-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathNoetherianOperators.m2
3960 lines (3549 loc) · 145 KB
/
NoetherianOperators.m2
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
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
-- -*- coding: utf-8 -*-
newPackage(
"NoetherianOperators",
Version => "2.2.1",
Date => "Nov 2 2021",
Authors => {
{Name => "Robert Krone",
Email => "krone@math.gatech.edu"},
{Name => "Justin Chen",
Email => "justin.chen@math.gatech.edu"},
{Name => "Marc Harkonen",
Email => "harkonen@gatech.edu"},
{Name => "Yairon Cid-Ruiz",
Email => "Yairon.CidRuiz@UGent.be"},
{Name => "Anton Leykin",
Email => "anton.leykin@gmail.com"}
},
Headline => "algorithms for computing local dual spaces and sets of Noetherian operators",
PackageExports => {"Bertini", "NumericalLinearAlgebra", "NAGtypes"},
PackageImports => {"PrimaryDecomposition"},
AuxiliaryFiles => false,
DebuggingMode => false,
Keywords => {"Numerical Algebraic Geometry", "Commutative Algebra"}
)
debug NumericalLinearAlgebra
export {
"truncatedDual",
"zeroDimensionalDual",
"gCorners",
"localHilbertRegularity",
"eliminatingDual",
"pairingMatrix",
"orthogonalInSubspace",
"Rational",
"StandardBasis",
"DiffOp",
"diffOp",
"diffOpRing",
"normalize",
"noetherianOperators",
"specializedNoetherianOperators",
"numericalNoetherianOperators",
"noethOpsFromComponents",
"coordinateChangeOps",
"rationalInterpolation",
"InterpolationTolerance",
"InterpolationDegreeLimit",
"NoetherianDegreeLimit",
"DependentSet",
"KernelStrategy",
"IntegralStrategy",
"Sampler",
"TrustedPoint",
--functions from punctual Hilb approach
"getIdealFromNoetherianOperators",
"getModuleFromNoetherianOperators",
"joinIdeals",
"mapToPunctualHilbertScheme",
"amult",
"solvePDE",
"differentialPrimaryDecomposition"
}
export { "isPointEmbedded", "isPointEmbeddedInCurve", "AllVisible", "Regularity" }
--TruncDualData private keys
protect \ {
Igens,
syl,
strategy,
deg,
dBasis,
hIgens,
BMintegrals,
BMcoefs,
BMbasis,
Seeds,
Op
}
-----------------------------------------------------------------------------------------
-- Create a dual space from a list of Noetherian operators
-- Caveat: if Noetherian operators have non-constant coefficients,
-- behavior is undefined.
dualSpace (List, AbstractPoint) := (L, p) -> (
if #L == 0 then error"expected nonempty list";
S := ring first L;
R := coefficientRing S;
L' := select(L, op -> op != 0);
gens := matrix{ L' / (op -> matrix op)};
if #L' == 0 then gens = sub(gens, S);
dualSpace((map(R,S, vars R)) gens, p)
)
TEST ///
R = CC[x,y]
S = diffOpRing R
foo1 = diffOp(12*dx + 3*dx^2*dy)
foo2 = diffOp(4*dx*dy + 4)
pt = point{{1_CC,2}}
a = dualSpace({foo1,foo2}, pt)
b = dualSpace(matrix{{12*x + 3*x^2*y, 4*x*y + 4}}, pt)
assert(gens a.Space - gens b.Space == 0)
assert(point a == point b)
///
shiftGens := (p,Igens) -> (
R := ring Igens;
sub(Igens, matrix{gens R + apply(p.Coordinates,c->sub(c,R))})
)
listFactorial = L -> product(L, l->l!)
-- Legacy methods
----------------------------------
truncatedDual = method(Options => {Tolerance => null})
truncatedDual (AbstractPoint,Ideal,ZZ) := o -> (p,I,d) -> (
depVars := gens (ring I);
t := getTolerance(ring first coordinates p,o);
L := numNoethOpsAtPoint(I,p, DependentSet => depVars, DegreeLimit => d, Tolerance => t);
dualSpace(L, p)
)
truncatedDual (Matrix, Ideal, ZZ) := o -> (p, I, d) -> truncatedDual(point p, I, d, o)
truncatedDual (AbstractPoint, Matrix, ZZ) := o -> (p, I, d) -> truncatedDual(p, ideal I, d, o)
truncatedDual (Matrix, Matrix, ZZ) := o -> (p, I, d) -> truncatedDual(point p, ideal I, d, o)
zeroDimensionalDual = method(TypicalValue => DualSpace, Options => {Tolerance => null})
zeroDimensionalDual (AbstractPoint,Ideal) := o -> (p,I) -> (
depVars := gens (ring I);
t := getTolerance(ring first coordinates p,o);
L := numNoethOpsAtPoint(I,p, DependentSet => depVars, Tolerance => t);
dualSpace(L, p)
)
zeroDimensionalDual (Matrix,Ideal) := o -> (p,I) -> zeroDimensionalDual(point p, I, o)
zeroDimensionalDual (AbstractPoint,Matrix) := o -> (p,I) -> zeroDimensionalDual(p, ideal I, o)
zeroDimensionalDual (Matrix,Matrix) := o -> (p,I) -> zeroDimensionalDual(point p, ideal I, o)
----------------------------------
--An object that stores the data for an ongoing iterative tuncated dual space computation
TruncDualData = new Type of MutableHashTable
initializeDualData = method(Options => {KernelStrategy => "Default"})
initializeDualData (Matrix,Boolean,Number) := opts -> (Igens,syl,t) -> (
H := new MutableHashTable;
R := ring Igens;
F := coefficientRing R;
H.Igens = Igens;
H.syl = syl;
H.deg = 0;
h := symbol h;
S := F(monoid[{h}|gens R, MonomialOrder => {Weights => (numgens R+1):1, 1, (options R).MonomialOrder}]); --projectivization of R
h = S_0;
H.hIgens = homogenize(sub(Igens,S), h);
T := if syl then S else R;
H.Seeds = dualSpace(matrix{{1_T}},origin(T));
H.BMmatrix = pairingMatrix(polySpace if syl then H.hIgens else H.Igens, H.Seeds);
H.BMmatrix = sub(H.BMmatrix,F);
H.BMintegrals = gens H.Seeds;
H.BMcoefs = myKernel(H.BMmatrix, opts, Tolerance=>t);
H.BMbasis = H.BMcoefs;
--print(H.BMmatrix,H.BMcoefs);
H.dBasis = H.BMintegrals * H.BMcoefs;
if H.syl then H.dBasis = (map(R,S,{1_R} | gens R)) H.dBasis;
new TruncDualData from H
)
truncDualData = initializeDualData
-- advances the truncated dual computation from whatever was stored in parameters up to degree d
nextTDD = method(Options => {KernelStrategy => "Default"})
nextTDD (TruncDualData,Number) := opts -> (H,t) -> nextTDD(H.deg + 1,H,t, opts)
nextTDD (ZZ,TruncDualData,Number) := opts -> (d,H,t) -> (
R := ring H.Igens;
S := ring H.hIgens;
dehomog := map(R,S,{1_R} | gens R);
for e from H.deg+1 to d do (
(M,E) := BMmatrix H;
H.BMmatrix = M; H.BMintegrals = E;
H.BMcoefs = myKernel(M, opts, Tolerance=>t);
--print(M,H.BMcoefs);
I := basisIndices(last coefficients E*H.BMcoefs, t);
H.BMbasis = submatrix(H.BMcoefs, I);
H.dBasis = if H.syl then dehomog(E*H.BMcoefs) else H.dBasis | E*H.BMcoefs;
if numcols H.BMcoefs == 0 then break;
--print (e, numrows M, numcols M, numcols H.dBasis, dim reduceSpace polySpace H.dBasis);
);
H.deg = d;
H
)
polySpace TruncDualData := o-> H -> polySpace(H.dBasis, Reduced=>false)
dualSpace (TruncDualData,AbstractPoint) := (H,p) -> dualSpace(polySpace H,p)
homogPolySpace = method()
homogPolySpace TruncDualData := H -> polySpace(H.BMintegrals*H.BMcoefs)
-- this version gets a piece of the eliminating DS from the "usual" truncated DS
eliminatingDual = method(Options => {Tolerance => null})
eliminatingDual (AbstractPoint,Ideal,List,ZZ) := o -> (p,I,ind,d) -> eliminatingDual (p,gens I,ind,d,o)
eliminatingDual (AbstractPoint,Matrix,List,ZZ) := o -> (p,Igens,ind,d) -> (
R := ring Igens;
if d < 0 then return dualSpace(map(R^1,R^0,0),p);
t := getTolerance(R,o);
Igens = shiftGens(p,Igens);
n := numgens R;
if not all(ind, i->class i === ZZ) or not all(ind, i -> i>=0 and i<n)
then error ("expected a list of nonnegative integers in the range [0," | n | "] as 2nd parameter");
TDD := initializeDualData(Igens,false,t);
RdBasis := dualSpace(TDD,p);
dBold := 0;
while dBold != dim RdBasis do (
dBold = dim RdBasis;
TDD = nextTDD(TDD,t);
RdBasis = truncate(dualSpace(TDD,p),ind,d);
--print(dualSpace(TDD,p),RdBasis);
);
RdBasis
)
truncate (PolySpace, ZZ) := {} >> o -> (L,d) -> (
R := ring L;
if not L.Reduced then L = reduceSpace L;
tGens := select(flatten entries gens L, q -> first degree q <= d);
if #tGens == 0 then polySpace map(R^1,R^0,0) else polySpace matrix{tGens}
)
truncate (DualSpace, ZZ) := {} >> o -> (L,d) -> dualSpace(truncate(L.Space,d),L.BasePoint)
truncate (PolySpace, List, ZZ) := {} >> o -> (L,ind,d) -> (
R := ring L;
n := numgens R;
if not all(ind, i->class i === ZZ) or not all(ind, i -> i>=0 and i<n)
then error ("expected a list of nonnegative integers in the range [0," | n | "] as 2nd parameter");
indC := flatten entries submatrix'(matrix{{0..n-1}},ind);
T := newRing(R,MonomialOrder=>{#ind,#indC},Degrees=>{(#ind:{1,0})|(#indC:{0,1})});
TtoR := map(R,T, (vars R)_ind | (vars R)_indC);
varList := new MutableList from n:0;
scan(#ind, i->(varList#(ind#i) = T_i));
scan(#indC, i->(varList#(indC#i) = T_(#ind + i)));
RtoT := map(T,R, matrix{toList varList});
TL := reduceSpace polySpace RtoT gens L;
TL = truncate(TL,d);
polySpace(TtoR gens TL)
)
truncate (DualSpace, List, ZZ) := {} >> o -> (L,ind,d) -> dualSpace(truncate(L.Space,ind,d),L.BasePoint)
gCorners = method(TypicalValue => Matrix, Options => {Tolerance => null, StandardBasis => false})
gCorners (AbstractPoint,Ideal) := o -> (p,I) -> gCorners(p,gens I,o)
gCorners (AbstractPoint,Matrix) := o -> (p,Igens) -> (
R := ring Igens;
t := getTolerance(R,o);
Igens = sub(Igens, matrix{gens R + apply(p.Coordinates,c->sub(c,R))});
ecart := max apply(flatten entries Igens, g->(gDegree g - lDegree g)); --max ecart of generators
GCs := {}; -- g-corners (as pairs: monomial, degree in homogenization)
SBs := {}; -- standard basis elements (if o.StandardBasis)
finalDegree := max(flatten entries Igens / gDegree);
d := 0;
dBasis := dBasisReduced := polySpace map(R^1,R^0,0); -- Sylvester truncated dual space
TDD := initializeDualData(Igens,true,t); -- initial parameters for computing truncated duals
while d <= finalDegree do (
TDD = nextTDD(d,TDD,t);
dBasis = polySpace TDD;
dBasisReduced = reduceSpace(dBasis,Tolerance=>t);
-- Find new g-corners based on what monomials are missing from dBasis.
newGCs := newGCorners(dBasisReduced,GCs,d,ecart);
GCs = GCs|newGCs;
-- If o.StandardBasis then compute a standard basis element for each new g-corner.
if o.StandardBasis and #newGCs > 0 then SBs = SBs|newSBasis(dBasis,newGCs,d,t);
-- Update stopping degree if there were new g-corners found.
if #newGCs > 0 then (
topLCMdegree := max apply(subsets(#GCs,2),s->(
(a,b) := (GCs#(s#0), GCs#(s#1));
tdegree := max(a#1 - first degree a#0, b#1 - first degree b#0);
(first degree mylcm {a#0,b#0}) + tdegree
));
finalDegree = max(finalDegree,topLCMdegree);
);
if debugLevel >= 1 then << "-" << "- at degree " << d << ": dim " << dim dBasisReduced << ", new corners " << newGCs/first << endl;
--print(d, finalDegree, dim dBasisReduced, newGCs/first);
d = d+1;
);
GCs = if o.StandardBasis then SBs else GCs/first;
sbReduce matrix {GCs}
)
mylcm = L -> (
k := #(first exponents L#0);
exps := apply(k, i->max apply(L, l->(first exponents l)#i));
(ring L#0)_exps
)
-- computes s-corners from the g-corners
-- i.e., the maximal monomials not in the ideal generated by the g-corners
socles = method(TypicalValue => Matrix)
socles MonomialIdeal := I -> mingens((I : ideal gens ring I)/I)
socles Matrix := M -> socles monomialIdeal M
sCorners = socles
hilbertFunction DualSpace := L -> (
if not L.Space.Reduced then L = reduceSpace L;
tally(flatten entries gens L / first @@ degree)
)
hilbertFunction(List,DualSpace) := (LL,L) -> (
h := hilbertFunction L;
apply(LL, d->(if h#?d then h#d else 0))
)
hilbertFunction(ZZ,DualSpace) := (d,L) -> first hilbertFunction({d},L)
localHilbertRegularity = method(TypicalValue => ZZ, Options=>{Tolerance => null})
localHilbertRegularity(AbstractPoint, Ideal) := o -> (p,I) -> localHilbertRegularity(p,gens I,o)
localHilbertRegularity(AbstractPoint, Matrix) := o -> (p,Igens) -> (
n := numgens ring Igens;
gCs := gCorners(p,Igens,o);
gCLists := apply(flatten entries gCs, l -> (listForm l)#0#0);
LCMexp := apply(n, i -> max(apply(gCLists, l->l#i)));
s := max{sum LCMexp - n + 1, 0}; -- an upperbound
J := ideal gCs;
HP := hilbertPolynomial(J, Projective=>false);
while hilbertFunction(s-1,J) == sub(HP,(ring HP)_0 => s-1) do s = s-1;
s
)
-- Implementation of algorithm from 1996 paper of Bernard Mourrain.
-- M is the main matrix
-- E is the row matrix of all dual basis integrals (including 1)
-- B contains the most recently found generators (as coefficients in terms of E)
BMmatrix = H -> (
Igens := if H.syl then H.hIgens else H.Igens;
(M,E,B,Bfull,homogeneous,Seeds) := (H.BMmatrix,H.BMintegrals,H.BMbasis,H.BMcoefs,H.syl,H.Seeds);
--print(M,E,(numcols B, numrows B),(numcols Bfull, numrows Bfull));
R := ring Igens;
F := coefficientRing R;
n := numgens R;
m := numcols Igens;
snew := numcols B;
offset := if homogeneous then 0 else dim Seeds;
s := (numcols E - offset)//n; --number of dual space generators
npairs := subsets(n,2);
newMEs := apply(snew, i -> (
bcol := B_{i};
bpoly := (E*bcol)_(0,0);
E' := matrix {apply(n, k->(
subs := matrix{apply(n, l->(if l > k then 0_R else (gens R)#l))};
(gens R)#k * sub(bpoly,subs)))};
M' := pairingMatrix(polySpace Igens, polySpace E');
if not homogeneous then M' = map(R^(s+(numcols Bfull)),R^n,0) || M';
for j from 0 to s-1 do (
w := apply(n,k->(bcol_(offset + j*n + k,0)));
v := mutableMatrix(R,#npairs,n);
for i from 0 to #npairs-1 do (
v_(i,npairs#i#0) = w#(npairs#i#1);
v_(i,npairs#i#1) = -w#(npairs#i#0);
);
M' = M' || new Matrix from v;
);
(M',E')
));
if not homogeneous then
M = transpose Bfull || M || map(R^(m + s*(1+#npairs) - numrows M),R^(numcols E),0)
else M = map(R^(m + s*#npairs),R^0,0);
(Mnew, Enew) := (matrix{newMEs/first}, matrix{newMEs/last});
if #newMEs == 0 then (Mnew, Enew) = (map(R^(numrows M),R^0,0),map(R^(numrows E),R^0,0));
M = sub(M | Mnew, F);
E = if homogeneous then Enew else E | Enew;
(M,E)
);
newGCorners = method()
newGCorners (PolySpace,List,ZZ,ZZ) := (dBasis,GCs,d,ecart) -> (
R := ring dBasis;
mons := flatten entries sort basis(d-ecart,d,R);
dBasisMons := sort(flatten entries gens dBasis / gLeadMonomial);
newGCs := {};
i := 0;
for m in mons do (
while i < #dBasisMons and m > dBasisMons#i do i = i+1;
if i < #dBasisMons and m == dBasisMons#i then (i = i+1; continue);
if not any(GCs, g->(isDivisible(m,g#0) and gDegree m - gDegree(g#0) <= d - g#1)) then newGCs = append(newGCs, (m,d));
);
newGCs
)
newSBasis = (dBasis,newGCs,d,t) -> (
R := ring dBasis;
Rd := sort basis(0,d,R);
Id := orthogonalInSubspace(dBasis, polySpace Rd, t);
iBasis := flatten entries gens reduceSpace(Id,Monomials=>Rd);
new List from apply(newGCs, n->first select(1,iBasis, b->(lLeadMonomial b == n#0)))
)
-- PolySpace of ideal basis through degree d-1.
idealBasis = method(Options => true)
idealBasis(Ideal, ZZ) := true >> opts -> (I,d) -> (
R := ring I;
if d < 1 then return map(R^1,R^0,0);
V := if opts.?DependentSet then opts.DependentSet else gens R;
matrix{ unique flatten entries (gens I ** basis(0,d-1,R, Variables => V))}
)
--lead monomial and lead monomial degree according to ordering associated with
--the ring (local) and reverse ordering (global)
lLeadMonomial = f -> leadMonomial last terms f
gLeadMonomial = f -> leadMonomial first terms f
lDegree = f -> first degree lLeadMonomial f
gDegree = f -> first degree gLeadMonomial f
--remove non-minimal standard basis elements
sbReduce = L -> (
n:= numcols L;
goodi := select(n, i->(
all(0..n-1, j->(j == i or not isDivisible(lLeadMonomial L_(0,i), lLeadMonomial L_(0,j))))
));
L_goodi
)
contract (PolySpace, PolySpace) := (S, T) -> (
T = gens T;
S = gens S;
cols := for s in flatten entries S list transpose contract(s,T);
matrix {cols}
)
diff (PolySpace, PolySpace) := (S, T) -> (
T = gens T;
S = gens S;
cols := for s in flatten entries S list transpose diff(s,T);
matrix {cols}
)
-- Matrix of natural pairings
-- PolySpace generators as rows, DualSpace generators as columns
pairingMatrix = method()
pairingMatrix (PolySpace, PolySpace) := (S, T) -> (
M := last coefficients(gens S | gens T);
Svec := submatrix(M,0..dim S-1);
Tvec := submatrix'(M,0..dim S-1);
(transpose Svec)*Tvec
)
pairingMatrix (PolySpace, DualSpace) := (S, L) -> (
Sshift := polySpace sub(gens S, matrix{(gens ring L) + coordinates L.BasePoint});
pairingMatrix(Sshift, L.Space)
)
pairingMatrix (RingElement, DualSpace) := (f, L) -> pairingMatrix(polySpace matrix{{f}}, L)
pairingMatrix (PolySpace, RingElement) := (S, l) -> pairingMatrix(S, polySpace matrix{{l}})
pairingMatrix (RingElement, RingElement) := (f, l) -> (
M := last coefficients(matrix{{f,l}});
((transpose M_{0})*M_{1})_(0,0)
)
orthogonalInSubspace = method()
orthogonalInSubspace (DualSpace, PolySpace, Number) := (D,S,t) -> (
R := ring S;
F := coefficientRing R;
M := sub(pairingMatrix(S,D),F);
K := myKernel(transpose M,Tolerance=>t);
polySpace((gens S)*K, Reduced=>false)
)
orthogonalInSubspace (PolySpace, PolySpace, Number) := (T,S,t) -> (
T' := dualSpace(T, origin(ring S));
orthogonalInSubspace(T',S,t)
)
isDivisible = (a, b) -> (
dif := (listForm a)#0#0 - (listForm b)#0#0;
all(dif, i->(i >= 0))
)
check PolySpace := S -> (
assert(numrows gens S == 1);
M := last coefficients gens S;
assert(numgens numericalImage M == numcols M);
)
check DualSpace := D -> check polySpace gens D
-------------------------------------------------------
-- functions related to numerical primary decomposition
-- (previously residing in NumericalAlgebraicGeometry)
-------------------------------------------------------
isPointEmbedded = method(Options=>{AllVisible=>false})
isPointEmbedded(AbstractPoint, Ideal, List) := o -> (p,I,C) -> ( -- C is a list of witness sets for irreducible components
R := ring I;
time gCs := gCorners(p,I); -- assume Robert's
-- algorithm a DualSpace D,
-- and g-corners
d := 0;
l := random(1,ring I); -- a generic linear form
while true do (
-- FIRST PART: returns true if embeddedness is certified
if o.AllVisible then (
Jd := interpolate(p,I,C,d); -- a function that returns
-- a list of polynomials forming a basis of J_d (vector
-- space), where J is the "part" of the decomposition of I
-- corresponding to the components given in C.
if dim R_d - dim Jd != hilbertFunction(d, monomialIdeal gCs) then return true; -- is there a better way?
)
else (
if debugLevel>0 then print "-- double truncation...";
time Jdd := doubleTruncation(I,C,d,d);
if debugLevel>0 then print (d,dim Jdd);
for d' from 1 to d do (
g := random(d', Jdd);
if debugLevel>0 then << "-- witness poly: (d',d) = " << (d',d) << endl;
time if isWitnessPolynomial(p,I,g,d)
then (
if debugLevel>0 then print toString g;
return true;
)
)
);
--SECOND PART: returns false if deemed not embedded
if debugLevel>0 then print "-- colon(truncated dual)...";
time Sd := colon(truncatedDual(p,I,d), l);
sCs := flatten entries sCorners gCs;
colonLMs := leadMonomial \ flatten entries gens reduceSpace Sd;
if debugLevel>0 then << "-- s-corners: " << sCs << endl << "-- LM(dual of colon ideal): " << colonLMs << endl;
if isSubset(sCs, colonLMs) then return false;
d = d+1;
)
)
colon = method(TypicalValue => DualSpace, Options => {Tolerance=>1e-6})
colon (DualSpace, RingElement) := o-> (L,g) -> (
(gmons,gcoefs) := coefficients g;
(Lmons,Lcoefs) := coefficients gens L;
M := matrix apply(flatten entries gmons, gm->(
apply(flatten entries Lmons, Lm->(
d := diff(gm,Lm);
if d == 0 then d else leadMonomial d
))
));
if numcols M == 0 then M = map((ring L)^1,(ring L)^0,0);
M = (transpose gcoefs)*M*Lcoefs;
(Mmons,Mcoefs) := coefficients M;
M = Mmons*sub(numericalImage(Mcoefs,o.Tolerance),ring Mmons);
dualSpace(polySpace M, L.BasePoint)
)
colon (DualSpace, Ideal) := (L,J) -> error "not implemented"
interpolate = method()
interpolate (AbstractPoint,Ideal,List,ZZ) := (p,I,C,d) -> error "not implemented"
TEST ///
setRandomSeed 0
needsPackage "NumericalAlgebraicGeometry"
-- NPD2.8: pseudo-component at the origin
RQQ = QQ[x_1..x_3]
M = matrix{{x_1^2,x_1*x_2*x_3}}
-- NPD3.10: all components are embedded
RQQ = QQ[x_1..x_3]
M = matrix{{x_1^2,x_1*x_2^2*x_3,x_1*x_2*x_3^3}}
I = ideal M
RCC = CC[x_1..x_3]
C = drop(flatten flatten (ass I / values@@numericalIrreducibleDecomposition),-1)
O = origin RCC
assert isPointEmbedded(O,sub(I,RCC),C)
///
doubleTruncation = method()
doubleTruncation (Ideal,List,ZZ,ZZ) := (I,C,d,e) -> (
R := ring I;
S := polySpace basis(0,d,R);
orthogonalInSubspace(I,C,e,S)
)
orthogonalInSubspace (Ideal,List,ZZ,PolySpace) := (I,C,e,S) -> (
t := 1e-6;
done := false;
while not done do done = all(C, V->(
p := random V;
D := truncatedDual(p,I,e); -- D_p^e[I]
S' := orthogonalInSubspace(D,S,t);
nothing'new := (dim S' == dim S);
S = S';
nothing'new
));
S
)
isWitnessPolynomial = method()
isWitnessPolynomial (AbstractPoint, Ideal, RingElement, ZZ) := (p,I,g,dStop) -> (
t := 1e-6;
R := ring g;
n := numgens R;
if g == 0 then return false;
Igens := sub(gens I, matrix{gens R + apply(p.Coordinates,c->sub(c,R))});
g = sub(g, matrix{gens R + apply(p.Coordinates,c->sub(c,R))});
e := first degree g;
Iplusg := Igens|matrix{{g}};
IP := truncDualData(Igens,false,t);
IplusgP := truncDualData(Iplusg,false,t);
IhP := truncDualData(Igens,true,t);
ID := IhD := IplusgD := polySpace map(R^1,R^0,0);
S := ring homogPolySpace IhP;
dehomog := map(R,S,{1_R} | gens R);
gh := homogenize(sub(g,S),S_0);
d := 0;
GCs := {};
varList := new MutableList from n:false;
ginI := true;
ginJ := false;
while d <= dStop do (
IP = nextTDD(d,IP,t);
IplusgP = nextTDD(d,IplusgP,t);
if dim polySpace IP != dim polySpace IplusgP then ginI = false;
IhP = nextTDD(d+e,IhP,t);
IhD = dualSpace(homogPolySpace IhP, point {toList (numgens S:0)});
IhcolonghD := reduceSpace polySpace dehomog gens colon(IhD,gh);
newGCs := newGCorners(IhcolonghD,GCs,d,d);
if any(newGCs, g->(g#0 == 1_R)) then return false;
for g in newGCs do (
l := (listForm(g#0))#0#0;
ls := select(n, i->(l#i != 0));
if #ls == 1 then varList#(first ls) = true;
);
if all(varList, v->v) then ginJ = true;
GCs = GCs|newGCs;
d = d+1;
if ginJ and not ginI then return true;
);
false
)
TEST ///
debug needsPackage "NoetherianOperators"
R = CC[x,y]
p = point {{0,0}}
I = ideal {x^2,y*x}
g = x^2
assert(not isWitnessPolynomial(p,I,g,10))
h = x
assert(isWitnessPolynomial(p,I,h,10))
///
isPointEmbeddedInCurve = method(Options=>{Regularity => -1})
isPointEmbeddedInCurve (AbstractPoint,Ideal) := o-> (p,I) -> (
R := ring I;
I' := ideal sub(gens I, matrix{gens R + apply(p.Coordinates,c->sub(c,R))});
p' := origin(R);
m := matrix{apply(gens R, v->random(1,R))}; -- matrix for random linear change of coordinates
I' = (map(R,R,m)) I'; -- I with new coordinates
r := o#Regularity;
if r == -1 then (
r1 := localHilbertRegularity(p',I');
r2 := dim truncatedDual(p',I',r);
r = max{r1,r2-1};
);
E := eliminatingDual(p',I',{0},r); -- assume I is in general position w.r.t. x = R_0
E1 := truncate(E,{0},r-1);
E2 := colon(E,R_0);
print (dim E1, dim E2, areEqual(E1,E2));
dim E1 != dim E2 -- "truncate" extracts a lesser truncated eliminating dual from E
-- can we make a less expensive test than above? look at the leading terms of the dual basis (and the quotient)?
)
--
----- Noetherian operator code
--
-- DiffOp is a type of hash table, which contains one key, Op. This is to make inheritance with NoethOp work.
-- The value of Op is a HashTable, with keys corresponding to partial monomials,
-- and values corresponding to coefficients.
-- Constructors
DiffOp = new Type of Vector
DiffOp.synonym = "differential operator"
DiffOp#{Standard,AfterPrint} = x -> (
<< endl; -- double space
<< concatenate(interpreterDepth:"o") << lineNumber;
y := class x;
<< " : DiffOp in " << y;
<< endl;
)
diffOpRing = (cacheValue "DiffOpRing") (R -> R(monoid [gens R / toString / (v -> "d" | v) / value]))
diffOpModule = memoize((S, k) -> new Module of DiffOp from S^k)
diffOp = method()
diffOp Matrix := m -> (
S := ring m;
if S.?cache and S.cache#?"DiffOpRing" then (S = diffOpRing S; m = sub(m,S);)
else if S =!= diffOpRing coefficientRing S then error"expected ring element in diffOpRing";
SS := diffOpModule(S, numRows m);
new SS from vector m
)
diffOp RingElement := f -> diffOp matrix f
DiffOp ? DiffOp := (a,b) -> (
if #(entries a) != #(entries b) then incomparable
else if (class a) != (class b) then incomparable;
delta := a - b;
i := position(entries delta, j -> j != 0);
if instance(i, Nothing) then symbol ==
else ((entries a)#i) ? ((entries b)#i)
)
DiffOp == DiffOp := (a,b) -> (
if ((a ? b) === incomparable) then error("expected comparable differential operators")
else if ((a ? b) === (symbol ==)) then true
else false
)
DiffOp == ZZ := (d,i) -> matrix d == i
ZZ == DiffOp := (i,d) -> d == i
coefficients DiffOp := opts -> d -> coefficients (matrix d, opts)
-- Operations
DiffOp + DiffOp := DiffOp => (a,b) -> (
A := class a;
B := class b;
if A === B then diffOp(matrix a+matrix b)
else error"expected pair to have a method for '+'"
)
DiffOp - DiffOp := DiffOp => (a,b) -> (
A := class a;
B := class b;
if A === B then diffOp(matrix a-matrix b)
else error"expected pair to have a method for '-'"
)
--Left action
RingElement * DiffOp := DiffOp => (r,d) -> (
R := class r;
S := class d;
if ring r === ring d or member(ring r, (ring d).baseRings) then new S from diffOp(promote(r,ring d)*matrix d)
else error"expected pair to have a method for '*'"
)
- DiffOp := DiffOp => d -> ((-1)*d)
-- Application
DiffOp SPACE Matrix := (D,m) -> (
if numColumns m != 1 then error"expected column matrix";
if numRows m != #(entries D) then error"expected differential operator and column matrix of same length";
S := ring D;
R := coefficientRing S;
try mm := lift(m, R) else error"expected matrix in the coefficient ring of operator";
DD := entries D;
mm = flatten entries mm;
sum(DD, mm, (d,f) -> (
(mon,coe) := coefficients d;
phi := map(R,S,gens R);
(diff(phi mon, f) * lift(matrix entries coe,R))_(0,0) -- `matrix entries (...)` clears the degree data
))
)
DiffOp SPACE RingElement := (D, f) -> D (matrix f)
-- other functions
normalize = method()
normalize DiffOp := DiffOp => d -> (
R := coefficientRing ring d;
KK := coefficientRing R;
lt := leadTerm matrix d;
lc := last coefficients lt;
lt = leadTerm sub(lc, R);
lc = (last coefficients lt)_(0,0);
(1/sub(lc,KK)) * d
)
evaluate(DiffOp, Matrix) := (D, pt) -> (
S := ring D;
R := coefficientRing S;
(mon, coe) := coefficients D;
diffOp (mon * evaluate(sub(coe, R) , pt))
)
evaluate(DiffOp, AbstractPoint) := (D, pt) -> evaluate(D, matrix pt)
TEST ///
debug NoetherianOperators
R = QQ[x,y]
S = diffOpRing(R)
use S
SS = diffOpModule(S, 2)
d = (y^3*dx*dy - x)*SS_0 + dx^2*dy*y * SS_1
assert(instance(SS_0, DiffOp))
assert(instance(dx*SS_0 + 3*SS_1, DiffOp))
assert(instance(d, DiffOp))
m = matrix transpose {{x*y + 5, x^2*y^2}}
assert(d m == y^3 -x^2*y - 5*x + y*4*y )
m = matrix transpose{{dx, y*dy*dx+dx^2+dy}}
a = diffOp m
b = (dx*SS_0 + (y*dy*dx+dx^2+dy)*SS_1)
assert(a==b)
SS = new Module of DiffOp from S^1
assert(diffOp (x^2*dx) == x^2*dx*SS_0)
assert(diffOp(x_R) == x*SS_0)
assert(diffOp(x_S) == x*SS_0)
///
TEST ///
--DiffOp
R = QQ[x,y,z]
S = diffOpRing R
foo = diffOp(y*dx + 2*x*dy)
bar = diffOp((z*x+3)*dx^2 + dy*x)
foobar = diffOp(dx*y + 3*x*dy + (z*x+3)*dx^2)
foo2 = diffOp(x*dx^2*dy)
foo3 = diffOp(0_S)
assert(foobar == foo + bar)
assert(foo (x^2) == 2*x*y)
assert(foo3 - foo == diffOp(-y*dx - 2*x*dy))
assert(foo3 - foo == diffOp(-y*dx - 2*x*dy))
assert(foo2 > foo)
assert(foo3 == 0)
assert(foo2 - foo2 == 0)
assert(not foo == 0)
///
sanityCheck = (nops, I) -> (
all(flatten table(nops, I_*, (N,i) -> (N i)%(radical I) == 0), identity)
)
myKernel = method(Options => {Tolerance => null, KernelStrategy => "Default"})
myKernel Matrix := Matrix => opts -> MM -> (
if precision MM < infinity then return colReduce(numericalKernel(MM,Tolerance => opts.Tolerance),Tolerance => opts.Tolerance);
if opts.KernelStrategy == "Default" then return colReduce gens kernel MM;
R := ring MM;
M := transpose colReduce(transpose MM, Reverse=>true);
(m,n) := (numrows M, numcols M);
es := entries M;
pivs := apply(m, i -> position(es#i, e -> (e != 0)));
nonPivs := toList(0..<n) - set pivs;
if #nonPivs == 0 then return map(R^n, R^0, 0);
transpose matrix apply(nonPivs, j -> (
apply(n, i -> (
pivRow := position(pivs, k -> i==k);
if pivRow =!= null then -M_(pivRow,j) / M_(pivRow,i)
else if i == j then 1_R else 0
))
))
)
TEST ///
debug NoetherianOperators
M = random(QQ^4,QQ^2) * random(QQ^2,QQ^4)
assert((myKernel M - gens kernel M) == 0)
///
-- dispatcher method
noetherianOperators = method(Options => true)
noetherianOperators (Ideal) := List => true >> opts -> I -> (
strats := new HashTable from {
"PunctualQuot" => noetherianOperatorsPunctual,
"PunctualHilbert" => noetherianOperatorsPunctual,
"Hybrid" => hybridNoetherianOperators,
"MacaulayMatrix" => noetherianOperatorsViaMacaulayMatrix,
};
strat := if opts.?Strategy then opts.Strategy else "PunctualQuot";
if strats#?strat then strats#strat(I, opts)
else error ("expected Strategy to be one of: \"" | demark("\", \"", sort keys strats) | "\"")
)
noetherianOperators (Ideal, Ideal) := List => true >> opts -> (I,P) -> (
strats := new HashTable from {
"PunctualQuot" => noetherianOperatorsPunctual,
"PunctualHilbert" => noetherianOperatorsPunctual,
"Hybrid" => hybridNoetherianOperators,
"MacaulayMatrix" => noetherianOperatorsViaMacaulayMatrix,
};
strat := if opts.?Strategy then opts.Strategy else "MacaulayMatrix";
if strats#?strat then strats#strat(I, P, opts)
else error ("expected Strategy to be one of: \"" | demark("\", \"", sort keys strats) | "\"")
)
noetherianOperators (Module) := List => true >> opts -> M -> (
strats := new HashTable from {
"PunctualQuot" => noetherianOperatorsPunctual
};
strat := if opts.?Strategy then opts.Strategy else "PunctualQuot";
if strats#?strat then strats#strat(M, opts)
else error ("expected Strategy to be one of: \"" | demark("\", \"", sort keys strats) | "\"")
)
noetherianOperators (Module, Ideal) := List => true >> opts -> (M,P) -> (
strats := new HashTable from {
"PunctualQuot" => noetherianOperatorsPunctual
};
strat := if opts.?Strategy then opts.Strategy else "PunctualQuot";
if strats#?strat then strats#strat(M, P, opts)
else error ("expected Strategy to be one of: \"" | demark("\", \"", sort keys strats) | "\"")
)
-- End dispatcher method
macaulayMatrixKernel := true >> opts -> (I, kP) -> (
S := ring I;
-- option handling
if debugLevel > 1 then <<opts<<endl;
tol := if not opts.?Tolerance then getTolerance(S,opts) else opts.Tolerance;
degLim := if not opts.?DegreeLimit then -1 else opts.DegreeLimit;
if instance(degLim, InfiniteNumber) then degLim = -1;
rat := if not opts.?Rational then false else opts.Rational;
kerStrat := if not opts.?KernelStrategy then "Default" else opts.KernelStrategy;
useBM := if not opts.?IntegralStrategy then null else opts.IntegralStrategy;
rat = rat or all(gens S, v -> isConstant sub(sub(v, kP), S));
if not rat or useBM === false then (
if debugLevel > 1 then <<"macaulayMatrixKernel: using DZ strategy"<<endl;
L := (map(S^1,S^1,0), map(kP^1,kP^0,0));
d := max(degLim, 1);
while true do (
Ldim := numcols last L;
dBasis := basis(0,d,S);
polys := transpose idealBasis(I,d);
M' := diff(dBasis, polys);
M := (map(kP,S)) M';
if debugLevel >= 1 then <<"Cols: "<<numColumns M<<", rows: "<<numRows M<<endl;
K := myKernel(M,Tolerance => tol, KernelStrategy => kerStrat);
L = (dBasis, K);
if degLim >=0 or Ldim == numcols last L then break;
d = d+1;
);
L
)
else if not rat and useBM === true then error"expected rational point when IntegralStrategy => true"
else (
if debugLevel > 1 then <<"macaulayMatrixKernel: using BM strategy"<<endl;
pt := sub(sub(vars S, kP), S);
if degLim < 0 then degLim = infinity;
igens := sub(gens I, vars S + pt);
H := truncDualData(igens,false,tol);
DB := map(S^1,S^0,0);
d = -1;
DBdim := -1;
while DBdim != numcols DB and d < degLim do (
d = d+1;
DBdim = numcols DB;
H = nextTDD(d,H,tol, KernelStrategy => kerStrat);
DB = H.dBasis;
);
(M,L) = coefficients DB;
L = apply(flatten entries M / exponents / first / listFactorial, entries L, (c, l) -> 1/c * l);
(M, if #L == 0 then map(kP^0, kP^0, 0) else sub(matrix L, kP))
)
)
-- returns a list of Diff ops based on matrices M, db
-- currently db must be a row-vector
matrixToDiffOps = (M, db) -> (
R := ring M;
S := diffOpRing R;
phi := map(S,R, gens S);
if M == 0 then {diffOp(0_S)}
else transpose entries M /
(c -> phi db * transpose matrix {c}) /
diffOp //
sort
)
noetherianOperatorsViaMacaulayMatrix = method(Options => true)
noetherianOperatorsViaMacaulayMatrix (Ideal, Ideal) := List => true >> opts -> (I, P) -> (
R := ring I;
m := if opts.?DegreeLimit then opts.DegreeLimit else -1;
if instance(m, InfiniteNumber) then m = -1;
rat := if opts.?Rational then opts.Rational else false;
t := getTolerance(R,opts);
(depVars,indVars) := getDepIndVars(P, opts);
-- use the original coefficient field if appropriate, else a rational function field
F := if #indVars == 0 then coefficientRing R else frac((coefficientRing R)(monoid[indVars]));
S := F(monoid[depVars]);
PS := sub(P,S); IS := sub(I,S);
-- extend the field only if the point is not specified to be rational
kP := if rat then F else toField(S/PS);
L := macaulayMatrixKernel(IS, kP, opts, Tolerance => t, DegreeLimit => m, Rational => rat);
-- Clear denominators, create list of DiffOps
matrixToDiffOps(liftColumns(lift(last L,S), R), sub(first L,R))
)
TEST ///
debug NoetherianOperators
R = QQ[x,y,t]
I = ideal(x^2, y^2 - t*x)
nops = noetherianOperatorsViaMacaulayMatrix(I) / normalize
S = diffOpRing R
correct = {diffOp(1_S), diffOp(dy), diffOp(t*dy^2+2*dx), diffOp(t*dy^3+6*dx*dy)}
assert(all(nops, correct, (i,j) -> i == j))
R = QQ[x,y]