1 | // IB/IG/JJS, last modified: 10.07.2007 |
---|
2 | /////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
3 | version = "$Id: CIMonomialCurve.lib"; |
---|
4 | category="Commutative Algebra"; |
---|
5 | info=" |
---|
6 | LIBRARY: cimonom.lib Determines if the toric ideal of an affine monomial curve is a complete intersection |
---|
7 | |
---|
8 | AUTHORS: I.Bermejo, ibermejo@ull.es |
---|
9 | @* I.Garcia-Marco, iggarcia@ull.es |
---|
10 | @* J.-J.Salazar-Gonzalez, jjsalaza@ull.es |
---|
11 | |
---|
12 | OVERVIEW: |
---|
13 | A library for determining if the toric ideal of an affine monomial curve is a complete intersection with NO |
---|
14 | NEED of computing explicitly a system of generators of such ideal. It also contains procedures to obtain the |
---|
15 | minimum positive multiple of an integer which is in a semigroup of positive integers. |
---|
16 | The procedures are based on a paper by Isabel Bermejo, Ignacio Garcia and Juan Jose Salazar-Gonzalez: 'An |
---|
17 | algorithm to check whether the toric ideal of an affine monomial curve is a complete intersection', Preprint. |
---|
18 | |
---|
19 | SEE ALSO: Integer programming |
---|
20 | |
---|
21 | PROCEDURES: |
---|
22 | BelongSemig(n,v[,sup]); checks whether n is in the semigroup generated by v; |
---|
23 | MinMult(a,b); computes k, the minimum positive integer such that k*a is in the semigroup of |
---|
24 | positive integers generated by the elements in b. |
---|
25 | CompInt(d); checks wether I(d) is a complete intersection or not. |
---|
26 | "; |
---|
27 | |
---|
28 | LIB "general.lib"; |
---|
29 | |
---|
30 | /////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
31 | // |
---|
32 | proc BelongSemig(bigint n, intvec v, list #) |
---|
33 | " |
---|
34 | USAGE: BelongSemig (n,v[,sup]); n bigint, v and sup intvec |
---|
35 | RETURN: In the default form, it returns 1 if n is in the semigroup generated by |
---|
36 | the elements of v or 0 otherwise. If the argument sup is added and in case |
---|
37 | n belongs to the semigroup generated by the elements of v, it returns |
---|
38 | a monomial in the variables {x(i) | i in sup} of degree n if we set |
---|
39 | deg(x(sup[j])) = v[j]. |
---|
40 | ASSUME: v and sup positive integer vectors of same size, sup has no |
---|
41 | repeated entries, x(i) has to be an indeterminate in the current ring for |
---|
42 | all i in sup. |
---|
43 | EXAMPLE: example BelongSemig; shows some examples |
---|
44 | " |
---|
45 | { |
---|
46 | //--------------------------- initialisation --------------------------------- |
---|
47 | int i, j, num; |
---|
48 | bigint PartialSum; |
---|
49 | num = size(v); |
---|
50 | int e = size(#); |
---|
51 | |
---|
52 | if (e > 0) |
---|
53 | { |
---|
54 | intvec sup = #[1]; |
---|
55 | poly mon; |
---|
56 | } |
---|
57 | |
---|
58 | for (i = 1; i <= nrows(v); i++) |
---|
59 | { |
---|
60 | if ((n % v[i]) == 0) |
---|
61 | { |
---|
62 | // ---- n is multiple of v[i] |
---|
63 | if (e) |
---|
64 | { |
---|
65 | mon = x(sup[i])^(int(n/v[i])); |
---|
66 | return(mon); |
---|
67 | } |
---|
68 | else |
---|
69 | { |
---|
70 | return (1); |
---|
71 | } |
---|
72 | } |
---|
73 | } |
---|
74 | |
---|
75 | if (num == 1) |
---|
76 | { |
---|
77 | // ---- num = 1 and n is not multiple of v[1] --> FALSE |
---|
78 | return(0); |
---|
79 | } |
---|
80 | |
---|
81 | intvec counter; |
---|
82 | counter[num] = 0; |
---|
83 | PartialSum = 0; |
---|
84 | |
---|
85 | intvec w = sort(v)[1]; |
---|
86 | intvec cambio = sort(v)[2]; |
---|
87 | |
---|
88 | // ---- Iterative procedure to determine if n is in the semigroup generated by v |
---|
89 | while (1) |
---|
90 | { |
---|
91 | if (n >= PartialSum) |
---|
92 | { |
---|
93 | if (((n - PartialSum) % w[1]) == 0) |
---|
94 | { |
---|
95 | // ---- n belongs to the semigroup generated by v, |
---|
96 | if (e) |
---|
97 | { |
---|
98 | // ---- obtain the monomial. |
---|
99 | mon = x(sup[cambio[1]])^(int((n - PartialSum) / w[1])); |
---|
100 | for (j = 2; j <= num; j++) |
---|
101 | { |
---|
102 | mon = mon * x(sup[cambio[j]])^(counter[j]); |
---|
103 | } |
---|
104 | return(mon); |
---|
105 | } |
---|
106 | else |
---|
107 | { |
---|
108 | // ---- returns true. |
---|
109 | return (1); |
---|
110 | } |
---|
111 | } |
---|
112 | } |
---|
113 | i = num; |
---|
114 | while (!defined(end)) |
---|
115 | { |
---|
116 | if (i == 1) |
---|
117 | { |
---|
118 | // ---- Stop, n is not in the semigroup |
---|
119 | return(0); |
---|
120 | } |
---|
121 | if (i > 1) |
---|
122 | { |
---|
123 | // counters control |
---|
124 | if (counter[i] >= ((n - PartialSum) / w[i])) |
---|
125 | { |
---|
126 | PartialSum = PartialSum - (counter[i]*w[i]); |
---|
127 | counter[i] = 0; |
---|
128 | i--; |
---|
129 | } |
---|
130 | else |
---|
131 | { |
---|
132 | counter[i] = counter[i] + 1; |
---|
133 | PartialSum = PartialSum + w[i]; |
---|
134 | int end; |
---|
135 | } |
---|
136 | } |
---|
137 | } |
---|
138 | kill end; |
---|
139 | } |
---|
140 | } |
---|
141 | example |
---|
142 | { "EXAMPLE:"; |
---|
143 | ring r=0,x(1..5),dp; |
---|
144 | int a = 125; |
---|
145 | intvec v = 13,17,51; |
---|
146 | intvec sup = 2,4,1; |
---|
147 | BelongSemig(a,v,sup); |
---|
148 | BelongSemig(a,v); |
---|
149 | } |
---|
150 | |
---|
151 | /////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
152 | /////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
153 | /////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
154 | |
---|
155 | proc MinMult(int a, intvec b) |
---|
156 | " |
---|
157 | USAGE: MinMult (a, b); a integer, b integer vector. |
---|
158 | RETURN: an integer k, the minimum positive integer such that ka belongs to the |
---|
159 | semigroup generated by the integers in b. |
---|
160 | ASSUME: a is a positive integer, b is a positive integers vector. |
---|
161 | EXAMPLE: example MinMult; shows some examples. |
---|
162 | " |
---|
163 | { |
---|
164 | //--------------------------- initialisation --------------------------------- |
---|
165 | int i, j, min, max; |
---|
166 | int n = nrows(b); |
---|
167 | |
---|
168 | if (n == 1) |
---|
169 | { |
---|
170 | // ---- trivial case |
---|
171 | return(b[1]/gcd(a,b[1])); |
---|
172 | } |
---|
173 | |
---|
174 | max = b[1]; |
---|
175 | for (i = 2; i <= n; i++) |
---|
176 | { |
---|
177 | if (b[i] > max) |
---|
178 | { |
---|
179 | max = b[i]; |
---|
180 | } |
---|
181 | } |
---|
182 | int NumNodes = a + max; //----Number of nodes in the graph |
---|
183 | |
---|
184 | int dist = 1; |
---|
185 | // ---- Auxiliary structures to obtain the shortest path between the nodes 1 and a+1 of this graph |
---|
186 | intvec queue = 1; |
---|
187 | intvec queue2; |
---|
188 | |
---|
189 | // ---- Control vector: |
---|
190 | // control[i] = 0 -> node not reached yet |
---|
191 | // control[i] = 1 -> node in queue1 |
---|
192 | // control[i] = 2 -> node in queue2 |
---|
193 | // control[i] = 3 -> node already processed |
---|
194 | intvec control; |
---|
195 | control[1] = 3; // Starting node |
---|
196 | control[a + max] = 0; // Ending node |
---|
197 | int current = 1; // Current node |
---|
198 | int next; // Node connected to corrent by arc (current, next) |
---|
199 | |
---|
200 | int ElemQueue, ElemQueue2; |
---|
201 | int PosQueue = 1; |
---|
202 | |
---|
203 | // Algoritmo de Dijkstra |
---|
204 | while (1) |
---|
205 | { |
---|
206 | if (current <= a) |
---|
207 | { |
---|
208 | // ---- current <= a, arcs are (current, current + b[i]) |
---|
209 | for (i = 1; i <= n; i++) |
---|
210 | { |
---|
211 | next = current + b[i]; |
---|
212 | if (next == a+1) |
---|
213 | { |
---|
214 | kill control; |
---|
215 | kill queue; |
---|
216 | kill queue2; |
---|
217 | return (dist); |
---|
218 | } |
---|
219 | if ((control[next] == 0)||(control[next] == 2)) |
---|
220 | { |
---|
221 | control[next] = 1; |
---|
222 | queue = queue, next; |
---|
223 | } |
---|
224 | } |
---|
225 | } |
---|
226 | if (current > a) |
---|
227 | { |
---|
228 | // ---- current > a, the only possible ars is (current, current - a) |
---|
229 | next = current - a; |
---|
230 | if (control[next] == 0) |
---|
231 | { |
---|
232 | control[next] = 2; |
---|
233 | queue2[nrows(queue2) + 1] = next; |
---|
234 | } |
---|
235 | } |
---|
236 | PosQueue++; |
---|
237 | if (PosQueue <= nrows(queue)) |
---|
238 | { |
---|
239 | current = queue[PosQueue]; |
---|
240 | } |
---|
241 | else |
---|
242 | { |
---|
243 | dist++; |
---|
244 | if (control[a+1] == 2) |
---|
245 | { |
---|
246 | return(dist); |
---|
247 | } |
---|
248 | queue = queue2[2..nrows(queue2)]; |
---|
249 | current = queue[1]; |
---|
250 | PosQueue = 1; |
---|
251 | queue2 = 0; |
---|
252 | } |
---|
253 | control[current] = 3; |
---|
254 | } |
---|
255 | } |
---|
256 | example |
---|
257 | { "EXAMPLE:"; |
---|
258 | "int a = 46;"; |
---|
259 | "intvec b = 13,17,59;"; |
---|
260 | "MinMult(a,b);"; |
---|
261 | int a = 46; |
---|
262 | intvec b = 13,17,59; |
---|
263 | MinMult(a,b); |
---|
264 | "// 3*a = 8*b[1] + 2*b[2]" |
---|
265 | } |
---|
266 | |
---|
267 | /////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
268 | /////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
269 | /////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
270 | |
---|
271 | proc CompInt(intvec d) |
---|
272 | " |
---|
273 | USAGE: CompInt(d); d intvec. |
---|
274 | RETURN: 1 if the toric ideal I(d) is a complete intersection or 0 otherwise. |
---|
275 | ASSUME: d is a vector of positive integers. |
---|
276 | NOTE: If printlevel > 0, additional info is displayed in case |
---|
277 | I(d) is a complete intersection: |
---|
278 | if printlevel >= 1, it displays a minimal set of generators of the toric |
---|
279 | ideal formed by quasihomogeneous binomials. Moreover, if printlevel >= 2 |
---|
280 | and gcd(d) = 1, it also shows the Frobenius number of the semigroup |
---|
281 | generated by the elements in d. |
---|
282 | EXAMPLE: example CompInt; shows some examples |
---|
283 | " |
---|
284 | { |
---|
285 | //--------------------------- initialisation --------------------------------- |
---|
286 | |
---|
287 | int i,j,k,l,divide,equal,possible; |
---|
288 | |
---|
289 | int n = nrows(d); |
---|
290 | int max = 2*n - 1; |
---|
291 | ring r = 0, x(1..n), dp; |
---|
292 | |
---|
293 | int level = printlevel - voice + 2; |
---|
294 | // ---- To decide how much extra information calculate and display |
---|
295 | if (level > 1) |
---|
296 | { |
---|
297 | int e = d[1]; |
---|
298 | for (i = 2; i <= n; i++) |
---|
299 | { |
---|
300 | e = gcd(e,d[i]); |
---|
301 | } |
---|
302 | if (e <> 1) |
---|
303 | { |
---|
304 | print ("// Semigroup generated by d is not numerical!"); |
---|
305 | } |
---|
306 | } |
---|
307 | if (level > 0) |
---|
308 | { |
---|
309 | ideal id; |
---|
310 | vector mon; |
---|
311 | mon[max] = 0; |
---|
312 | if ((level > 1)&&(e == 1)) |
---|
313 | { |
---|
314 | bigint frob = 0; |
---|
315 | } |
---|
316 | } |
---|
317 | |
---|
318 | // ---- Trivial cases: n = 1,2 (it is a complete intersection). |
---|
319 | if (n == 1) |
---|
320 | { |
---|
321 | print("// Ideal is (0)"); |
---|
322 | return (1); |
---|
323 | } |
---|
324 | |
---|
325 | if (n == 2) |
---|
326 | { |
---|
327 | if (level > 0) |
---|
328 | { |
---|
329 | intvec d1 = d[1]; |
---|
330 | intvec d2 = d[2]; |
---|
331 | int f1 = MinMult(d[1],d2); |
---|
332 | int f2 = MinMult(d[2],d1); |
---|
333 | id = x(1)^(f1) - x(2)^(f2); |
---|
334 | print ("// Toric ideal:"); |
---|
335 | id; |
---|
336 | if ((level > 1)&&(e == 1)) |
---|
337 | { |
---|
338 | frob = d[1]*f1 - d[1] - d[2]; |
---|
339 | print ("// Frobenius number of the numerical semigroup:"); |
---|
340 | frob; |
---|
341 | } |
---|
342 | } |
---|
343 | return (1); |
---|
344 | } |
---|
345 | |
---|
346 | // ---- For n >= 3 (non-trivial cases) |
---|
347 | matrix mat[max][n]; |
---|
348 | intvec using, bound, multiple; |
---|
349 | multiple[max] = 0; |
---|
350 | bound[max] = 0; |
---|
351 | using[max] = 0; |
---|
352 | |
---|
353 | for (i = 1; i <= n; i++) |
---|
354 | { |
---|
355 | using[i] = 1; |
---|
356 | multiple[i] = 0; |
---|
357 | mat[i,i] = 1; |
---|
358 | } |
---|
359 | if (level > 1) |
---|
360 | { |
---|
361 | if (e == 1) |
---|
362 | { |
---|
363 | for (i = 1; i <= n; i++) |
---|
364 | { |
---|
365 | frob = frob - d[i]; |
---|
366 | } |
---|
367 | } |
---|
368 | } |
---|
369 | |
---|
370 | int new, new1, new2; |
---|
371 | for (i = 1; i <= n; i++) |
---|
372 | { |
---|
373 | for (j = 1; j < i; j++) |
---|
374 | { |
---|
375 | if (i <> j) |
---|
376 | { |
---|
377 | new = gcd(d[i],d[j]); |
---|
378 | new1 = d[j]/new; |
---|
379 | new2 = d[i]/new; |
---|
380 | if (!bound[i] ||(new1 < bound[i])) |
---|
381 | { |
---|
382 | bound[i] = new1; |
---|
383 | } |
---|
384 | if (!bound[j] ||(new2 < bound[j])) |
---|
385 | { |
---|
386 | bound[j] = new2; |
---|
387 | } |
---|
388 | } |
---|
389 | } |
---|
390 | } |
---|
391 | |
---|
392 | // ---- Begins the inductive part |
---|
393 | for (i = 1; i < n; i++) |
---|
394 | { |
---|
395 | // ---- n-1 stages |
---|
396 | for (j = 1; j < n + i; j++) |
---|
397 | { |
---|
398 | if ((using[j])&&(multiple[j] == 0)) |
---|
399 | { |
---|
400 | possible = 0; |
---|
401 | for (k = 1; (k < n + i)&&(!possible); k++) |
---|
402 | { |
---|
403 | if ((using[k])&&(k != j)&&(bigint(bound[k])*d[k] == bigint(bound[j])*d[j])) |
---|
404 | { |
---|
405 | possible = 1; |
---|
406 | } |
---|
407 | } |
---|
408 | if (possible) |
---|
409 | { |
---|
410 | // ---- If possible == 1, then c_j has to be computed |
---|
411 | intvec aux; |
---|
412 | // ---- auxiliary vector containing all d[l] in use except d[j] |
---|
413 | k = 1; |
---|
414 | for (l = 1; l < n + i; l++) |
---|
415 | { |
---|
416 | if (using[l] && (l != j)) |
---|
417 | { |
---|
418 | aux[k] = d[l]; |
---|
419 | k++; |
---|
420 | } |
---|
421 | } |
---|
422 | |
---|
423 | multiple[j] = MinMult(d[j], aux); |
---|
424 | kill aux; |
---|
425 | |
---|
426 | if (j <= n) |
---|
427 | { |
---|
428 | if (level > 0) |
---|
429 | { |
---|
430 | mon = mon + (x(j)^multiple[j])*gen(j); |
---|
431 | } |
---|
432 | } |
---|
433 | else |
---|
434 | { |
---|
435 | // ---- if j > n, it has to be checked if c_j belongs to a certain semigroup |
---|
436 | intvec aux, sup; |
---|
437 | k = 1; |
---|
438 | for (l = 1; l <= n; l++) |
---|
439 | { |
---|
440 | if (mat[j, l] <> 0) |
---|
441 | { |
---|
442 | sup[k] = l; |
---|
443 | aux[k] = d[l]; |
---|
444 | k++; |
---|
445 | } |
---|
446 | } |
---|
447 | if (level > 0) |
---|
448 | { |
---|
449 | mon = mon + (BelongSemig(bigint(multiple[j])*d[j], aux, sup))*gen(j); |
---|
450 | if (mon[j] == 0) |
---|
451 | { |
---|
452 | // ---- multiple[j]*d[j] does not belong to the semigroup generated by aux, |
---|
453 | // ---- then it is NOT a complete intersection |
---|
454 | return (0); |
---|
455 | } |
---|
456 | } |
---|
457 | else |
---|
458 | { |
---|
459 | if (!BelongSemig(bigint(multiple[j])*d[j], aux)) |
---|
460 | { |
---|
461 | // ---- multiple[j]*d[j] does not belong to the semigroup generated by aux, |
---|
462 | // ---- then it is NOT a complete intersection |
---|
463 | return (0); |
---|
464 | } |
---|
465 | } |
---|
466 | kill sup; |
---|
467 | kill aux; |
---|
468 | } |
---|
469 | |
---|
470 | // ---- Searching if there exist k such that multiple[k]*d[k]= multiple[j]*d[j] |
---|
471 | equal = 0; |
---|
472 | for (k = 1; k < n+i; k++) |
---|
473 | { |
---|
474 | if ((k <> j) && multiple[k] && using[k]) |
---|
475 | { |
---|
476 | if (d[j]*bigint(multiple[j]) == d[k]*bigint(multiple[k])) |
---|
477 | { |
---|
478 | // found |
---|
479 | equal = k; |
---|
480 | break; |
---|
481 | } |
---|
482 | } |
---|
483 | } |
---|
484 | // ---- if equal = 0 no coincidence |
---|
485 | if (!equal) |
---|
486 | { |
---|
487 | if (j == n + i - 1) |
---|
488 | { |
---|
489 | // ---- All multiple[k]*d[k] in use are different -> NOT complete intersection |
---|
490 | return (0); |
---|
491 | } |
---|
492 | } |
---|
493 | else |
---|
494 | { |
---|
495 | // ---- Next stage is prepared |
---|
496 | if (level > 0) |
---|
497 | { |
---|
498 | //---- New generator of the toric ideal |
---|
499 | id[i] = mon[j] - mon[equal]; |
---|
500 | if ((level > 1)&&(e == 1)) |
---|
501 | { |
---|
502 | frob = frob + bigint(multiple[j])*d[j]; |
---|
503 | } |
---|
504 | } |
---|
505 | //---- Two exponents are removed and one is added |
---|
506 | using[j] = 0; |
---|
507 | using[equal] = 0; |
---|
508 | using[n + i] = 1; |
---|
509 | d[n + i] = gcd(d[j], d[equal]); //---- new exponent |
---|
510 | for (l = 1; l <= n; l++) |
---|
511 | { |
---|
512 | mat[n + i, l] = mat[j, l] + mat[equal, l]; |
---|
513 | } |
---|
514 | |
---|
515 | // Bounds are reestablished |
---|
516 | for (l = 1; l < n+i; l++) |
---|
517 | { |
---|
518 | if (using[l]) |
---|
519 | { |
---|
520 | divide = gcd(d[l],d[n+i]); |
---|
521 | new = d[n+i] / divide; |
---|
522 | if ((multiple[l])&&(multiple[l] > new)) |
---|
523 | { |
---|
524 | return (0); |
---|
525 | } |
---|
526 | if (new < bound[l]) |
---|
527 | { |
---|
528 | bound[l] = new; |
---|
529 | } |
---|
530 | new = d[l] / divide; |
---|
531 | if ( !bound[n+i] || (new < bound[n+i])) |
---|
532 | { |
---|
533 | bound[n+i] = new; |
---|
534 | } |
---|
535 | } |
---|
536 | } |
---|
537 | break; |
---|
538 | } |
---|
539 | } |
---|
540 | } |
---|
541 | if (j == n + i - 1) |
---|
542 | { |
---|
543 | // ---- All multiple[k]*d[k] in use are different -> NOT complete intersection |
---|
544 | return (0); |
---|
545 | } |
---|
546 | } |
---|
547 | } |
---|
548 | if (level > 0) |
---|
549 | { |
---|
550 | "// Toric ideal: "; |
---|
551 | id; |
---|
552 | if ((level > 1)&&(e == 1)) |
---|
553 | { |
---|
554 | "// Frobenius number of the numerical semigroup: "; |
---|
555 | frob; |
---|
556 | } |
---|
557 | } |
---|
558 | return(1); |
---|
559 | } |
---|
560 | example |
---|
561 | { "EXAMPLE:"; |
---|
562 | "printlevel = 0;"; |
---|
563 | printlevel = 0; |
---|
564 | "intvec d = 14,15,10,21;"; |
---|
565 | intvec d = 14,15,10,21; |
---|
566 | "CompInt(d);"; |
---|
567 | CompInt(d); |
---|
568 | " "; |
---|
569 | "printlevel = 2;"; |
---|
570 | printlevel = 3; |
---|
571 | "d = 36,54,125,150,225;"; |
---|
572 | d = 36,54,125,150,225; |
---|
573 | "CompInt(d);"; |
---|
574 | CompInt(d); |
---|
575 | " "; |
---|
576 | "d = 45,70,75,98,147;"; |
---|
577 | d = 45,70,75,98,147; |
---|
578 | "CompInt(d);"; |
---|
579 | CompInt(d); |
---|
580 | }; |
---|
581 | /////////////////////////////////////////////////////////////////////////////// |
---|
582 | /////////////////////////////////////////////////////////////////////////////// |
---|