1 | /////////////////////////////////////////////////////// |
---|
2 | version="$Id$"; |
---|
3 | category="Noncommutative"; |
---|
4 | info=" |
---|
5 | LIBRARY: fpadim.lib Algorithms for quotient algebras in the letterplace case |
---|
6 | AUTHORS: Grischa Studzinski, grischa.studzinski@rwth-aachen.de |
---|
7 | |
---|
8 | Support: Joint projects LE 2697/2-1 and KR 1907/3-1 of the Priority Programme SPP 1489: |
---|
9 | @* 'Algorithmische und Experimentelle Methoden in Algebra, Geometrie und Zahlentheorie' |
---|
10 | @* of the German DFG |
---|
11 | |
---|
12 | OVERVIEW: Given the free algebra A = K<x_1,...,x_n> and a (finite) Groebner basis |
---|
13 | @* GB = {g_1,..,g_w}, one is interested in the K-dimension and in the |
---|
14 | @* explicit K-basis of A/<GB>. |
---|
15 | @* Therefore one is interested in the following data: |
---|
16 | @* - the Ufnarovskij graph induced by GB |
---|
17 | @* - the mistletoes of A/<GB> |
---|
18 | @* - the K-dimension of A/<GB> |
---|
19 | @* - the Hilbert series of A/<GB> |
---|
20 | @* |
---|
21 | @* The Ufnarovskij graph is used to determine whether A/<GB> has finite |
---|
22 | @* K-dimension. One has to check if the graph contains cycles. |
---|
23 | @* For the whole theory we refer to [ufna]. Given a |
---|
24 | @* reduced set of monomials GB one can define the basis tree, whose vertex |
---|
25 | @* set V consists of all normal monomials w.r.t. GB. For every two |
---|
26 | @* monomials m_1, m_2 in V there is a direct edge from m_1 to m_2, if and |
---|
27 | @* only if there exists x_k in {x_1,..,x_n}, such that m_1*x_k = m_2. The |
---|
28 | @* set M = {m in V | there is no edge from m to another monomial in V} is |
---|
29 | @* called the set of mistletoes. As one can easily see it consists of |
---|
30 | @* the endpoints of the graph. Since there is a unique path to every |
---|
31 | @* monomial in V the whole graph can be described only from the knowledge |
---|
32 | @* of the mistletoes. Note that V corresponds to a basis of A/<GB>, so |
---|
33 | @* knowing the mistletoes we know a K-basis. The name mistletoes was given |
---|
34 | @* to those points because of these miraculous value and the algorithm is |
---|
35 | @* named sickle, because a sickle is the tool to harvest mistletoes. |
---|
36 | @* For more details see [studzins]. This package uses the Letterplace |
---|
37 | @* format introduced by [lls]. The algebra can either be represented as a |
---|
38 | @* Letterplace ring or via integer vectors: Every variable will only be |
---|
39 | @* represented by its number, so variable one is represented as 1, |
---|
40 | @* variable two as 2 and so on. The monomial x_1*x_3*x_2 for example will |
---|
41 | @* be stored as (1,3,2). Multiplication is concatenation. Note that there |
---|
42 | @* is no algorithm for computing the normal form needed for our case. |
---|
43 | @* Note that the name fpadim.lib is short for dimensions of finite |
---|
44 | @* presented algebras. |
---|
45 | |
---|
46 | References: |
---|
47 | |
---|
48 | @* [ufna] Ufnarovskij: Combinatorical and asymptotic methods in algebra, 1990 |
---|
49 | @* [lls] Levandovskyy, La Scala: Letterplace ideals and non-commutative |
---|
50 | Groebner bases, 2009 |
---|
51 | @* [studzins] Studzinski: Dimension computations in non-commutative, |
---|
52 | associative algebras, Diploma thesis, RWTH Aachen, 2010 |
---|
53 | |
---|
54 | Assumptions: |
---|
55 | @* - basering is always a Letterplace ring |
---|
56 | @* - all intvecs correspond to Letterplace monomials |
---|
57 | @* - if you specify a different degree bound d, |
---|
58 | d <= attrib(basering,uptodeg) should hold. |
---|
59 | @* In the procedures below, 'iv' stands for intvec representation |
---|
60 | and 'lp' for the letterplace representation of monomials |
---|
61 | |
---|
62 | PROCEDURES: |
---|
63 | |
---|
64 | ivDHilbert(L,n[,d]); computes the K-dimension and the Hilbert series |
---|
65 | ivDHilbertSickle(L,n[,d]); computes mistletoes, K-dimension and Hilbert series |
---|
66 | ivDimCheck(L,n); checks if the K-dimension of A/<L> is infinite |
---|
67 | ivHilbert(L,n[,d]); computes the Hilbert series of A/<L> in intvec format |
---|
68 | ivKDim(L,n[,d]); computes the K-dimension of A/<L> in intvec format |
---|
69 | ivMis2Base(M); computes a K-basis of the factor algebra |
---|
70 | ivMis2Dim(M); computes the K-dimension of the factor algebra |
---|
71 | ivOrdMisLex(M); orders a list of intvecs lexicographically |
---|
72 | ivSickle(L,n[,d]); computes the mistletoes of A/<L> in intvec format |
---|
73 | ivSickleHil(L,n[,d]); computes the mistletoes and Hilbert series of A/<L> |
---|
74 | ivSickleDim(L,n[,d]); computes the mistletoes and the K-dimension of A/<L> |
---|
75 | lpDHilbert(G[,d,n]); computes the K-dimension and Hilbert series of A/<G> |
---|
76 | lpDHilbertSickle(G[,d,n]); computes mistletoes, K-dimension and Hilbert series |
---|
77 | lpHilbert(G[,d,n]); computes the Hilbert series of A/<G> in lp format |
---|
78 | lpDimCheck(G); checks if the K-dimension of A/<G> is infinite |
---|
79 | lpKDim(G[,d,n]); computes the K-dimension of A/<G> in lp format |
---|
80 | lpMis2Base(M); computes a K-basis of the factor algebra |
---|
81 | lpMis2Dim(M); computes the K-dimension of the factor algebra |
---|
82 | lpOrdMisLex(M); orders an ideal of lp-monomials lexicographically |
---|
83 | lpSickle(G[,d,n]); computes the mistletoes of A/<G> in lp format |
---|
84 | lpSickleHil(G[,d,n]); computes the mistletoes and Hilbert series of A/<G> |
---|
85 | lpSickleDim(G[,d,n]); computes the mistletoes and the K-dimension of A/<G> |
---|
86 | sickle(G[,m,d,h]); can be used to access all lp main procedures |
---|
87 | |
---|
88 | |
---|
89 | ivL2lpI(L); transforms a list of intvecs into an ideal of lp monomials |
---|
90 | iv2lp(I); transforms an intvec into the corresponding monomial |
---|
91 | iv2lpList(L); transforms a list of intmats into an ideal of lp monomials |
---|
92 | iv2lpMat(M); transforms an intmat into an ideal of lp monomials |
---|
93 | lp2iv(p); transforms a polynomial into the corresponding intvec |
---|
94 | lp2ivId(G); transforms an ideal into the corresponding list of intmats |
---|
95 | lpId2ivLi(G); transforms a lp-ideal into the corresponding list of intvecs |
---|
96 | |
---|
97 | SEE ALSO: freegb_lib |
---|
98 | "; |
---|
99 | |
---|
100 | LIB "freegb.lib"; //for letterplace rings |
---|
101 | LIB "general.lib";//for sorting mistletoes |
---|
102 | |
---|
103 | ///////////////////////////////////////////////////////// |
---|
104 | |
---|
105 | |
---|
106 | //--------------- auxiliary procedures ------------------ |
---|
107 | |
---|
108 | static proc allVars(list L, intvec P, int n) |
---|
109 | "USAGE: allVars(L,P,n); L a list of intmats, P an intvec, n an integer |
---|
110 | RETURN: int, 0 if all variables are contained in the quotient algebra, 1 otherwise |
---|
111 | " |
---|
112 | {int i,j,r; |
---|
113 | intvec V; |
---|
114 | for (i = 1; i <= size(P); i++) {if (P[i] == 1){ j = i; break;}} |
---|
115 | V = L[j][1..nrows(L[j]),1]; |
---|
116 | for (i = 1; i <= n; i++) {if (isInVec(i,V) == 0) {r = 1; break;}} |
---|
117 | if (r == 0) {return(1);} |
---|
118 | else {return(0);} |
---|
119 | } |
---|
120 | |
---|
121 | static proc checkAssumptions(int d, list L) |
---|
122 | "PURPOSE: Checks, if all the Assumptions are holding |
---|
123 | " |
---|
124 | {if (typeof(attrib(basering,"isLetterplaceRing"))=="string") {ERROR("Basering is not a Letterplace ring!");} |
---|
125 | if (d > attrib(basering,"uptodeg")) {ERROR("Specified degree bound exceeds ring parameter!");} |
---|
126 | int i; |
---|
127 | for (i = 1; i <= size(L); i++) |
---|
128 | {if (entryViolation(L[i], attrib(basering,"lV"))) |
---|
129 | {ERROR("Not allowed monomial/intvec found!");} |
---|
130 | } |
---|
131 | return(); |
---|
132 | } |
---|
133 | |
---|
134 | static proc createStartMat(int d, int n) |
---|
135 | "USAGE: createStartMat(d,n); d, n integers |
---|
136 | RETURN: intmat |
---|
137 | PURPOSE:Creating the intmat with all normal monomials in n variables and of degree d to start with |
---|
138 | NOTE: d has to be > 0 |
---|
139 | " |
---|
140 | {intmat M[(n^d)][d]; |
---|
141 | int i1,i2,i3,i4; |
---|
142 | for (i1 = 1; i1 <= d; i1++) //Spalten |
---|
143 | {i2 = 1; //durchlaeuft Zeilen |
---|
144 | while (i2 <= (n^d)) |
---|
145 | {for (i3 = 1; i3 <= n; i3++) |
---|
146 | {for (i4 = 1; i4 <= (n^(i1-1)); i4++) |
---|
147 | {M[i2,i1] = i3; |
---|
148 | i2 = i2 + 1; |
---|
149 | } |
---|
150 | } |
---|
151 | } |
---|
152 | } |
---|
153 | return(M); |
---|
154 | } |
---|
155 | |
---|
156 | static proc createStartMat1(int n, intmat M) |
---|
157 | "USAGE: createStartMat1(n,M); n an integer, M an intmat |
---|
158 | RETURN: intmat, with all variables except those in M |
---|
159 | " |
---|
160 | {int i; |
---|
161 | intvec V,Vt; |
---|
162 | V = M[(1..nrows(M)),1]; |
---|
163 | for (i = 1; i <= size(V); i++) {if (isInVec(i,V) == 0) {Vt = Vt,i;}} |
---|
164 | if (Vt == 0) {intmat S; return(S);} |
---|
165 | else {Vt = Vt[2..size(Vt)]; intmat S [size(Vt)][1]; S[1..size(Vt),1] = Vt; return(S);} |
---|
166 | } |
---|
167 | |
---|
168 | static proc entryViolation(intmat M, int n) |
---|
169 | "PURPOSE:checks, if all entries in M are variable-related |
---|
170 | " |
---|
171 | {if ((nrows(M) == 1) && (ncols(M) == 1)) {if (M[1,1] == 0){return(0);}} |
---|
172 | int i,j; |
---|
173 | for (i = 1; i <= nrows(M); i++) |
---|
174 | {for (j = 1; j <= ncols(M); j++) |
---|
175 | {if(!((1<=M[i,j])&&(M[i,j]<=n))) {return(1);}} |
---|
176 | } |
---|
177 | return(0); |
---|
178 | } |
---|
179 | |
---|
180 | static proc findDimen(intvec V, int n, list L, intvec P, list #) |
---|
181 | "USAGE: findDimen(V,n,L,P,degbound); V,P intvecs, n, an integer, L a list, |
---|
182 | @* degbound an optional integer |
---|
183 | RETURN: int |
---|
184 | PURPOSE:Computing the K-dimension of the quotient algebra |
---|
185 | " |
---|
186 | {int degbound = 0; |
---|
187 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
188 | int dimen,i,j,w,it; |
---|
189 | intvec Vt,Vt2; |
---|
190 | module M; |
---|
191 | if (degbound == 0) |
---|
192 | {for (i = 1; i <= n; i++) |
---|
193 | {Vt = V, i; w = 0; |
---|
194 | for (j = 1; j<= size(P); j++) |
---|
195 | {if (P[j] <= size(Vt)) |
---|
196 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
197 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
198 | } |
---|
199 | } |
---|
200 | if (w == 0) |
---|
201 | {vector Vtt; |
---|
202 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
203 | M = M,Vtt; |
---|
204 | kill Vtt; |
---|
205 | } |
---|
206 | } |
---|
207 | if (size(M) == 0) {return(0);} |
---|
208 | else |
---|
209 | {M = simplify(M,2); |
---|
210 | for (i = 1; i <= size(M); i++) |
---|
211 | {kill Vt; intvec Vt; |
---|
212 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
213 | dimen = dimen + 1 + findDimen(Vt,n,L,P); |
---|
214 | } |
---|
215 | return(dimen); |
---|
216 | } |
---|
217 | } |
---|
218 | else |
---|
219 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
220 | if (size(V) == degbound) {return(0);} |
---|
221 | for (i = 1; i <= n; i++) |
---|
222 | {Vt = V, i; w = 0; |
---|
223 | for (j = 1; j<= size(P); j++) |
---|
224 | {if (P[j] <= size(Vt)) |
---|
225 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
226 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
227 | } |
---|
228 | } |
---|
229 | if (w == 0) {vector Vtt; |
---|
230 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
231 | M = M,Vtt; |
---|
232 | kill Vtt; |
---|
233 | } |
---|
234 | } |
---|
235 | if (size(M) == 0) {return(0);} |
---|
236 | else |
---|
237 | {M = simplify(M,2); |
---|
238 | for (i = 1; i <= size(M); i++) |
---|
239 | {kill Vt; intvec Vt; |
---|
240 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
241 | dimen = dimen + 1 + findDimen(Vt,n,L,P,degbound); |
---|
242 | } |
---|
243 | return(dimen); |
---|
244 | } |
---|
245 | } |
---|
246 | } |
---|
247 | |
---|
248 | static proc findCycle(intvec V, list L, intvec P, int n, int ld, module M) |
---|
249 | "USAGE: |
---|
250 | RETURN: int, 1 if Ufn-graph contains a cycle, or 0 otherwise |
---|
251 | PURPOSE:Searching the Ufnarovskij graph for cycles |
---|
252 | " |
---|
253 | {int i,j,w,r;intvec Vt,Vt2; |
---|
254 | int it, it2; |
---|
255 | if (size(V) < ld) |
---|
256 | {for (i = 1; i <= n; i++) |
---|
257 | {Vt = V,i; w = 0; |
---|
258 | for (j = 1; j <= size(P); j++) |
---|
259 | {if (P[j] <= size(Vt)) |
---|
260 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
261 | if (isInMat(Vt2,L[j]) > 0) |
---|
262 | {w = 1; break;} |
---|
263 | } |
---|
264 | } |
---|
265 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
---|
266 | if (r == 1) {break;} |
---|
267 | } |
---|
268 | return(r); |
---|
269 | } |
---|
270 | else |
---|
271 | {j = size(M); |
---|
272 | if (j > 0) |
---|
273 | { |
---|
274 | intmat Mt[j][nrows(M)]; |
---|
275 | for (it = 1; it <= j; it++) |
---|
276 | { for(it2 = 1; it2 <= nrows(M);it2++) |
---|
277 | {Mt[it,it2] = int(leadcoef(M[it2,it]));} |
---|
278 | } |
---|
279 | Vt = V[(size(V)-ld+1)..size(V)]; |
---|
280 | //Mt; type(Mt);Vt;type(Vt); |
---|
281 | if (isInMat(Vt,Mt) > 0) {return(1);} |
---|
282 | else |
---|
283 | {vector Vtt; |
---|
284 | for (it =1; it <= size(Vt); it++) |
---|
285 | {Vtt = Vtt + Vt[it]*gen(it);} |
---|
286 | M = M,Vtt; |
---|
287 | kill Vtt; |
---|
288 | for (i = 1; i <= n; i++) |
---|
289 | {Vt = V,i; w = 0; |
---|
290 | for (j = 1; j <= size(P); j++) |
---|
291 | {if (P[j] <= size(Vt)) |
---|
292 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
293 | //L[j]; type(L[j]);Vt2;type(Vt2); |
---|
294 | if (isInMat(Vt2,L[j]) > 0) |
---|
295 | {w = 1; break;} |
---|
296 | } |
---|
297 | } |
---|
298 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
---|
299 | if (r == 1) {break;} |
---|
300 | } |
---|
301 | return(r); |
---|
302 | } |
---|
303 | } |
---|
304 | else |
---|
305 | { Vt = V[(size(V)-ld+1)..size(V)]; |
---|
306 | vector Vtt; |
---|
307 | for (it = 1; it <= size(Vt); it++) |
---|
308 | {Vtt = Vtt + Vt[it]*gen(it);} |
---|
309 | M = Vtt; |
---|
310 | kill Vtt; |
---|
311 | for (i = 1; i <= n; i++) |
---|
312 | {Vt = V,i; w = 0; |
---|
313 | for (j = 1; j <= size(P); j++) |
---|
314 | {if (P[j] <= size(Vt)) |
---|
315 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
316 | //L[j]; type(L[j]);Vt2;type(Vt2); |
---|
317 | if (isInMat(Vt2,L[j]) > 0) |
---|
318 | {w = 1; break;} |
---|
319 | } |
---|
320 | } |
---|
321 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
---|
322 | if (r == 1) {break;} |
---|
323 | } |
---|
324 | return(r); |
---|
325 | } |
---|
326 | } |
---|
327 | } |
---|
328 | |
---|
329 | static proc findHCoeff(intvec V,int n,list L,intvec P,intvec H,list #) |
---|
330 | "USAGE: findHCoeff(V,n,L,P,H,degbound); L a list of intmats, degbound an integer |
---|
331 | RETURN: intvec |
---|
332 | PURPOSE:Computing the coefficient of the Hilbert series (upto degree degbound) |
---|
333 | NOTE: Starting with a part of the Hilbert series we change the coefficient |
---|
334 | @* depending on how many basis elements we found on the actual branch |
---|
335 | " |
---|
336 | {int degbound = 0; |
---|
337 | if (size(#) > 0){if (#[1] > 0){degbound = #[1];}} |
---|
338 | int i,w,j,it; |
---|
339 | int h1 = 0; |
---|
340 | intvec Vt,Vt2,H1; |
---|
341 | module M; |
---|
342 | if (degbound == 0) |
---|
343 | {for (i = 1; i <= n; i++) |
---|
344 | {Vt = V, i; w = 0; |
---|
345 | for (j = 1; j<= size(P); j++) |
---|
346 | {if (P[j] <= size(Vt)) |
---|
347 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
348 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
349 | } |
---|
350 | } |
---|
351 | if (w == 0) |
---|
352 | {vector Vtt; |
---|
353 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
354 | M = M,Vtt; |
---|
355 | kill Vtt; |
---|
356 | } |
---|
357 | } |
---|
358 | if (size(M) == 0) {return(H);} |
---|
359 | else |
---|
360 | {M = simplify(M,2); |
---|
361 | for (i = 1; i <= size(M); i++) |
---|
362 | {kill Vt; intvec Vt; |
---|
363 | for (j =1; j <= size(M[i]); j++) {Vt[j] = int(leadcoef(M[i][j]));} |
---|
364 | h1 = h1 + 1; H1 = findHCoeff(Vt,n,L,P,H1); |
---|
365 | } |
---|
366 | if (size(H1) < (size(V)+2)) {H1[(size(V)+2)] = h1;} |
---|
367 | else {H1[(size(V)+2)] = H1[(size(V)+2)] + h1;} |
---|
368 | H1 = H1 + H; |
---|
369 | return(H1); |
---|
370 | } |
---|
371 | } |
---|
372 | else |
---|
373 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
374 | if (size(V) == degbound) {return(H);} |
---|
375 | for (i = 1; i <= n; i++) |
---|
376 | {Vt = V, i; w = 0; |
---|
377 | for (j = 1; j<= size(P); j++) |
---|
378 | {if (P[j] <= size(Vt)) |
---|
379 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
380 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
381 | } |
---|
382 | } |
---|
383 | if (w == 0) |
---|
384 | {vector Vtt; |
---|
385 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
386 | M = M,Vtt; |
---|
387 | kill Vtt; |
---|
388 | } |
---|
389 | } |
---|
390 | if (size(M) == 0) {return(H);} |
---|
391 | else |
---|
392 | {M = simplify(M,2); |
---|
393 | for (i = 1; i <= size(M); i++) |
---|
394 | {kill Vt; intvec Vt; |
---|
395 | for (j =1; j <= size(M[i]); j++) |
---|
396 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
397 | h1 = h1 + 1; H1 = findHCoeff(Vt,n,L,P,H1,degbound); |
---|
398 | } |
---|
399 | if (size(H1) < (size(V)+2)) { H1[(size(V)+2)] = h1;} |
---|
400 | else {H1[(size(V)+2)] = H1[(size(V)+2)] + h1;} |
---|
401 | H1 = H1 + H; |
---|
402 | return(H1); |
---|
403 | } |
---|
404 | } |
---|
405 | } |
---|
406 | |
---|
407 | static proc findHCoeffMis(intvec V, int n, list L, intvec P, list R,list #) |
---|
408 | "USAGE: findHCoeffMis(V,n,L,P,R,degbound); degbound an optional integer, L a |
---|
409 | @* list of Intmats, R |
---|
410 | RETURN: list |
---|
411 | PURPOSE:Computing the coefficients of the Hilbert series and the Mistletoes all |
---|
412 | @* at once |
---|
413 | " |
---|
414 | {int degbound = 0; |
---|
415 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
416 | int i,w,j,h1; |
---|
417 | intvec Vt,Vt2,H1; int it; |
---|
418 | module M; |
---|
419 | if (degbound == 0) |
---|
420 | {for (i = 1; i <= n; i++) |
---|
421 | {Vt = V, i; w = 0; |
---|
422 | for (j = 1; j<= size(P); j++) |
---|
423 | {if (P[j] <= size(Vt)) |
---|
424 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
425 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
426 | } |
---|
427 | } |
---|
428 | if (w == 0) |
---|
429 | {vector Vtt; |
---|
430 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
431 | M = M,Vtt; |
---|
432 | kill Vtt; |
---|
433 | } |
---|
434 | } |
---|
435 | if (size(M) == 0) {if (size(R) < 2){R[2] = list(V);} else {R[2] = R[2] + list(V);} return(R);} |
---|
436 | else |
---|
437 | {M = simplify(M,2); |
---|
438 | for (i = 1; i <= size(M); i++) |
---|
439 | {kill Vt; intvec Vt; |
---|
440 | for (j =1; j <= size(M[i]); j++) |
---|
441 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
442 | if (size(R[1]) < (size(V)+2)) { R[1][(size(V)+2)] = 1;} |
---|
443 | else |
---|
444 | {R[1][(size(V)+2)] = R[1][(size(V)+2)] + 1;} |
---|
445 | R = findHCoeffMis(Vt,n,L,P,R); |
---|
446 | } |
---|
447 | return(R); |
---|
448 | } |
---|
449 | } |
---|
450 | else |
---|
451 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
452 | if (size(V) == degbound) |
---|
453 | {if (size(R) < 2){R[2] = list (V);} |
---|
454 | else{R[2] = R[2] + list (V);} |
---|
455 | return(R); |
---|
456 | } |
---|
457 | for (i = 1; i <= n; i++) |
---|
458 | {Vt = V, i; w = 0; |
---|
459 | for (j = 1; j<= size(P); j++) |
---|
460 | {if (P[j] <= size(Vt)) |
---|
461 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
462 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
463 | } |
---|
464 | } |
---|
465 | if (w == 0) |
---|
466 | {vector Vtt; |
---|
467 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
468 | M = M,Vtt; |
---|
469 | kill Vtt; |
---|
470 | } |
---|
471 | } |
---|
472 | if (size(M) == 0) {if (size(R) < 2){R[2] = list(V);} else {R[2] = R[2] + list(V);} return(R);} |
---|
473 | else |
---|
474 | {M = simplify(M,2); |
---|
475 | for (i = 1; i <= ncols(M); i++) |
---|
476 | {kill Vt; intvec Vt; |
---|
477 | for (j =1; j <= size(M[i]); j++) |
---|
478 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
479 | if (size(R[1]) < (size(V)+2)) { R[1][(size(V)+2)] = 1;} |
---|
480 | else |
---|
481 | {R[1][(size(V)+2)] = R[1][(size(V)+2)] + 1;} |
---|
482 | R = findHCoeffMis(Vt,n,L,P,R,degbound); |
---|
483 | } |
---|
484 | return(R); |
---|
485 | } |
---|
486 | } |
---|
487 | } |
---|
488 | |
---|
489 | |
---|
490 | static proc findMisDim(intvec V,int n,list L,intvec P,list R,list #) |
---|
491 | "USAGE: |
---|
492 | RETURN: list |
---|
493 | PURPOSE:Computing the K-dimension and the Mistletoes all at once |
---|
494 | " |
---|
495 | {int degbound = 0; |
---|
496 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
497 | int dimen,i,j,w; |
---|
498 | intvec Vt,Vt2; int it; |
---|
499 | module M; |
---|
500 | if (degbound == 0) |
---|
501 | {for (i = 1; i <= n; i++) |
---|
502 | {Vt = V, i; w = 0; |
---|
503 | for (j = 1; j<= size(P); j++) |
---|
504 | {if (P[j] <= size(Vt)) |
---|
505 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
506 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
507 | } |
---|
508 | } |
---|
509 | if (w == 0) |
---|
510 | {vector Vtt; |
---|
511 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
512 | M = M,Vtt; |
---|
513 | kill Vtt; |
---|
514 | } |
---|
515 | } |
---|
516 | if (size(M) == 0) |
---|
517 | {if (size(R) < 2){R[2] = list (V);} |
---|
518 | else{R[2] = R[2] + list(V);} |
---|
519 | return(R); |
---|
520 | } |
---|
521 | else |
---|
522 | {M = simplify(M,2); |
---|
523 | for (i = 1; i <= size(M); i++) |
---|
524 | {kill Vt; intvec Vt; |
---|
525 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
526 | R[1] = R[1] + 1; R = findMisDim(Vt,n,L,P,R); |
---|
527 | } |
---|
528 | return(R); |
---|
529 | } |
---|
530 | } |
---|
531 | else |
---|
532 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
533 | if (size(V) == degbound) |
---|
534 | {if (size(R) < 2){R[2] = list (V);} |
---|
535 | else{R[2] = R[2] + list (V);} |
---|
536 | return(R); |
---|
537 | } |
---|
538 | for (i = 1; i <= n; i++) |
---|
539 | {Vt = V, i; w = 0; |
---|
540 | for (j = 1; j<= size(P); j++) |
---|
541 | {if (P[j] <= size(Vt)) |
---|
542 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
543 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
544 | } |
---|
545 | } |
---|
546 | if (w == 0) |
---|
547 | {vector Vtt; |
---|
548 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
549 | M = M,Vtt; |
---|
550 | kill Vtt; |
---|
551 | } |
---|
552 | } |
---|
553 | if (size(M) == 0) |
---|
554 | {if (size(R) < 2){R[2] = list (V);} |
---|
555 | else{R[2] = R[2] + list(V);} |
---|
556 | return(R); |
---|
557 | } |
---|
558 | else |
---|
559 | {M = simplify(M,2); |
---|
560 | for (i = 1; i <= size(M); i++) |
---|
561 | {kill Vt; intvec Vt; |
---|
562 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
563 | R[1] = R[1] + 1; R = findMisDim(Vt,n,L,P,R,degbound); |
---|
564 | } |
---|
565 | return(R); |
---|
566 | |
---|
567 | } |
---|
568 | } |
---|
569 | } |
---|
570 | |
---|
571 | |
---|
572 | static proc findmistletoes(intvec V, int n, list L, intvec P, list #) |
---|
573 | "USAGE: findmistletoes(V,n,L,P,degbound); V a normal word, n the number of |
---|
574 | @* variables, L the GB, P the occuring degrees, |
---|
575 | @* and degbound the (optional) degreebound |
---|
576 | RETURN: list |
---|
577 | PURPOSE:Computing mistletoes starting in V |
---|
578 | NOTE: V has to be normal w.r.t. L, it will not be checked for being so |
---|
579 | " |
---|
580 | {int degbound = 0; |
---|
581 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
582 | list R; intvec Vt,Vt2; int it; |
---|
583 | int i,j; |
---|
584 | module M; |
---|
585 | if (degbound == 0) |
---|
586 | {int w; |
---|
587 | for (i = 1; i <= n; i++) |
---|
588 | {Vt = V,i; w = 0; |
---|
589 | for (j = 1; j <= size(P); j++) |
---|
590 | {if (P[j] <= size(Vt)) |
---|
591 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
592 | if (isInMat(Vt2,L[j]) > 0) |
---|
593 | {w = 1; break;} |
---|
594 | } |
---|
595 | } |
---|
596 | if (w == 0) |
---|
597 | {vector Vtt; |
---|
598 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
599 | M = M,Vtt; |
---|
600 | kill Vtt; |
---|
601 | } |
---|
602 | } |
---|
603 | if (size(M)==0) {R = V; return(R);} |
---|
604 | else |
---|
605 | {M = simplify(M,2); |
---|
606 | for (i = 1; i <= size(M); i++) |
---|
607 | {kill Vt; intvec Vt; |
---|
608 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
609 | R = R + findmistletoes(Vt,n,L,P); |
---|
610 | } |
---|
611 | return(R); |
---|
612 | } |
---|
613 | } |
---|
614 | else |
---|
615 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
616 | if (size(V) == degbound) {R = V; return(R);} |
---|
617 | int w; |
---|
618 | for (i = 1; i <= n; i++) |
---|
619 | {Vt = V,i; w = 0; |
---|
620 | for (j = 1; j <= size(P); j++) |
---|
621 | {if (P[j] <= size(Vt)) |
---|
622 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
623 | if (isInMat(Vt2,L[j]) > 0){w = 1; break;} |
---|
624 | } |
---|
625 | } |
---|
626 | if (w == 0) |
---|
627 | {vector Vtt; |
---|
628 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
629 | M = M,Vtt; |
---|
630 | kill Vtt; |
---|
631 | } |
---|
632 | } |
---|
633 | if (size(M) == 0) {R = V; return(R);} |
---|
634 | else |
---|
635 | {M = simplify(M,2); |
---|
636 | for (i = 1; i <= ncols(M); i++) |
---|
637 | {kill Vt; intvec Vt; |
---|
638 | for (j =1; j <= size(M[i]); j++) |
---|
639 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
640 | //Vt; typeof(Vt); size(Vt); |
---|
641 | R = R + findmistletoes(Vt,n,L,P,degbound); |
---|
642 | } |
---|
643 | return(R); |
---|
644 | } |
---|
645 | } |
---|
646 | } |
---|
647 | |
---|
648 | static proc isInList(intvec V, list L) |
---|
649 | "USAGE: isInList(V,L); V an intvec, L a list of intvecs |
---|
650 | RETURN: int |
---|
651 | PURPOSE:Finding the position of V in L, returns 0, if V is not in M |
---|
652 | " |
---|
653 | {int i,n; |
---|
654 | n = 0; |
---|
655 | for (i = 1; i <= size(L); i++) {if (L[i] == V) {n = i; break;}} |
---|
656 | return(n); |
---|
657 | } |
---|
658 | |
---|
659 | static proc isInMat(intvec V, intmat M) |
---|
660 | "USAGE: isInMat(V,M);V an intvec, M an intmat |
---|
661 | RETURN: int |
---|
662 | PURPOSE:Finding the position of V in M, returns 0, if V is not in M |
---|
663 | " |
---|
664 | {if (size(V) <> ncols(M)) {return(0);} |
---|
665 | int i; |
---|
666 | intvec Vt; |
---|
667 | for (i = 1; i <= nrows(M); i++) |
---|
668 | {Vt = M[i,1..ncols(M)]; |
---|
669 | if ((V-Vt) == 0){return(i);} |
---|
670 | } |
---|
671 | return(0); |
---|
672 | } |
---|
673 | |
---|
674 | static proc isInVec(int v,intvec V) |
---|
675 | "USAGE: isInVec(v,V); v an integer,V an intvec |
---|
676 | RETURN: int |
---|
677 | PURPOSE:Finding the position of v in V, returns 0, if v is not in V |
---|
678 | " |
---|
679 | {int i,n; |
---|
680 | n = 0; |
---|
681 | for (i = 1; i <= size(V); i++) {if (V[i] == v) {n = i; break;}} |
---|
682 | return(n); |
---|
683 | } |
---|
684 | |
---|
685 | proc ivL2lpI(list L) |
---|
686 | "USAGE: ivL2lpI(L); L a list of intvecs |
---|
687 | RETURN: ideal |
---|
688 | PURPOSE:Transforming a list of intvecs into an ideal of Letterplace monomials. |
---|
689 | @* For the encoding of the variables see the overview. |
---|
690 | ASSUME: - Intvec corresponds to a Letterplace monomial |
---|
691 | @* - basering has to be a Letterplace ring |
---|
692 | EXAMPLE: example ivL2lpI; shows examples |
---|
693 | " |
---|
694 | {checkAssumptions(0,L); |
---|
695 | int i; ideal G; |
---|
696 | poly p; |
---|
697 | for (i = 1; i <= size(L); i++) |
---|
698 | {p = iv2lp(L[i]); |
---|
699 | G[(size(G) + 1)] = p; |
---|
700 | } |
---|
701 | return(G); |
---|
702 | } |
---|
703 | example |
---|
704 | { |
---|
705 | "EXAMPLE:"; echo = 2; |
---|
706 | ring r = 0,(x,y,z),dp; |
---|
707 | def R = makeLetterplaceRing(5);// constructs a Letterplace ring |
---|
708 | setring R; //sets basering to Letterplace ring |
---|
709 | intvec u = 1,1,2; intvec v = 2,1,3; intvec w = 3,1,1; |
---|
710 | // u = x^2y, v = yxz, w = zx^2 in intvec representation |
---|
711 | list L = u,v,w; |
---|
712 | ivL2lpI(L);// invokes the procedure, returns the ideal containing u,v,w |
---|
713 | } |
---|
714 | |
---|
715 | proc iv2lp(intvec I) |
---|
716 | "USAGE: iv2lp(I); I an intvec |
---|
717 | RETURN: poly |
---|
718 | PURPOSE:Transforming an intvec into the corresponding Letterplace polynomial |
---|
719 | @* For the encoding of the variables see the overview. |
---|
720 | ASSUME: - Intvec corresponds to a Letterplace monomial |
---|
721 | @* - basering has to be a Letterplace ring |
---|
722 | NOTE: - Assumptions will not be checked! |
---|
723 | EXAMPLE: example iv2lp; shows examples |
---|
724 | " |
---|
725 | {if (I[1] == 0) {return(1);} |
---|
726 | int i = size(I); |
---|
727 | if (i > attrib(basering,"uptodeg")) {ERROR("polynomial exceeds degreebound");} |
---|
728 | int j; poly p = 1; |
---|
729 | for (j = 1; j <= i; j++) {if (I[j] > 0) { p = lpMult(p,var(I[j]));}} //ignore zeroes, because they correspond to 1 |
---|
730 | return(p); |
---|
731 | } |
---|
732 | example |
---|
733 | { |
---|
734 | "EXAMPLE:"; echo = 2; |
---|
735 | ring r = 0,(x,y,z),dp; |
---|
736 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
737 | setring R; //sets basering to Letterplace ring |
---|
738 | intvec u = 1,1,2; intvec v = 2,1,3; intvec w = 3,1,1; |
---|
739 | // u = x^2y, v = yxz, w = zx^2 in intvec representation |
---|
740 | iv2lp(u); // invokes the procedure and returns the corresponding poly |
---|
741 | iv2lp(v); |
---|
742 | iv2lp(w); |
---|
743 | } |
---|
744 | |
---|
745 | proc iv2lpList(list L) |
---|
746 | "USAGE: iv2lpList(L); L a list of intmats |
---|
747 | RETURN: ideal |
---|
748 | PURPOSE:Converting a list of intmats into an ideal of corresponding monomials |
---|
749 | @* The rows of the intmat corresponds to an intvec, which stores the |
---|
750 | @* monomial. |
---|
751 | @* For the encoding of the variables see the overview. |
---|
752 | ASSUME: - The rows of each intmat in L must correspond to a Letterplace monomial |
---|
753 | @* - basering has to be a Letterplace ring |
---|
754 | EXAMPLE: example iv2lpList; shows examples |
---|
755 | " |
---|
756 | {checkAssumptions(0,L); |
---|
757 | ideal G; |
---|
758 | int i; |
---|
759 | for (i = 1; i <= size(L); i++){G = G + iv2lpMat(L[i]);} |
---|
760 | return(G); |
---|
761 | } |
---|
762 | example |
---|
763 | { |
---|
764 | "EXAMPLE:"; echo = 2; |
---|
765 | ring r = 0,(x,y,z),dp; |
---|
766 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
767 | setring R; // sets basering to Letterplace ring |
---|
768 | intmat u[3][1] = 1,1,2; intmat v[1][3] = 2,1,3; intmat w[2][3] = 3,1,1,2,3,1; |
---|
769 | // defines intmats of different size containing intvec representations of |
---|
770 | // monomials as rows |
---|
771 | list L = u,v,w; |
---|
772 | print(u); print(v); print(w); // shows the intmats contained in L |
---|
773 | iv2lpList(L); // returns the corresponding monomials as an ideal |
---|
774 | } |
---|
775 | |
---|
776 | |
---|
777 | proc iv2lpMat(intmat M) |
---|
778 | "USAGE: iv2lpMat(M); M an intmat |
---|
779 | RETURN: ideal |
---|
780 | PURPOSE:Converting an intmat into an ideal of the corresponding monomials. |
---|
781 | @* The rows of the intmat corresponds to an intvec, which stores the |
---|
782 | @* monomial. |
---|
783 | @* For the encoding of the variables see the overview. |
---|
784 | ASSUME: - The rows of M must correspond to Letterplace monomials |
---|
785 | @* - basering has to be a Letterplace ring |
---|
786 | EXAMPLE: example iv2lpMat; shows examples |
---|
787 | " |
---|
788 | {list L = M; |
---|
789 | checkAssumptions(0,L); |
---|
790 | kill L; |
---|
791 | ideal G; poly p; |
---|
792 | int i; intvec I; |
---|
793 | for (i = 1; i <= nrows(M); i++) |
---|
794 | { I = M[i,1..ncols(M)]; |
---|
795 | p = iv2lp(I); |
---|
796 | G[size(G)+1] = p; |
---|
797 | } |
---|
798 | return(G); |
---|
799 | } |
---|
800 | example |
---|
801 | { |
---|
802 | "EXAMPLE:"; echo = 2; |
---|
803 | ring r = 0,(x,y,z),dp; |
---|
804 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
805 | setring R; // sets basering to Letterplace ring |
---|
806 | intmat u[3][1] = 1,1,2; intmat v[1][3] = 2,1,3; intmat w[2][3] = 3,1,1,2,3,1; |
---|
807 | // defines intmats of different size containing intvec representations of |
---|
808 | // monomials as rows |
---|
809 | iv2lpMat(u); // returns the monomials contained in u |
---|
810 | iv2lpMat(v); // returns the monomials contained in v |
---|
811 | iv2lpMat(w); // returns the monomials contained in w |
---|
812 | } |
---|
813 | |
---|
814 | proc lpId2ivLi(ideal G) |
---|
815 | "USAGE: lpId2ivLi(G); G an ideal |
---|
816 | RETURN: list |
---|
817 | PURPOSE:Transforming an ideal into the corresponding list of intvecs. |
---|
818 | @* For the encoding of the variables see the overview. |
---|
819 | ASSUME: - basering has to be a Letterplace ring |
---|
820 | EXAMPLE: example lpId2ivLi; shows examples |
---|
821 | " |
---|
822 | {int i,j,k; |
---|
823 | list M; |
---|
824 | checkAssumptions(0,M); |
---|
825 | for (i = 1; i <= size(G); i++) {M[i] = lp2iv(G[i]);} |
---|
826 | return(M); |
---|
827 | } |
---|
828 | example |
---|
829 | { |
---|
830 | "EXAMPLE:"; echo = 2; |
---|
831 | ring r = 0,(x,y),dp; |
---|
832 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
833 | setring R; // sets basering to Letterplace ring |
---|
834 | ideal L = x(1)*x(2),y(1)*y(2),x(1)*y(2)*x(3); |
---|
835 | lpId2ivLi(L); // returns the corresponding intvecs as a list |
---|
836 | } |
---|
837 | |
---|
838 | proc lp2iv(poly p) |
---|
839 | "USAGE: lp2iv(p); p a poly |
---|
840 | RETURN: intvec |
---|
841 | PURPOSE:Transforming a monomial into the corresponding intvec. |
---|
842 | @* For the encoding of the variables see the overview. |
---|
843 | ASSUME: - basering has to be a Letterplace ring |
---|
844 | NOTE: - Assumptions will not be checked! |
---|
845 | EXAMPLE: example lp2iv; shows examples |
---|
846 | " |
---|
847 | {p = normalize(lead(p)); |
---|
848 | intvec I; |
---|
849 | int i,j; |
---|
850 | if (deg(p) > attrib(basering,"uptodeg")) {ERROR("Monomial exceeds degreebound");} |
---|
851 | if (p == 1) {return(I);} |
---|
852 | if (p == 0) {ERROR("Monomial is not allowed to equal zero");} |
---|
853 | intvec lep = leadexp(p); |
---|
854 | for ( i = 1; i <= attrib(basering,"lV"); i++) {if (lep[i] == 1) {I = i; break;}} |
---|
855 | for (i = (attrib(basering,"lV")+1); i <= size(lep); i++) |
---|
856 | {if (lep[i] == 1) |
---|
857 | { j = (i mod attrib(basering,"lV")); |
---|
858 | if (j == 0) {I = I,attrib(basering,"lV");} |
---|
859 | else {I = I,j;} |
---|
860 | } |
---|
861 | else { if (lep[i] > 1) {ERROR("monomial has a not allowed multidegree");}} |
---|
862 | } |
---|
863 | if (I[1] == 0) {ERROR("monomial has a not allowed multidegree");} |
---|
864 | |
---|
865 | return(I); |
---|
866 | } |
---|
867 | example |
---|
868 | { |
---|
869 | "EXAMPLE:"; echo = 2; |
---|
870 | ring r = 0,(x,y,z),dp; |
---|
871 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
872 | setring R; // sets basering to Letterplace ring |
---|
873 | poly p = x(1)*x(2)*z(3); poly q = y(1)*y(2)*x(3)*x(4); |
---|
874 | poly w= z(1)*y(2)*x(3)*z(4)*z(5); |
---|
875 | // p,q,w are some polynomials we want to transform into their |
---|
876 | // intvec representation |
---|
877 | lp2iv(p); lp2iv(q); lp2iv(w); |
---|
878 | } |
---|
879 | |
---|
880 | proc lp2ivId(ideal G) |
---|
881 | "USAGE: lp2ivId(G); G an ideal |
---|
882 | RETURN: list |
---|
883 | PURPOSE:Converting an ideal into an list of intmats, |
---|
884 | @* the corresponding intvecs forming the rows. |
---|
885 | @* For the encoding of the variables see the overview. |
---|
886 | ASSUME: - basering has to be a Letterplace ring |
---|
887 | EXAMPLE: example lp2ivId; shows examples |
---|
888 | " |
---|
889 | {G = normalize(lead(G)); |
---|
890 | intvec I; list L; |
---|
891 | checkAssumptions(0,L); |
---|
892 | int i,md; |
---|
893 | for (i = 1; i <= size(G); i++) { if (md <= deg(G[i])) {md = deg(G[i]);}} |
---|
894 | while (size(G) > 0) |
---|
895 | {ideal Gt; |
---|
896 | for (i = 1; i <= ncols(G); i++) {if (md == deg(G[i])) {Gt = Gt + G[i]; G[i] = 0;}} |
---|
897 | if (size(Gt) > 0) |
---|
898 | {G = simplify(G,2); |
---|
899 | intmat M [size(Gt)][md]; |
---|
900 | for (i = 1; i <= size(Gt); i++) {M[i,1..md] = lp2iv(Gt[i]);} |
---|
901 | L = insert(L,M); |
---|
902 | kill M; kill Gt; |
---|
903 | md = md - 1; |
---|
904 | } |
---|
905 | else {kill Gt; md = md - 1;} |
---|
906 | } |
---|
907 | return(L); |
---|
908 | } |
---|
909 | example |
---|
910 | { |
---|
911 | "EXAMPLE:"; echo = 2; |
---|
912 | ring r = 0,(x,y,z),dp; |
---|
913 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
914 | setring R; // sets basering to Letterplace ring |
---|
915 | poly p = x(1)*x(2)*z(3); poly q = y(1)*y(2)*x(3)*x(4); |
---|
916 | poly w = z(1)*y(2)*x(3)*z(4); |
---|
917 | // p,q,w are some polynomials we want to transform into their |
---|
918 | // intvec representation |
---|
919 | ideal G = p,q,w; |
---|
920 | // define the ideal containing p,q and w |
---|
921 | lp2ivId(G); // and return the list of intmats for this ideal |
---|
922 | } |
---|
923 | |
---|
924 | // -----------------main procedures---------------------- |
---|
925 | |
---|
926 | proc ivDHilbert(list L, int n, list #) |
---|
927 | "USAGE: ivDHilbert(L,n[,degbound]); L a list of intmats, n an integer, |
---|
928 | @* degbound an optional integer |
---|
929 | RETURN: list |
---|
930 | PURPOSE:Computing the K-dimension and the Hilbert series |
---|
931 | ASSUME: - basering is a Letterplace ring |
---|
932 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
933 | @* for the encoding of the variables see the overview |
---|
934 | @* - if you specify a different degree bound degbound, |
---|
935 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
936 | NOTE: - If L is the list returned, then L[1] is an integer corresponding to the |
---|
937 | @* dimension, L[2] is an intvec which contains the coefficients of the |
---|
938 | @* Hilbert series |
---|
939 | @* - If degbound is set, there will be a degree bound added. By default there |
---|
940 | @* is no degree bound |
---|
941 | @* - n is the number of variables |
---|
942 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th coefficient of |
---|
943 | @* the Hilbert series. |
---|
944 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
945 | EXAMPLE: example ivDHilbert; shows examples |
---|
946 | " |
---|
947 | {int degbound = 0; |
---|
948 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
949 | checkAssumptions(degbound,L); |
---|
950 | intvec H; int i,dimen; |
---|
951 | H = ivHilbert(L,n,degbound); |
---|
952 | for (i = 1; i <= size(H); i++){dimen = dimen + H[i];} |
---|
953 | L = dimen,H; |
---|
954 | return(L); |
---|
955 | } |
---|
956 | example |
---|
957 | { |
---|
958 | "EXAMPLE:"; echo = 2; |
---|
959 | ring r = 0,(x,y),dp; |
---|
960 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
961 | R; |
---|
962 | setring R; // sets basering to Letterplace ring |
---|
963 | //some intmats, which contain monomials in intvec representation as rows |
---|
964 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
965 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
966 | print(I1); |
---|
967 | print(I2); |
---|
968 | print(J1); |
---|
969 | print(J2); |
---|
970 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
971 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
972 | //the procedure without a degree bound |
---|
973 | ivDHilbert(G,2); |
---|
974 | // the procedure with degree bound 5 |
---|
975 | ivDHilbert(I,2,5); |
---|
976 | } |
---|
977 | |
---|
978 | proc ivDHilbertSickle(list L, int n, list #) |
---|
979 | "USAGE: ivDHilbertSickle(L,n[,degbound]); L a list of intmats, n an integer, |
---|
980 | @* degbound an optional integer |
---|
981 | RETURN: list |
---|
982 | PURPOSE:Computing K-dimension, Hilbert series and mistletoes |
---|
983 | ASSUME: - basering is a Letterplace ring. |
---|
984 | @* - All rows of each intmat correspond to a Letterplace monomial. |
---|
985 | @* for the encoding of the variables see the overview |
---|
986 | @* - If you specify a different degree bound degbound, |
---|
987 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
988 | NOTE: - If L is the list returned, then L[1] is an integer, L[2] is an intvec |
---|
989 | @* which contains the coefficients of the Hilbert series and L[3] |
---|
990 | @* is a list, containing the mistletoes as intvecs. |
---|
991 | @* - If degbound is set, a degree bound will be added. By default there |
---|
992 | @* is no degree bound. |
---|
993 | @* - n is the number of variables. |
---|
994 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th |
---|
995 | @* coefficient of the Hilbert series. |
---|
996 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
997 | EXAMPLE: example ivDHilbertSickle; shows examples |
---|
998 | " |
---|
999 | {int degbound = 0; |
---|
1000 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
1001 | checkAssumptions(degbound,L); |
---|
1002 | int i,dimen; list R; |
---|
1003 | R = ivSickleHil(L,n,degbound); |
---|
1004 | for (i = 1; i <= size(R[1]); i++){dimen = dimen + R[1][i];} |
---|
1005 | R[3] = R[2]; R[2] = R[1]; R[1] = dimen; |
---|
1006 | return(R); |
---|
1007 | } |
---|
1008 | example |
---|
1009 | { |
---|
1010 | "EXAMPLE:"; echo = 2; |
---|
1011 | ring r = 0,(x,y),dp; |
---|
1012 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1013 | R; |
---|
1014 | setring R; // sets basering to Letterplace ring |
---|
1015 | //some intmats, which contain monomials in intvec representation as rows |
---|
1016 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1017 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1018 | print(I1); |
---|
1019 | print(I2); |
---|
1020 | print(J1); |
---|
1021 | print(J2); |
---|
1022 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
1023 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1024 | ivDHilbertSickle(G,2); // invokes the procedure without a degree bound |
---|
1025 | ivDHilbertSickle(I,2,3); // invokes the procedure with degree bound 3 |
---|
1026 | } |
---|
1027 | |
---|
1028 | proc ivDimCheck(list L, int n) |
---|
1029 | "USAGE: ivDimCheck(L,n); L a list of intmats, n an integer |
---|
1030 | RETURN: int, 0 if the dimension is finite, or 1 otherwise |
---|
1031 | PURPOSE:Decides, whether the K-dimension is finite or not |
---|
1032 | ASSUME: - basering is a Letterplace ring |
---|
1033 | @* - All rows of each intmat correspond to a Letterplace monomial |
---|
1034 | @* For the encoding of the variables see the overview. |
---|
1035 | NOTE: - n is the number of variables |
---|
1036 | EXAMPLE: example ivDimCheck; shows examples |
---|
1037 | " |
---|
1038 | {checkAssumptions(0,L); |
---|
1039 | int i,r; |
---|
1040 | intvec P,H; |
---|
1041 | for (i = 1; i <= size(L); i++) |
---|
1042 | {P[i] = ncols(L[i]); |
---|
1043 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
1044 | } |
---|
1045 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1046 | kill H; |
---|
1047 | intmat S; int sd,ld; intvec V; |
---|
1048 | sd = P[1]; ld = P[1]; |
---|
1049 | for (i = 2; i <= size(P); i++) |
---|
1050 | {if (P[i] < sd) {sd = P[i];} |
---|
1051 | if (P[i] > ld) {ld = P[i];} |
---|
1052 | } |
---|
1053 | sd = (sd - 1); ld = ld - 1; |
---|
1054 | if (ld == 0) { return(allVars(L,P,n));} |
---|
1055 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1056 | else {S = createStartMat(sd,n);} |
---|
1057 | module M; |
---|
1058 | for (i = 1; i <= nrows(S); i++) |
---|
1059 | {V = S[i,1..ncols(S)]; |
---|
1060 | if (findCycle(V,L,P,n,ld,M)) {r = 1; break;} |
---|
1061 | } |
---|
1062 | return(r); |
---|
1063 | } |
---|
1064 | example |
---|
1065 | { |
---|
1066 | "EXAMPLE:"; echo = 2; |
---|
1067 | ring r = 0,(x,y),dp; |
---|
1068 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1069 | R; |
---|
1070 | setring R; // sets basering to Letterplace ring |
---|
1071 | //some intmats, which contain monomials in intvec representation as rows |
---|
1072 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1073 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1074 | print(I1); |
---|
1075 | print(I2); |
---|
1076 | print(J1); |
---|
1077 | print(J2); |
---|
1078 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
1079 | list I = J1,J2; // ideal, which is already a Groebner basis and which |
---|
1080 | ivDimCheck(G,2); // invokes the procedure, factor is of finite K-dimension |
---|
1081 | ivDimCheck(I,2); // invokes the procedure, factor is not of finite K-dimension |
---|
1082 | } |
---|
1083 | |
---|
1084 | proc ivHilbert(list L, int n, list #) |
---|
1085 | "USAGE: ivHilbert(L,n[,degbound]); L a list of intmats, n an integer, |
---|
1086 | @* degbound an optional integer |
---|
1087 | RETURN: intvec, containing the coefficients of the Hilbert series |
---|
1088 | PURPOSE:Computing the Hilbert series |
---|
1089 | ASSUME: - basering is a Letterplace ring. |
---|
1090 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
1091 | @* for the encoding of the variables see the overview |
---|
1092 | @* - if you specify a different degree bound degbound, |
---|
1093 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1094 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
1095 | @* is no degree bound. |
---|
1096 | @* - n is the number of variables. |
---|
1097 | @* - If I is returned, then I[k] is the (k-1)-th coefficient of the Hilbert |
---|
1098 | @* series. |
---|
1099 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1100 | EXAMPLE: example ivHilbert; shows examples |
---|
1101 | " |
---|
1102 | {int degbound = 0; |
---|
1103 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
1104 | intvec P,H; int i; |
---|
1105 | for (i = 1; i <= size(L); i++) |
---|
1106 | {P[i] = ncols(L[i]); |
---|
1107 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
1108 | } |
---|
1109 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1110 | H[1] = 1; |
---|
1111 | checkAssumptions(degbound,L); |
---|
1112 | if (degbound == 0) |
---|
1113 | {int sd; |
---|
1114 | intmat S; |
---|
1115 | sd = P[1]; |
---|
1116 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1117 | sd = (sd - 1); |
---|
1118 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1119 | else {S = createStartMat(sd,n);} |
---|
1120 | if (intvec(S) == 0) {return(H);} |
---|
1121 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
1122 | for (i = 1; i <= nrows(S); i++) |
---|
1123 | {intvec St = S[i,1..ncols(S)]; |
---|
1124 | H = findHCoeff(St,n,L,P,H); |
---|
1125 | kill St; |
---|
1126 | } |
---|
1127 | return(H); |
---|
1128 | } |
---|
1129 | else |
---|
1130 | {for (i = 1; i <= size(P); i++) |
---|
1131 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
1132 | int sd; |
---|
1133 | intmat S; |
---|
1134 | sd = P[1]; |
---|
1135 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1136 | sd = (sd - 1); |
---|
1137 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1138 | else {S = createStartMat(sd,n);} |
---|
1139 | if (intvec(S) == 0) {return(H);} |
---|
1140 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
1141 | for (i = 1; i <= nrows(S); i++) |
---|
1142 | {intvec St = S[i,1..ncols(S)]; |
---|
1143 | H = findHCoeff(St,n,L,P,H,degbound); |
---|
1144 | kill St; |
---|
1145 | } |
---|
1146 | return(H); |
---|
1147 | } |
---|
1148 | } |
---|
1149 | example |
---|
1150 | { |
---|
1151 | "EXAMPLE:"; echo = 2; |
---|
1152 | ring r = 0,(x,y),dp; |
---|
1153 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1154 | R; |
---|
1155 | setring R; // sets basering to Letterplace ring |
---|
1156 | //some intmats, which contain monomials in intvec representation as rows |
---|
1157 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1158 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1159 | print(I1); |
---|
1160 | print(I2); |
---|
1161 | print(J1); |
---|
1162 | print(J2); |
---|
1163 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
1164 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1165 | ivHilbert(G,2); // invokes the procedure without any degree bound |
---|
1166 | ivHilbert(I,2,5); // invokes the procedure with degree bound 5 |
---|
1167 | } |
---|
1168 | |
---|
1169 | |
---|
1170 | proc ivKDim(list L, int n, list #) |
---|
1171 | "USAGE: ivKDim(L,n[,degbound]); L a list of intmats, |
---|
1172 | @* n an integer, degbound an optional integer |
---|
1173 | RETURN: int, the K-dimension of A/<L> |
---|
1174 | PURPOSE:Computing the K-dimension of A/<L> |
---|
1175 | ASSUME: - basering is a Letterplace ring. |
---|
1176 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
1177 | @* for the encoding of the variables see the overview |
---|
1178 | @* - if you specify a different degree bound degbound, |
---|
1179 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1180 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
1181 | @* is no degree bound. |
---|
1182 | @* - n is the number of variables. |
---|
1183 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1184 | EXAMPLE: example ivKDim; shows examples |
---|
1185 | " |
---|
1186 | {int degbound = 0; |
---|
1187 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
1188 | intvec P,H; int i; |
---|
1189 | for (i = 1; i <= size(L); i++) |
---|
1190 | {P[i] = ncols(L[i]); |
---|
1191 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
1192 | } |
---|
1193 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1194 | kill H; |
---|
1195 | checkAssumptions(degbound,L); |
---|
1196 | if (degbound == 0) |
---|
1197 | {int sd; int dimen = 1; |
---|
1198 | intmat S; |
---|
1199 | sd = P[1]; |
---|
1200 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1201 | sd = (sd - 1); |
---|
1202 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1203 | else {S = createStartMat(sd,n);} |
---|
1204 | if (intvec(S) == 0) {return(dimen);} |
---|
1205 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
1206 | for (i = 1; i <= nrows(S); i++) |
---|
1207 | {intvec St = S[i,1..ncols(S)]; |
---|
1208 | dimen = dimen + findDimen(St,n,L,P); |
---|
1209 | kill St; |
---|
1210 | } |
---|
1211 | return(dimen); |
---|
1212 | } |
---|
1213 | else |
---|
1214 | {for (i = 1; i <= size(P); i++) |
---|
1215 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
1216 | int sd; int dimen = 1; |
---|
1217 | intmat S; |
---|
1218 | sd = P[1]; |
---|
1219 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1220 | sd = (sd - 1); |
---|
1221 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1222 | else {S = createStartMat(sd,n);} |
---|
1223 | if (intvec(S) == 0) {return(dimen);} |
---|
1224 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
1225 | for (i = 1; i <= nrows(S); i++) |
---|
1226 | {intvec St = S[i,1..ncols(S)]; |
---|
1227 | dimen = dimen + findDimen(St,n,L,P, degbound); |
---|
1228 | kill St; |
---|
1229 | } |
---|
1230 | return(dimen); |
---|
1231 | } |
---|
1232 | } |
---|
1233 | example |
---|
1234 | { |
---|
1235 | "EXAMPLE:"; echo = 2; |
---|
1236 | ring r = 0,(x,y),dp; |
---|
1237 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1238 | R; |
---|
1239 | setring R; // sets basering to Letterplace ring |
---|
1240 | //some intmats, which contain monomials in intvec representation as rows |
---|
1241 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1242 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1243 | print(I1); |
---|
1244 | print(I2); |
---|
1245 | print(J1); |
---|
1246 | print(J2); |
---|
1247 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
1248 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1249 | ivKDim(G,2); // invokes the procedure without any degree bound |
---|
1250 | ivKDim(I,2,5); // invokes the procedure with degree bound 5 |
---|
1251 | } |
---|
1252 | |
---|
1253 | proc ivMis2Base(list M) |
---|
1254 | "USAGE: ivMis2Base(M); M a list of intvecs |
---|
1255 | RETURN: ideal, a K-base of the given algebra |
---|
1256 | PURPOSE:Computing the K-base out of given mistletoes |
---|
1257 | ASSUME: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
1258 | @* Otherwise there might some elements missing. |
---|
1259 | @* - basering is a Letterplace ring. |
---|
1260 | @* - mistletoes are stored as intvecs, as described in the overview |
---|
1261 | EXAMPLE: example ivMis2Base; shows examples |
---|
1262 | " |
---|
1263 | { |
---|
1264 | //checkAssumptions(0,M); |
---|
1265 | intvec L,A; |
---|
1266 | if (size(M) == 0){ERROR("There are no mistletoes, so it appears your dimension is infinite!");} |
---|
1267 | if (isInList(L,M) > 0) {print("1 is a mistletoe, therefore 1 is the only basis element"); return(list(intvec(0)));} |
---|
1268 | int i,j,d,s; |
---|
1269 | list Rt; |
---|
1270 | Rt[1] = intvec(0); |
---|
1271 | L = M[1]; |
---|
1272 | for (i = size(L); 1 <= i; i--) {Rt = insert(Rt,intvec(L[1..i]));} |
---|
1273 | for (i = 2; i <= size(M); i++) |
---|
1274 | {A = M[i]; L = M[i-1]; |
---|
1275 | s = size(A); |
---|
1276 | if (s > size(L)) |
---|
1277 | {d = size(L); |
---|
1278 | for (j = s; j > d; j--) {Rt = insert(Rt,intvec(A[1..j]));} |
---|
1279 | A = A[1..d]; |
---|
1280 | } |
---|
1281 | if (size(L) > s){L = L[1..s];} |
---|
1282 | while (A <> L) |
---|
1283 | {Rt = insert(Rt, intvec(A)); |
---|
1284 | if (size(A) > 1) |
---|
1285 | {A = A[1..(size(A)-1)]; |
---|
1286 | L = L[1..(size(L)-1)]; |
---|
1287 | } |
---|
1288 | else {break;} |
---|
1289 | } |
---|
1290 | } |
---|
1291 | return(Rt); |
---|
1292 | } |
---|
1293 | example |
---|
1294 | { |
---|
1295 | "EXAMPLE:"; echo = 2; |
---|
1296 | ring r = 0,(x,y),dp; |
---|
1297 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1298 | R; |
---|
1299 | setring R; // sets basering to Letterplace ring |
---|
1300 | intvec i1 = 1,2; intvec i2 = 2,1,2; |
---|
1301 | // the mistletoes are xy and yxy, which are already ordered lexicographically |
---|
1302 | list L = i1,i2; |
---|
1303 | ivMis2Base(L); // returns the basis of the factor algebra |
---|
1304 | } |
---|
1305 | |
---|
1306 | |
---|
1307 | proc ivMis2Dim(list M) |
---|
1308 | "USAGE: ivMis2Dim(M); M a list of intvecs |
---|
1309 | RETURN: int, the K-dimension of the given algebra |
---|
1310 | PURPOSE:Computing the K-dimension out of given mistletoes |
---|
1311 | ASSUME: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
1312 | @* Otherwise the returned value may differ from the K-dimension. |
---|
1313 | @* - basering is a Letterplace ring. |
---|
1314 | @* - mistletoes are stored as intvecs, as described in the overview |
---|
1315 | EXAMPLE: example ivMis2Dim; shows examples |
---|
1316 | " |
---|
1317 | {checkAssumptions(0,M); |
---|
1318 | intvec L; |
---|
1319 | if (size(M) == 0){ERROR("There are no mistletoes, so it appears your dimension is infinite!");} |
---|
1320 | if (isInList(L,M) > 0) {print("1 is a mistletoe, therefore dim = 1"); return(1);} |
---|
1321 | int i,j,d,s; |
---|
1322 | d = 1 + size(M[1]); |
---|
1323 | for (i = 1; i < size(M); i++) |
---|
1324 | {j = 1; |
---|
1325 | s = size(M[i]); if (s > size(M[i+1])){s = size(M[i+1]);} |
---|
1326 | while ((M[i][j] == M[i+1][j]) && (j <= s)){j = j + 1;} |
---|
1327 | d = d + size(M[i+1])- j + 1; |
---|
1328 | } |
---|
1329 | return(d); |
---|
1330 | } |
---|
1331 | example |
---|
1332 | { |
---|
1333 | "EXAMPLE:"; echo = 2; |
---|
1334 | ring r = 0,(x,y),dp; |
---|
1335 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1336 | R; |
---|
1337 | setring R; // sets basering to Letterplace ring |
---|
1338 | intvec i1 = 1,2; intvec i2 = 2,1,2; |
---|
1339 | // the mistletoes are xy and yxy, which are already ordered lexicographically |
---|
1340 | list L = i1,i2; |
---|
1341 | ivMis2Dim(L); // returns the dimension of the factor algebra |
---|
1342 | } |
---|
1343 | |
---|
1344 | proc ivOrdMisLex(list M) |
---|
1345 | "USAGE: ivOrdMisLex(M); M a list of intvecs |
---|
1346 | RETURN: list, containing the ordered intvecs of M |
---|
1347 | PURPOSE:Orders a given set of mistletoes lexicographically |
---|
1348 | ASSUME: - basering is a Letterplace ring. |
---|
1349 | @* - intvecs correspond to monomials, as explained in the overview |
---|
1350 | NOTE: - This is preprocessing, it's not needed if the mistletoes are returned |
---|
1351 | @* from the sickle algorithm. |
---|
1352 | @* - Each entry of the list returned is an intvec. |
---|
1353 | EXAMPLE: example ivOrdMisLex; shows examples |
---|
1354 | " |
---|
1355 | {checkAssumptions(0,M); |
---|
1356 | return(sort(M)[1]); |
---|
1357 | } |
---|
1358 | example |
---|
1359 | { |
---|
1360 | "EXAMPLE:"; echo = 2; |
---|
1361 | ring r = 0,(x,y),dp; |
---|
1362 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1363 | setring R; // sets basering to Letterplace ring |
---|
1364 | intvec i1 = 1,2,1; intvec i2 = 2,2,1; intvec i3 = 1,1; intvec i4 = 2,1,1,1; |
---|
1365 | // the corresponding monomials are xyx,y^2x,x^2,yx^3 |
---|
1366 | list M = i1,i2,i3,i4; |
---|
1367 | M; |
---|
1368 | ivOrdMisLex(M);// orders the list of monomials |
---|
1369 | } |
---|
1370 | |
---|
1371 | proc ivSickle(list L, int n, list #) |
---|
1372 | "USAGE: ivSickle(L,n,[degbound]); L a list of intmats, n an int, degbound an |
---|
1373 | @* optional integer |
---|
1374 | RETURN: list, containing intvecs, the mistletoes of A/<L> |
---|
1375 | PURPOSE:Computing the mistletoes for a given Groebner basis L, given by intmats |
---|
1376 | ASSUME: - basering is a Letterplace ring. |
---|
1377 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
1378 | @* as explained in the overview |
---|
1379 | @* - if you specify a different degree bound degbound, |
---|
1380 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1381 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
1382 | @* is no degree bound. |
---|
1383 | @* - n is the number of variables. |
---|
1384 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1385 | EXAMPLE: example ivSickle; shows examples |
---|
1386 | " |
---|
1387 | {list M; |
---|
1388 | int degbound = 0; |
---|
1389 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
1390 | int i; |
---|
1391 | intvec P,H; |
---|
1392 | for (i = 1; i <= size(L); i++) |
---|
1393 | {P[i] = ncols(L[i]); |
---|
1394 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
1395 | } |
---|
1396 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1397 | kill H; |
---|
1398 | checkAssumptions(degbound,L); |
---|
1399 | if (degbound == 0) |
---|
1400 | {intmat S; int sd; |
---|
1401 | sd = P[1]; |
---|
1402 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1403 | sd = (sd - 1); |
---|
1404 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1405 | else {S = createStartMat(sd,n);} |
---|
1406 | if (intvec(S) == 0) {return(list (intvec(0)));} |
---|
1407 | for (i = 1; i <= nrows(S); i++) |
---|
1408 | {intvec St = S[i,1..ncols(S)]; |
---|
1409 | M = M + findmistletoes(St,n,L,P); |
---|
1410 | kill St; |
---|
1411 | } |
---|
1412 | return(M); |
---|
1413 | } |
---|
1414 | else |
---|
1415 | {for (i = 1; i <= size(P); i++) |
---|
1416 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
1417 | intmat S; int sd; |
---|
1418 | sd = P[1]; |
---|
1419 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1420 | sd = (sd - 1); |
---|
1421 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1422 | else {S = createStartMat(sd,n);} |
---|
1423 | if (intvec(S) == 0) {return(list (intvec(0)));} |
---|
1424 | for (i = 1; i <= nrows(S); i++) |
---|
1425 | {intvec St = S[i,1..ncols(S)]; |
---|
1426 | M = M + findmistletoes(St,n,L,P,degbound); |
---|
1427 | kill St; |
---|
1428 | } |
---|
1429 | return(M); |
---|
1430 | } |
---|
1431 | } |
---|
1432 | example |
---|
1433 | { |
---|
1434 | "EXAMPLE:"; echo = 2; |
---|
1435 | ring r = 0,(x,y),dp; |
---|
1436 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1437 | setring R; // sets basering to Letterplace ring |
---|
1438 | //some intmats, which contain monomials in intvec representation as rows |
---|
1439 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1440 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1441 | print(I1); |
---|
1442 | print(I2); |
---|
1443 | print(J1); |
---|
1444 | print(J2); |
---|
1445 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
1446 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1447 | ivSickle(G,2); // invokes the procedure without any degree bound |
---|
1448 | ivSickle(I,2,5); // invokes the procedure with degree bound 5 |
---|
1449 | } |
---|
1450 | |
---|
1451 | proc ivSickleDim(list L, int n, list #) |
---|
1452 | "USAGE: ivSickleDim(L,n[,degbound]); L a list of intmats, n an integer, degbound |
---|
1453 | @* an optional integer |
---|
1454 | RETURN: list |
---|
1455 | PURPOSE:Computing mistletoes and the K-dimension |
---|
1456 | ASSUME: - basering is a Letterplace ring. |
---|
1457 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
1458 | @* as explained in the overview |
---|
1459 | @* - if you specify a different degree bound degbound, |
---|
1460 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1461 | NOTE: - If L is the list returned, then L[1] is an integer, L[2] is a list, |
---|
1462 | @* containing the mistletoes as intvecs. |
---|
1463 | @* - If degbound is set, a degree bound will be added. By default there |
---|
1464 | @* is no degree bound. |
---|
1465 | @* - n is the number of variables. |
---|
1466 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1467 | EXAMPLE: example ivSickleDim; shows examples |
---|
1468 | " |
---|
1469 | {list M; |
---|
1470 | int degbound = 0; |
---|
1471 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
1472 | int i,dimen; list R; |
---|
1473 | intvec P,H; |
---|
1474 | for (i = 1; i <= size(L); i++) |
---|
1475 | {P[i] = ncols(L[i]); |
---|
1476 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial, dimension equals zero");}} |
---|
1477 | } |
---|
1478 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1479 | kill H; |
---|
1480 | checkAssumptions(degbound,L); |
---|
1481 | if (degbound == 0) |
---|
1482 | {int sd; dimen = 1; |
---|
1483 | intmat S; |
---|
1484 | sd = P[1]; |
---|
1485 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1486 | sd = (sd - 1); |
---|
1487 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1488 | else {S = createStartMat(sd,n);} |
---|
1489 | if (intvec(S) == 0) {return(list(dimen,list(intvec(0))));} |
---|
1490 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
1491 | R[1] = dimen; |
---|
1492 | for (i = 1; i <= nrows(S); i++) |
---|
1493 | {intvec St = S[i,1..ncols(S)]; |
---|
1494 | R = findMisDim(St,n,L,P,R); |
---|
1495 | kill St; |
---|
1496 | } |
---|
1497 | return(R); |
---|
1498 | } |
---|
1499 | else |
---|
1500 | {for (i = 1; i <= size(P); i++) |
---|
1501 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
1502 | int sd; dimen = 1; |
---|
1503 | intmat S; |
---|
1504 | sd = P[1]; |
---|
1505 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1506 | sd = (sd - 1); |
---|
1507 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1508 | else {S = createStartMat(sd,n);} |
---|
1509 | if (intvec(S) == 0) {return(list(dimen,list(intvec(0))));} |
---|
1510 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
1511 | R[1] = dimen; |
---|
1512 | for (i = 1; i <= nrows(S); i++) |
---|
1513 | {intvec St = S[i,1..ncols(S)]; |
---|
1514 | R = findMisDim(St,n,L,P,R,degbound); |
---|
1515 | kill St; |
---|
1516 | } |
---|
1517 | return(R); |
---|
1518 | } |
---|
1519 | } |
---|
1520 | example |
---|
1521 | { |
---|
1522 | "EXAMPLE:"; echo = 2; |
---|
1523 | ring r = 0,(x,y),dp; |
---|
1524 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1525 | setring R; // sets basering to Letterplace ring |
---|
1526 | //some intmats, which contain monomials in intvec representation as rows |
---|
1527 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1528 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1529 | print(I1); |
---|
1530 | print(I2); |
---|
1531 | print(J1); |
---|
1532 | print(J2); |
---|
1533 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
1534 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1535 | ivSickleDim(G,2); // invokes the procedure without any degree bound |
---|
1536 | ivSickleDim(I,2,5); // invokes the procedure with degree bound 5 |
---|
1537 | } |
---|
1538 | |
---|
1539 | proc ivSickleHil(list L, int n, list #) |
---|
1540 | "USAGE:ivSickleHil(L,n[,degbound]); L a list of intmats, n an integer, |
---|
1541 | @* degbound an optional integer |
---|
1542 | RETURN: list |
---|
1543 | PURPOSE:Computing the mistletoes and the Hilbert series |
---|
1544 | ASSUME: - basering is a Letterplace ring. |
---|
1545 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
1546 | @* as explained in the overview |
---|
1547 | @* - if you specify a different degree bound degbound, |
---|
1548 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1549 | NOTE: - If L is the list returned, then L[1] is an intvec, L[2] is a list, |
---|
1550 | @* containing the mistletoes as intvecs. |
---|
1551 | @* - If degbound is set, a degree bound will be added. By default there |
---|
1552 | @* is no degree bound. |
---|
1553 | @* - n is the number of variables. |
---|
1554 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
1555 | @* coefficient of the Hilbert series. |
---|
1556 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1557 | EXAMPLE: example ivSickleHil; shows examples |
---|
1558 | " |
---|
1559 | {int degbound = 0; |
---|
1560 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
1561 | intvec P,H; int i; list R; |
---|
1562 | for (i = 1; i <= size(L); i++) |
---|
1563 | {P[i] = ncols(L[i]); |
---|
1564 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
1565 | } |
---|
1566 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1567 | H[1] = 1; |
---|
1568 | checkAssumptions(degbound,L); |
---|
1569 | if (degbound == 0) |
---|
1570 | {int sd; |
---|
1571 | intmat S; |
---|
1572 | sd = P[1]; |
---|
1573 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1574 | sd = (sd - 1); |
---|
1575 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1576 | else {S = createStartMat(sd,n);} |
---|
1577 | if (intvec(S) == 0) {return(list(H,list(intvec (0))));} |
---|
1578 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
1579 | R[1] = H; kill H; |
---|
1580 | for (i = 1; i <= nrows(S); i++) |
---|
1581 | {intvec St = S[i,1..ncols(S)]; |
---|
1582 | R = findHCoeffMis(St,n,L,P,R); |
---|
1583 | kill St; |
---|
1584 | } |
---|
1585 | return(R); |
---|
1586 | } |
---|
1587 | else |
---|
1588 | {for (i = 1; i <= size(P); i++) |
---|
1589 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
1590 | int sd; |
---|
1591 | intmat S; |
---|
1592 | sd = P[1]; |
---|
1593 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1594 | sd = (sd - 1); |
---|
1595 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1596 | else {S = createStartMat(sd,n);} |
---|
1597 | if (intvec(S) == 0) {return(list(H,list(intvec(0))));} |
---|
1598 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
1599 | R[1] = H; kill H; |
---|
1600 | for (i = 1; i <= nrows(S); i++) |
---|
1601 | {intvec St = S[i,1..ncols(S)]; |
---|
1602 | R = findHCoeffMis(St,n,L,P,R,degbound); |
---|
1603 | kill St; |
---|
1604 | } |
---|
1605 | return(R); |
---|
1606 | } |
---|
1607 | } |
---|
1608 | example |
---|
1609 | { |
---|
1610 | "EXAMPLE:"; echo = 2; |
---|
1611 | ring r = 0,(x,y),dp; |
---|
1612 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1613 | setring R; // sets basering to Letterplace ring |
---|
1614 | //some intmats, which contain monomials in intvec representation as rows |
---|
1615 | intmat I1[2][2] = 1,1,2,2; intmat I2[1][3] = 1,2,1; |
---|
1616 | intmat J1[1][2] = 1,1; intmat J2[2][3] = 2,1,2,1,2,1; |
---|
1617 | print(I1); |
---|
1618 | print(I2); |
---|
1619 | print(J1); |
---|
1620 | print(J2); |
---|
1621 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
1622 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1623 | ivSickleHil(G,2); // invokes the procedure without any degree bound |
---|
1624 | ivSickleHil(I,2,5); // invokes the procedure with degree bound 5 |
---|
1625 | } |
---|
1626 | |
---|
1627 | proc lpDHilbert(ideal G, list #) |
---|
1628 | "USAGE: lpDHilbert(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
1629 | RETURN: list |
---|
1630 | PURPOSE:Computing K-dimension and Hilbert series, starting with a lp-ideal |
---|
1631 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1632 | @* - if you specify a different degree bound degbound, |
---|
1633 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1634 | NOTE: - If L is the list returned, then L[1] is an integer corresponding to the |
---|
1635 | @* dimension, L[2] is an intvec which contains the coefficients of the |
---|
1636 | @* Hilbert series |
---|
1637 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
1638 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
1639 | @* - n can be set to a different number of variables. |
---|
1640 | @* Default: n = attrib(basering, lV). |
---|
1641 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th |
---|
1642 | @* coefficient of the Hilbert series. |
---|
1643 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1644 | EXAMPLE: example lpDHilbert; shows examples |
---|
1645 | " |
---|
1646 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
1647 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
1648 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
1649 | list L; |
---|
1650 | L = lp2ivId(normalize(lead(G))); |
---|
1651 | return(ivDHilbert(L,n,degbound)); |
---|
1652 | } |
---|
1653 | example |
---|
1654 | { |
---|
1655 | "EXAMPLE:"; echo = 2; |
---|
1656 | ring r = 0,(x,y),dp; |
---|
1657 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1658 | setring R; // sets basering to Letterplace ring |
---|
1659 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
1660 | //Groebner basis |
---|
1661 | lpDHilbert(G,5,2); // invokes procedure with degree bound 5 and 2 variables |
---|
1662 | // note that the optional parameters are not necessary, due to the finiteness |
---|
1663 | // of the K-dimension of the factor algebra |
---|
1664 | lpDHilbert(G); // procedure with ring parameters |
---|
1665 | lpDHilbert(G,0); // procedure without degreebound |
---|
1666 | } |
---|
1667 | |
---|
1668 | proc lpDHilbertSickle(ideal G, list #) |
---|
1669 | "USAGE: lpDHilbertSickle(G[,degbound,n]); G an ideal, degbound, n optional |
---|
1670 | @* integers |
---|
1671 | RETURN: list |
---|
1672 | PURPOSE:Computing K-dimension, Hilbert series and mistletoes at once |
---|
1673 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1674 | @* - if you specify a different degree bound degbound, |
---|
1675 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1676 | NOTE: - If L is the list returned, then L[1] is an integer, the K-dimension, |
---|
1677 | @* L[2] is an intvec, the Hilbert series and L[3] is an ideal, |
---|
1678 | @* the mistletoes |
---|
1679 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
1680 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
1681 | @* - n can be set to a different number of variables. |
---|
1682 | @* Default: n = attrib(basering, lV). |
---|
1683 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
1684 | @* coefficient of the Hilbert series. |
---|
1685 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1686 | EXAMPLE: example lpDHilbertSickle; shows examples |
---|
1687 | " |
---|
1688 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
1689 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
1690 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
1691 | list L; |
---|
1692 | L = lp2ivId(normalize(lead(G))); |
---|
1693 | L = ivDHilbertSickle(L,n,degbound); |
---|
1694 | L[3] = ivL2lpI(L[3]); |
---|
1695 | return(L); |
---|
1696 | } |
---|
1697 | example |
---|
1698 | { |
---|
1699 | "EXAMPLE:"; echo = 2; |
---|
1700 | ring r = 0,(x,y),dp; |
---|
1701 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1702 | setring R; // sets basering to Letterplace ring |
---|
1703 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
1704 | //Groebner basis |
---|
1705 | lpDHilbertSickle(G,5,2); //invokes procedure with degree bound 5 and 2 variables |
---|
1706 | // note that the optional parameters are not necessary, due to the finiteness |
---|
1707 | // of the K-dimension of the factor algebra |
---|
1708 | lpDHilbertSickle(G); // procedure with ring parameters |
---|
1709 | lpDHilbertSickle(G,0); // procedure without degreebound |
---|
1710 | } |
---|
1711 | |
---|
1712 | proc lpHilbert(ideal G, list #) |
---|
1713 | "USAGE: lpHilbert(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
1714 | RETURN: intvec, containing the coefficients of the Hilbert series |
---|
1715 | PURPOSE:Computing the Hilbert series |
---|
1716 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1717 | @* - if you specify a different degree bound degbound, |
---|
1718 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1719 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
1720 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
1721 | @* - n is the number of variables, which can be set to a different number. |
---|
1722 | @* Default: attrib(basering, lV). |
---|
1723 | @* - If I is returned, then I[k] is the (k-1)-th coefficient of the Hilbert |
---|
1724 | @* series. |
---|
1725 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1726 | EXAMPLE: example lpHilbert; shows examples |
---|
1727 | " |
---|
1728 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
1729 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
1730 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
1731 | list L; |
---|
1732 | L = lp2ivId(normalize(lead(G))); |
---|
1733 | return(ivHilbert(L,n,degbound)); |
---|
1734 | } |
---|
1735 | example |
---|
1736 | { |
---|
1737 | "EXAMPLE:"; echo = 2; |
---|
1738 | ring r = 0,(x,y),dp; |
---|
1739 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1740 | setring R; // sets basering to Letterplace ring |
---|
1741 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
1742 | //Groebner basis |
---|
1743 | lpHilbert(G,5,2); // invokes procedure with degree bound 5 and 2 variables |
---|
1744 | // note that the optional parameters are not necessary, due to the finiteness |
---|
1745 | // of the K-dimension of the factor algebra |
---|
1746 | lpDHilbert(G); // procedure with ring parameters |
---|
1747 | lpDHilbert(G,0); // procedure without degreebound |
---|
1748 | } |
---|
1749 | |
---|
1750 | proc lpDimCheck(ideal G) |
---|
1751 | "USAGE: lpDimCheck(G); |
---|
1752 | RETURN: int, 1 if K-dimension of the factor algebra is infinite, 0 otherwise |
---|
1753 | PURPOSE:Checking a factor algebra for finiteness of the K-dimension |
---|
1754 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1755 | EXAMPLE: example lpDimCheck; shows examples |
---|
1756 | " |
---|
1757 | {int n = attrib(basering,"lV"); |
---|
1758 | list L; |
---|
1759 | ideal R; |
---|
1760 | R = normalize(lead(G)); |
---|
1761 | L = lp2ivId(R); |
---|
1762 | return(ivDimCheck(L,n)); |
---|
1763 | } |
---|
1764 | example |
---|
1765 | { |
---|
1766 | "EXAMPLE:"; echo = 2; |
---|
1767 | ring r = 0,(x,y),dp; |
---|
1768 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1769 | setring R; // sets basering to Letterplace ring |
---|
1770 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); |
---|
1771 | // Groebner basis |
---|
1772 | ideal I = x(1)*x(2), y(1)*x(2)*y(3), x(1)*y(2)*x(3); |
---|
1773 | // Groebner basis |
---|
1774 | lpDimCheck(G); // invokes procedure, factor algebra is of finite K-dimension |
---|
1775 | lpDimCheck(I); // invokes procedure, factor algebra is of infinite Kdimension |
---|
1776 | } |
---|
1777 | |
---|
1778 | proc lpKDim(ideal G, list #) |
---|
1779 | "USAGE: lpKDim(G[,degbound, n]); G an ideal, degbound, n optional integers |
---|
1780 | RETURN: int, the K-dimension of the factor algebra |
---|
1781 | PURPOSE:Computing the K-dimension of a factor algebra, given via an ideal |
---|
1782 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1783 | @* - if you specify a different degree bound degbound, |
---|
1784 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1785 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
1786 | @* degree bound. Default: attrib(basering, uptodeg). |
---|
1787 | @* - n is the number of variables, which can be set to a different number. |
---|
1788 | @* Default: attrib(basering, lV). |
---|
1789 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1790 | EXAMPLE: example lpKDim; shows examples |
---|
1791 | " |
---|
1792 | {int degbound = attrib(basering, "uptodeg");int n = attrib(basering, "lV"); |
---|
1793 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
1794 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
1795 | list L; |
---|
1796 | L = lp2ivId(normalize(lead(G))); |
---|
1797 | return(ivKDim(L,n,degbound)); |
---|
1798 | } |
---|
1799 | example |
---|
1800 | { |
---|
1801 | "EXAMPLE:"; echo = 2; |
---|
1802 | ring r = 0,(x,y),dp; |
---|
1803 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1804 | setring R; // sets basering to Letterplace ring |
---|
1805 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); |
---|
1806 | // ideal G contains a Groebner basis |
---|
1807 | lpKDim(G); //procedure invoked with ring parameters |
---|
1808 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
1809 | // ring is not necessary |
---|
1810 | lpKDim(G,0); // procedure without any degree bound |
---|
1811 | } |
---|
1812 | |
---|
1813 | proc lpMis2Base(ideal M) |
---|
1814 | "USAGE: lpMis2Base(M); M an ideal |
---|
1815 | RETURN: ideal, a K-basis of the factor algebra |
---|
1816 | PURPOSE:Computing a K-basis out of given mistletoes |
---|
1817 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1818 | @* - M contains only monomials |
---|
1819 | NOTE: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
1820 | EXAMPLE: example lpMis2Base; shows examples |
---|
1821 | " |
---|
1822 | {list L; |
---|
1823 | L = lpId2ivLi(M); |
---|
1824 | return(ivL2lpI(ivMis2Base(L))); |
---|
1825 | } |
---|
1826 | example |
---|
1827 | { |
---|
1828 | "EXAMPLE:"; echo = 2; |
---|
1829 | ring r = 0,(x,y),dp; |
---|
1830 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1831 | setring R; // sets basering to Letterplace ring |
---|
1832 | ideal L = x(1)*y(2),y(1)*x(2)*y(3); |
---|
1833 | // ideal containing the mistletoes |
---|
1834 | lpMis2Base(L); // returns the K-basis of the factor algebra |
---|
1835 | } |
---|
1836 | |
---|
1837 | proc lpMis2Dim(ideal M) |
---|
1838 | "USAGE: lpMis2Dim(M); M an ideal |
---|
1839 | RETURN: int, the K-dimension of the factor algebra |
---|
1840 | PURPOSE:Computing the K-dimension out of given mistletoes |
---|
1841 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1842 | @* - M contains only monomials |
---|
1843 | NOTE: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
1844 | EXAMPLE: example lpMis2Dim; shows examples |
---|
1845 | " |
---|
1846 | {list L; |
---|
1847 | L = lpId2ivLi(M); |
---|
1848 | return(ivMis2Dim(L)); |
---|
1849 | } |
---|
1850 | example |
---|
1851 | { |
---|
1852 | "EXAMPLE:"; echo = 2; |
---|
1853 | ring r = 0,(x,y),dp; |
---|
1854 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1855 | setring R; // sets basering to Letterplace ring |
---|
1856 | ideal L = x(1)*y(2),y(1)*x(2)*y(3); |
---|
1857 | // ideal containing the mistletoes |
---|
1858 | lpMis2Dim(L); // returns the K-dimension of the factor algebra |
---|
1859 | } |
---|
1860 | |
---|
1861 | proc lpOrdMisLex(ideal M) |
---|
1862 | "USAGE: lpOrdMisLex(M); M an ideal of mistletoes |
---|
1863 | RETURN: ideal, containing the mistletoes, ordered lexicographically |
---|
1864 | PURPOSE:A given set of mistletoes is ordered lexicographically |
---|
1865 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1866 | NOTE: This is preprocessing, it is not needed if the mistletoes are returned |
---|
1867 | @* from the sickle algorithm. |
---|
1868 | EXAMPLE: example lpOrdMisLex; shows examples |
---|
1869 | " |
---|
1870 | {return(ivL2lpI(sort(lpId2ivLi(M))[1]));} |
---|
1871 | example |
---|
1872 | { |
---|
1873 | "EXAMPLE:"; echo = 2; |
---|
1874 | ring r = 0,(x,y),dp; |
---|
1875 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1876 | setring R; // sets basering to Letterplace ring |
---|
1877 | ideal M = x(1)*y(2)*x(3), y(1)*y(2)*x(3), x(1)*x(2), y(1)*x(2)*x(3)*x(4); |
---|
1878 | // some monomials |
---|
1879 | lpOrdMisLex(M); // orders the monomials lexicographically |
---|
1880 | } |
---|
1881 | |
---|
1882 | proc lpSickle(ideal G, list #) |
---|
1883 | "USAGE: lpSickle(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
1884 | RETURN: ideal |
---|
1885 | PURPOSE:Computing the mistletoes of K[X]/<G> |
---|
1886 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1887 | @* - if you specify a different degree bound degbound, |
---|
1888 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1889 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
1890 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
1891 | @* - n is the number of variables, which can be set to a different number. |
---|
1892 | @* Default: attrib(basering, lV). |
---|
1893 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1894 | EXAMPLE: example lpSickle; shows examples |
---|
1895 | " |
---|
1896 | {int degbound = attrib(basering,"uptodeg"); int n = attrib(basering, "lV"); |
---|
1897 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
1898 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
1899 | list L; ideal R; |
---|
1900 | R = normalize(lead(G)); |
---|
1901 | L = lp2ivId(R); |
---|
1902 | L = ivSickle(L,n,degbound); |
---|
1903 | R = ivL2lpI(L); |
---|
1904 | return(R); |
---|
1905 | } |
---|
1906 | example |
---|
1907 | { |
---|
1908 | "EXAMPLE:"; echo = 2; |
---|
1909 | ring r = 0,(x,y),dp; |
---|
1910 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1911 | setring R; // sets basering to Letterplace ring |
---|
1912 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
1913 | //Groebner basis |
---|
1914 | lpSickle(G); //invokes the procedure with ring parameters |
---|
1915 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
1916 | // ring is not necessary |
---|
1917 | lpSickle(G,0); // procedure without any degree bound |
---|
1918 | } |
---|
1919 | |
---|
1920 | proc lpSickleDim(ideal G, list #) |
---|
1921 | "USAGE: lpSickleDim(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
1922 | RETURN: list |
---|
1923 | PURPOSE:Computing the K-dimension and the mistletoes |
---|
1924 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1925 | @* - if you specify a different degree bound degbound, |
---|
1926 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1927 | NOTE: - If L is the list returned, then L[1] is an integer, the K-dimension, |
---|
1928 | @* L[2] is an ideal, the mistletoes. |
---|
1929 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
1930 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
1931 | @* - n is the number of variables, which can be set to a different number. |
---|
1932 | @* Default: attrib(basering, lV). |
---|
1933 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1934 | EXAMPLE: example lpSickleDim; shows examples |
---|
1935 | " |
---|
1936 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
1937 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
1938 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
1939 | list L; |
---|
1940 | L = lp2ivId(normalize(lead(G))); |
---|
1941 | L = ivSickleDim(L,n,degbound); |
---|
1942 | L[2] = ivL2lpI(L[2]); |
---|
1943 | return(L); |
---|
1944 | } |
---|
1945 | example |
---|
1946 | { |
---|
1947 | "EXAMPLE:"; echo = 2; |
---|
1948 | ring r = 0,(x,y),dp; |
---|
1949 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1950 | setring R; // sets basering to Letterplace ring |
---|
1951 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
1952 | //Groebner basis |
---|
1953 | lpSickleDim(G); // invokes the procedure with ring parameters |
---|
1954 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
1955 | // ring is not necessary |
---|
1956 | lpSickleDim(G,0); // procedure without any degree bound |
---|
1957 | } |
---|
1958 | |
---|
1959 | proc lpSickleHil(ideal G, list #) |
---|
1960 | "USAGE: lpSickleHil(G); |
---|
1961 | RETURN: list |
---|
1962 | PURPOSE:Computing the Hilbert series and the mistletoes |
---|
1963 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
1964 | @* - if you specify a different degree bound degbound, |
---|
1965 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
1966 | NOTE: - If L is the list returned, then L[1] is an intvec, corresponding to the |
---|
1967 | @* Hilbert series, L[2] is an ideal, the mistletoes. |
---|
1968 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
1969 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
1970 | @* - n is the number of variables, which can be set to a different number. |
---|
1971 | @* Default: attrib(basering, lV). |
---|
1972 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
1973 | @* coefficient of the Hilbert series. |
---|
1974 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1975 | EXAMPLE: example lpSickleHil; shows examples |
---|
1976 | " |
---|
1977 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
1978 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
1979 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
1980 | list L; |
---|
1981 | L = lp2ivId(normalize(lead(G))); |
---|
1982 | L = ivSickleHil(L,n,degbound); |
---|
1983 | L[2] = ivL2lpI(L[2]); |
---|
1984 | return(L); |
---|
1985 | } |
---|
1986 | example |
---|
1987 | { |
---|
1988 | "EXAMPLE:"; echo = 2; |
---|
1989 | ring r = 0,(x,y),dp; |
---|
1990 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1991 | setring R; // sets basering to Letterplace ring |
---|
1992 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
1993 | //Groebner basis |
---|
1994 | lpSickleHil(G); // invokes the procedure with ring parameters |
---|
1995 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
1996 | // ring is not necessary |
---|
1997 | lpSickleHil(G,0); // procedure without any degree bound |
---|
1998 | } |
---|
1999 | |
---|
2000 | proc sickle(ideal G, list #) |
---|
2001 | "USAGE: sickle(G[,m, d, h, degbound]); G an ideal; m,d,h,degbound optional |
---|
2002 | @* integers |
---|
2003 | RETURN: list |
---|
2004 | PURPOSE:Allowing the user to access all procs with one command |
---|
2005 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
2006 | @* - if you specify a different degree bound degbound, |
---|
2007 | @* degbound <= attrib(basering,uptodeg) should hold. |
---|
2008 | NOTE: The returned object will always be a list, but the entries of the |
---|
2009 | @* returned list may be very different |
---|
2010 | @* case m=1,d=1,h=1: see lpDHilbertSickle |
---|
2011 | @* case m=1,d=1,h=0: see lpSickleDim |
---|
2012 | @* case m=1,d=0,h=1: see lpSickleHil |
---|
2013 | @* case m=1,d=0,h=0: see lpSickle (this is the default case) |
---|
2014 | @* case m=0,d=1,h=1: see lpDHilbert |
---|
2015 | @* case m=0,d=1,h=0: see lpKDim |
---|
2016 | @* case m=0,d=0,h=1: see lpHilbert |
---|
2017 | @* case m=0,d=0,h=0: returns an error |
---|
2018 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
2019 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
2020 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
2021 | EXAMPLE: example sickle; shows examples |
---|
2022 | " |
---|
2023 | {int m,d,h,degbound; |
---|
2024 | m = 1; d = 0; h = 0; degbound = attrib(basering,"uptodeg"); |
---|
2025 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] < 1) {m = 0;}}} |
---|
2026 | if (size(#) > 1) {if (typeof(#[1])=="int"){if (#[2] > 0) {d = 1;}}} |
---|
2027 | if (size(#) > 2) {if (typeof(#[1])=="int"){if (#[3] > 0) {h = 1;}}} |
---|
2028 | if (size(#) > 3) {if (typeof(#[1])=="int"){if (#[4] >= 0) {degbound = #[4];}}} |
---|
2029 | if (m == 1) |
---|
2030 | {if (d == 0) |
---|
2031 | {if (h == 0) {return(lpSickle(G,degbound,attrib(basering,"lV")));} |
---|
2032 | else {return(lpSickleHil(G,degbound,attrib(basering,"lV")));} |
---|
2033 | } |
---|
2034 | else |
---|
2035 | {if (h == 0) {return(lpSickleDim(G,degbound,attrib(basering,"lV")));} |
---|
2036 | else {return(lpDHilbertSickle(G,degbound,attrib(basering,"lV")));} |
---|
2037 | } |
---|
2038 | } |
---|
2039 | else |
---|
2040 | {if (d == 0) |
---|
2041 | {if (h == 0) {ERROR("You request to do nothing, so relax and do so");} |
---|
2042 | else {return(lpHilbert(G,degbound,attrib(basering,"lV")));} |
---|
2043 | } |
---|
2044 | else |
---|
2045 | {if (h == 0) {return(lpKDim(G,degbound,attrib(basering,"lV")));} |
---|
2046 | else {return(lpDHilbert(G,degbound,attrib(basering,"lV")));} |
---|
2047 | } |
---|
2048 | } |
---|
2049 | } |
---|
2050 | example |
---|
2051 | { |
---|
2052 | "EXAMPLE:"; echo = 2; |
---|
2053 | ring r = 0,(x,y),dp; |
---|
2054 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2055 | setring R; // sets basering to Letterplace ring |
---|
2056 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); |
---|
2057 | // G contains a Groebner basis |
---|
2058 | sickle(G,1,1,1); // computes mistletoes, K-dimension and the Hilbert series |
---|
2059 | sickle(G,1,0,0); // computes mistletoes only |
---|
2060 | sickle(G,0,1,0); // computes K-dimension only |
---|
2061 | sickle(G,0,0,1); // computes Hilbert series only |
---|
2062 | } |
---|
2063 | |
---|
2064 | /////////////////////////////////////////////////////////////////////////////// |
---|
2065 | |
---|
2066 | |
---|
2067 | proc tst_fpadim() |
---|
2068 | { |
---|
2069 | example ivDHilbert; |
---|
2070 | example ivDHilbertSickle; |
---|
2071 | example ivDimCheck; |
---|
2072 | example ivHilbert; |
---|
2073 | example ivKDim; |
---|
2074 | example ivMis2Base; |
---|
2075 | example ivMis2Dim; |
---|
2076 | example ivOrdMisLex; |
---|
2077 | example ivSickle; |
---|
2078 | example ivSickleHil; |
---|
2079 | example ivSickleDim; |
---|
2080 | example lpDHilbert; |
---|
2081 | example lpDHilbertSickle; |
---|
2082 | example lpHilbert; |
---|
2083 | example lpDimCheck; |
---|
2084 | example lpKDim; |
---|
2085 | example lpMis2Base; |
---|
2086 | example lpMis2Dim; |
---|
2087 | example lpOrdMisLex; |
---|
2088 | example lpSickle; |
---|
2089 | example lpSickleHil; |
---|
2090 | example lpSickleDim; |
---|
2091 | example sickle; |
---|
2092 | example ivL2lpI; |
---|
2093 | example iv2lp; |
---|
2094 | example iv2lpList; |
---|
2095 | example iv2lpMat; |
---|
2096 | example lp2iv; |
---|
2097 | example lp2ivId; |
---|
2098 | example lpId2ivLi; |
---|
2099 | } |
---|
2100 | |
---|
2101 | |
---|
2102 | |
---|
2103 | |
---|
2104 | |
---|
2105 | /* |
---|
2106 | Here are some examples one may try. Just copy them into your console. |
---|
2107 | These are relations for braid groups, up to degree d: |
---|
2108 | |
---|
2109 | |
---|
2110 | LIB "fpadim.lib"; |
---|
2111 | ring r = 0,(x,y,z),dp; |
---|
2112 | int d =10; // degree |
---|
2113 | def R = makeLetterplaceRing(d); |
---|
2114 | setring R; |
---|
2115 | ideal I = y(1)*x(2)*y(3) - z(1)*y(2)*z(3), x(1)*y(2)*x(3) - z(1)*x(2)*y(3), |
---|
2116 | z(1)*x(2)*z(3) - y(1)*z(2)*x(3), x(1)*x(2)*x(3) + y(1)*y(2)*y(3) + |
---|
2117 | z(1)*z(2)*z(3) + x(1)*y(2)*z(3); |
---|
2118 | option(prot); |
---|
2119 | option(redSB);option(redTail);option(mem); |
---|
2120 | ideal J = system("freegb",I,d,3); |
---|
2121 | lpDimCheck(J); |
---|
2122 | sickle(J,1,1,1,d);//Computes mistletoes, K-dimension and the Hilbert series |
---|
2123 | |
---|
2124 | |
---|
2125 | |
---|
2126 | LIB "fpadim.lib"; |
---|
2127 | ring r = 0,(x,y,z),dp; |
---|
2128 | int d =11; // degree |
---|
2129 | def R = makeLetterplaceRing(d); |
---|
2130 | setring R; |
---|
2131 | ideal I = y(1)*x(2)*y(3) - z(1)*y(2)*z(3), x(1)*y(2)*z(3) - z(1)*x(2)*y(3), |
---|
2132 | z(1)*x(2)*z(3) - y(1)*z(2)*x(3), x(1)*x(2)*x(3) + y(1)*y(2)*y(3) + |
---|
2133 | z(1)*z(2)*z(3) + x(1)*y(2)*z(3); |
---|
2134 | option(prot); |
---|
2135 | option(redSB);option(redTail);option(mem); |
---|
2136 | ideal J = system("freegb",I,d,3); |
---|
2137 | lpDimCheck(J); |
---|
2138 | sickle(J,1,1,1,d); |
---|
2139 | |
---|
2140 | |
---|
2141 | |
---|
2142 | LIB "fpadim.lib"; |
---|
2143 | ring r = 0,(x,y,z),dp; |
---|
2144 | int d = 6; // degree |
---|
2145 | def R = makeLetterplaceRing(d); |
---|
2146 | setring R; |
---|
2147 | ideal I = y(1)*x(2)*y(3) - z(1)*y(2)*z(3), x(1)*y(2)*x(3) - z(1)*x(2)*y(3), |
---|
2148 | z(1)*x(2)*z(3) - y(1)*z(2)*x(3), x(1)*x(2)*x(3) -2*y(1)*y(2)*y(3) + 3*z(1)*z(2)*z(3) -4*x(1)*y(2)*z(3) + 5*x(1)*z(2)*z(3)- 6*x(1)*y(2)*y(3) +7*x(1)*x(2)*z(3) - 8*x(1)*x(2)*y(3); |
---|
2149 | option(prot); |
---|
2150 | option(redSB);option(redTail);option(mem); |
---|
2151 | ideal J = system("freegb",I,d,3); |
---|
2152 | lpDimCheck(J); |
---|
2153 | sickle(J,1,1,1,d); |
---|
2154 | */ |
---|
2155 | |
---|
2156 | /* |
---|
2157 | Here are some examples, which can also be found in [studzins]: |
---|
2158 | |
---|
2159 | // takes up to 880Mb of memory |
---|
2160 | LIB "fpadim.lib"; |
---|
2161 | ring r = 0,(x,y,z),dp; |
---|
2162 | int d =10; // degree |
---|
2163 | def R = makeLetterplaceRing(d); |
---|
2164 | setring R; |
---|
2165 | ideal I = |
---|
2166 | z(1)*z(2)*z(3)*z(4) + y(1)*x(2)*y(3)*x(4) - x(1)*y(2)*y(3)*x(4) - 3*z(1)*y(2)*x(3)*z(4), x(1)*x(2)*x(3) + y(1)*x(2)*y(3) - x(1)*y(2)*x(3), z(1)*y(2)*x(3)-x(1)*y(2)*z(3) + z(1)*x(2)*z(3); |
---|
2167 | option(prot); |
---|
2168 | option(redSB);option(redTail);option(mem); |
---|
2169 | ideal J = system("freegb",I,d,nvars(r)); |
---|
2170 | lpDimCheck(J); |
---|
2171 | sickle(J,1,1,1,d); // dimension is 24872 |
---|
2172 | |
---|
2173 | |
---|
2174 | LIB "fpadim.lib"; |
---|
2175 | ring r = 0,(x,y,z),dp; |
---|
2176 | int d =10; // degree |
---|
2177 | def R = makeLetterplaceRing(d); |
---|
2178 | setring R; |
---|
2179 | ideal I = x(1)*y(2) + y(1)*z(2), x(1)*x(2) + x(1)*y(2) - y(1)*x(2) - y(1)*y(2); |
---|
2180 | option(prot); |
---|
2181 | option(redSB);option(redTail);option(mem); |
---|
2182 | ideal J = system("freegb",I,d,3); |
---|
2183 | lpDimCheck(J); |
---|
2184 | sickle(J,1,1,1,d); |
---|
2185 | */ |
---|