3x3 matrices with double eigenvalue
S = QQ[a_(0,0)..a_(2,2), lambda, mu]
B = transpose genericMatrix(S,3,3)
J = ideal(B*transpose(B) - id_(S^3), det(B)-1)
n = B*diagonalMatrix{lambda,lambda,mu}*transpose(B)
T = QQ[p_(0,0),p_(0,1),p_(0,2),p_(1,1),p_(1,2),p_(2,2)]
J = ker map(S/J,T,{n_(0,0),n_(0,1),n_(0,2),n_(1,1),n_(1,2),n_(2,2)})
R = CC[a_(0,0)..a_(2,2), lambda, mu]
A = transpose genericMatrix(R,3,3)
I = ideal(A*transpose(A) - id_(R^3), det(A)-1)
m = A*diagonalMatrix{lambda,lambda,mu}*transpose(A)
F = {m_(0,0),m_(0,1),m_(0,2),m_(1,1),m_(1,2),m_(2,2)}
p = point{flatten entries diagonalMatrix {-1,-1,1} | {-1_CC, 1}}
assert(numericalImageDim(F,I,p) == dim J)
assert((pseudoWitnessSet(F,I,p)).degree == degree J)
///
TEST /// -- Numerical nullity tests
assert(numericalNullity(matrix{{2, 1}, {0, 1e-7}}, Precondition => false) == 1)
assert(numericalNullity(map(CC^2,CC^2,0)) == 2)
assert(numericalNullity(id_(CC^2)) == 0)
assert(numericalNullity(random(CC^2,CC^2)) == 0)
assert(numericalNullity(random(CC^0,CC^2)) == 2)
assert(numericalNullity(random(CC^2,CC^0)) == 0)
assert(numericalNullity(random(CC^0,CC^0)) == 0)
///
TEST /// -- approxPoint tests
(n,r) = (4,5)
R = QQ[x_(1,1)..x_(n,r)]
A = transpose genericMatrix(R,r,n)
I1 = ideal(A*transpose A - (r/n)*id_(R^n))
I2 = ideal apply(entries transpose A, row -> sum(row, v -> v^2) - 1)
I = I1 + I2 -- funtf variety
-- setRandomSeed 5
elapsedTime q = first numericalSourceSample(I, Software => I -> realPoint(I, Iterations => 100))
assert(norm evaluate(gens I, q) < 1e-5)
///
|
