[f2b431] | 1 | /////////////////////////////////////////////////////// |
---|
| 2 | version="$Id$"; |
---|
[5e2dd1] | 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 | |
---|
[df9f881] | 8 | Support: Joint projects LE 2697/2-1 and KR 1907/3-1 of the Priority Programme SPP 1489: |
---|
[5e2dd1] | 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 |
---|
[f2b431] | 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: |
---|
[5e2dd1] | 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> |
---|
[f2b431] | 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. For more details see |
---|
| 34 | @* [studzins]. This package uses the Letterplace format introduced by |
---|
| 35 | @* [lls]. The algebra can either be represented as a Letterplace ring or |
---|
| 36 | @* via integer vectors: Every variable will only be represented by its |
---|
| 37 | @* number, so variable one is represented as 1, variable two as 2 and so |
---|
| 38 | @* on. The monomial x_1*x_3*x_2 for example will be stored as (1,3,2). |
---|
| 39 | @* Multiplication is concatenation. Note that there is no algorithm for |
---|
| 40 | @* computing the normal form yet, but for our case it is not needed. |
---|
| 41 | @* |
---|
| 42 | |
---|
| 43 | References: |
---|
[df9f881] | 44 | |
---|
[5e2dd1] | 45 | @* [ufna] Ufnarovskij: Combinatorical and asymptotic methods in algebra, 1990 |
---|
| 46 | @* [lls] Levandovskyy, La Scala: Letterplace ideals and non-commutative |
---|
[df9f881] | 47 | Groebner bases, 2009 |
---|
[5e2dd1] | 48 | @* [studzins] Studzinski: Dimension computations in non-commutative, |
---|
[df9f881] | 49 | associative algebras, Diploma thesis, RWTH Aachen, 2010 |
---|
[5e2dd1] | 50 | |
---|
[0a2f7d] | 51 | Assumptions: |
---|
[5e2dd1] | 52 | @* - basering is always a Letterplace ring |
---|
| 53 | @* - all intvecs correspond to Letterplace monomials |
---|
| 54 | @* - if you specify a different degree bound d, |
---|
[f2b431] | 55 | d <= attrib(basering,uptodeg) holds |
---|
[5e2dd1] | 56 | @* In the procedures below, 'iv' stands for intvec representation |
---|
[df9f881] | 57 | and 'lp' for the letterplace representation of monomials |
---|
[5e2dd1] | 58 | |
---|
| 59 | PROCEDURES: |
---|
| 60 | |
---|
[f2b431] | 61 | lpGkDim(G); computes the Gelfand Kirillov dimension of A/<G> |
---|
[5e2dd1] | 62 | ivDHilbert(L,n[,d]); computes the K-dimension and the Hilbert series |
---|
| 63 | ivDHilbertSickle(L,n[,d]); computes mistletoes, K-dimension and Hilbert series |
---|
| 64 | ivDimCheck(L,n); checks if the K-dimension of A/<L> is infinite |
---|
[f2b431] | 65 | lpGlDimBound(G); computes upper bound of global dimension of A/<G> |
---|
[5e2dd1] | 66 | ivHilbert(L,n[,d]); computes the Hilbert series of A/<L> in intvec format |
---|
| 67 | ivKDim(L,n[,d]); computes the K-dimension of A/<L> in intvec format |
---|
| 68 | ivMis2Dim(M); computes the K-dimension of the factor algebra |
---|
| 69 | ivOrdMisLex(M); orders a list of intvecs lexicographically |
---|
| 70 | ivSickle(L,n[,d]); computes the mistletoes of A/<L> in intvec format |
---|
| 71 | ivSickleHil(L,n[,d]); computes the mistletoes and Hilbert series of A/<L> |
---|
| 72 | ivSickleDim(L,n[,d]); computes the mistletoes and the K-dimension of A/<L> |
---|
| 73 | lpDHilbert(G[,d,n]); computes the K-dimension and Hilbert series of A/<G> |
---|
| 74 | lpDHilbertSickle(G[,d,n]); computes mistletoes, K-dimension and Hilbert series |
---|
| 75 | lpHilbert(G[,d,n]); computes the Hilbert series of A/<G> in lp format |
---|
| 76 | lpDimCheck(G); checks if the K-dimension of A/<G> is infinite |
---|
| 77 | lpKDim(G[,d,n]); computes the K-dimension of A/<G> in lp format |
---|
| 78 | lpMis2Dim(M); computes the K-dimension of the factor algebra |
---|
| 79 | lpOrdMisLex(M); orders an ideal of lp-monomials lexicographically |
---|
| 80 | lpSickle(G[,d,n]); computes the mistletoes of A/<G> in lp format |
---|
| 81 | lpSickleHil(G[,d,n]); computes the mistletoes and Hilbert series of A/<G> |
---|
| 82 | lpSickleDim(G[,d,n]); computes the mistletoes and the K-dimension of A/<G> |
---|
| 83 | sickle(G[,m,d,h]); can be used to access all lp main procedures |
---|
| 84 | |
---|
| 85 | |
---|
| 86 | ivL2lpI(L); transforms a list of intvecs into an ideal of lp monomials |
---|
| 87 | iv2lp(I); transforms an intvec into the corresponding monomial |
---|
| 88 | iv2lpList(L); transforms a list of intmats into an ideal of lp monomials |
---|
| 89 | iv2lpMat(M); transforms an intmat into an ideal of lp monomials |
---|
| 90 | lp2iv(p); transforms a polynomial into the corresponding intvec |
---|
| 91 | lp2ivId(G); transforms an ideal into the corresponding list of intmats |
---|
| 92 | lpId2ivLi(G); transforms a lp-ideal into the corresponding list of intvecs |
---|
| 93 | |
---|
| 94 | SEE ALSO: freegb_lib |
---|
| 95 | "; |
---|
| 96 | |
---|
| 97 | LIB "freegb.lib"; //for letterplace rings |
---|
| 98 | LIB "general.lib";//for sorting mistletoes |
---|
| 99 | |
---|
| 100 | ///////////////////////////////////////////////////////// |
---|
| 101 | |
---|
| 102 | |
---|
| 103 | //--------------- auxiliary procedures ------------------ |
---|
| 104 | |
---|
| 105 | static proc allVars(list L, intvec P, int n) |
---|
| 106 | "USAGE: allVars(L,P,n); L a list of intmats, P an intvec, n an integer |
---|
[5e8ee4c] | 107 | RETURN: int, 0 if all variables are contained in the quotient algebra, 1 otherwise |
---|
[5e2dd1] | 108 | " |
---|
| 109 | {int i,j,r; |
---|
[df9f881] | 110 | intvec V; |
---|
| 111 | for (i = 1; i <= size(P); i++) {if (P[i] == 1){ j = i; break;}} |
---|
| 112 | V = L[j][1..nrows(L[j]),1]; |
---|
| 113 | for (i = 1; i <= n; i++) {if (isInVec(i,V) == 0) {r = 1; break;}} |
---|
| 114 | if (r == 0) {return(1);} |
---|
| 115 | else {return(0);} |
---|
[5e2dd1] | 116 | } |
---|
| 117 | |
---|
| 118 | static proc checkAssumptions(int d, list L) |
---|
| 119 | "PURPOSE: Checks, if all the Assumptions are holding |
---|
| 120 | " |
---|
| 121 | {if (typeof(attrib(basering,"isLetterplaceRing"))=="string") {ERROR("Basering is not a Letterplace ring!");} |
---|
[df9f881] | 122 | if (d > attrib(basering,"uptodeg")) {ERROR("Specified degree bound exceeds ring parameter!");} |
---|
| 123 | int i; |
---|
| 124 | for (i = 1; i <= size(L); i++) |
---|
| 125 | {if (entryViolation(L[i], attrib(basering,"lV"))) |
---|
| 126 | {ERROR("Not allowed monomial/intvec found!");} |
---|
| 127 | } |
---|
| 128 | return(); |
---|
[5e2dd1] | 129 | } |
---|
| 130 | |
---|
| 131 | static proc createStartMat(int d, int n) |
---|
| 132 | "USAGE: createStartMat(d,n); d, n integers |
---|
[5e8ee4c] | 133 | RETURN: intmat |
---|
[5e2dd1] | 134 | PURPOSE:Creating the intmat with all normal monomials in n variables and of degree d to start with |
---|
| 135 | NOTE: d has to be > 0 |
---|
| 136 | " |
---|
| 137 | {intmat M[(n^d)][d]; |
---|
[df9f881] | 138 | int i1,i2,i3,i4; |
---|
| 139 | for (i1 = 1; i1 <= d; i1++) //Spalten |
---|
| 140 | {i2 = 1; //durchlaeuft Zeilen |
---|
| 141 | while (i2 <= (n^d)) |
---|
| 142 | {for (i3 = 1; i3 <= n; i3++) |
---|
| 143 | {for (i4 = 1; i4 <= (n^(i1-1)); i4++) |
---|
[f2b431] | 144 | {M[i2,i1] = i3; |
---|
| 145 | i2 = i2 + 1; |
---|
| 146 | } |
---|
[df9f881] | 147 | } |
---|
[5e2dd1] | 148 | } |
---|
| 149 | } |
---|
[df9f881] | 150 | return(M); |
---|
[5e2dd1] | 151 | } |
---|
| 152 | |
---|
| 153 | static proc createStartMat1(int n, intmat M) |
---|
| 154 | "USAGE: createStartMat1(n,M); n an integer, M an intmat |
---|
[5e8ee4c] | 155 | RETURN: intmat, with all variables except those in M |
---|
[5e2dd1] | 156 | " |
---|
| 157 | {int i; |
---|
[df9f881] | 158 | intvec V,Vt; |
---|
| 159 | V = M[(1..nrows(M)),1]; |
---|
| 160 | for (i = 1; i <= size(V); i++) {if (isInVec(i,V) == 0) {Vt = Vt,i;}} |
---|
| 161 | if (Vt == 0) {intmat S; return(S);} |
---|
| 162 | else {Vt = Vt[2..size(Vt)]; intmat S [size(Vt)][1]; S[1..size(Vt),1] = Vt; return(S);} |
---|
[5e2dd1] | 163 | } |
---|
| 164 | |
---|
| 165 | static proc entryViolation(intmat M, int n) |
---|
| 166 | "PURPOSE:checks, if all entries in M are variable-related |
---|
| 167 | " |
---|
[f2b431] | 168 | {int i,j; |
---|
[df9f881] | 169 | for (i = 1; i <= nrows(M); i++) |
---|
| 170 | {for (j = 1; j <= ncols(M); j++) |
---|
| 171 | {if(!((1<=M[i,j])&&(M[i,j]<=n))) {return(1);}} |
---|
| 172 | } |
---|
| 173 | return(0); |
---|
[5e2dd1] | 174 | } |
---|
| 175 | |
---|
| 176 | static proc findDimen(intvec V, int n, list L, intvec P, list #) |
---|
| 177 | "USAGE: findDimen(V,n,L,P,degbound); V,P intvecs, n, an integer, L a list, |
---|
| 178 | @* degbound an optional integer |
---|
[5e8ee4c] | 179 | RETURN: int |
---|
[5e2dd1] | 180 | PURPOSE:Computing the K-dimension of the quotient algebra |
---|
| 181 | " |
---|
| 182 | {int degbound = 0; |
---|
[df9f881] | 183 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
| 184 | int dimen,i,j,w,it; |
---|
| 185 | intvec Vt,Vt2; |
---|
| 186 | module M; |
---|
| 187 | if (degbound == 0) |
---|
| 188 | {for (i = 1; i <= n; i++) |
---|
| 189 | {Vt = V, i; w = 0; |
---|
| 190 | for (j = 1; j<= size(P); j++) |
---|
| 191 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 192 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 193 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 194 | } |
---|
[df9f881] | 195 | } |
---|
| 196 | if (w == 0) |
---|
| 197 | {vector Vtt; |
---|
[f2b431] | 198 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 199 | M = M,Vtt; |
---|
| 200 | kill Vtt; |
---|
[df9f881] | 201 | } |
---|
[5e2dd1] | 202 | } |
---|
[df9f881] | 203 | if (size(M) == 0) {return(0);} |
---|
| 204 | else |
---|
| 205 | {M = simplify(M,2); |
---|
| 206 | for (i = 1; i <= size(M); i++) |
---|
| 207 | {kill Vt; intvec Vt; |
---|
[f2b431] | 208 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 209 | dimen = dimen + 1 + findDimen(Vt,n,L,P); |
---|
[df9f881] | 210 | } |
---|
| 211 | return(dimen); |
---|
[5e2dd1] | 212 | } |
---|
| 213 | } |
---|
| 214 | else |
---|
[df9f881] | 215 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
| 216 | if (size(V) == degbound) {return(0);} |
---|
| 217 | for (i = 1; i <= n; i++) |
---|
| 218 | {Vt = V, i; w = 0; |
---|
| 219 | for (j = 1; j<= size(P); j++) |
---|
| 220 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 221 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 222 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 223 | } |
---|
[df9f881] | 224 | } |
---|
| 225 | if (w == 0) {vector Vtt; |
---|
[f2b431] | 226 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 227 | M = M,Vtt; |
---|
| 228 | kill Vtt; |
---|
[df9f881] | 229 | } |
---|
[5e2dd1] | 230 | } |
---|
[df9f881] | 231 | if (size(M) == 0) {return(0);} |
---|
| 232 | else |
---|
| 233 | {M = simplify(M,2); |
---|
| 234 | for (i = 1; i <= size(M); i++) |
---|
| 235 | {kill Vt; intvec Vt; |
---|
[f2b431] | 236 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 237 | dimen = dimen + 1 + findDimen(Vt,n,L,P,degbound); |
---|
[df9f881] | 238 | } |
---|
| 239 | return(dimen); |
---|
[5e2dd1] | 240 | } |
---|
| 241 | } |
---|
| 242 | } |
---|
| 243 | |
---|
| 244 | static proc findCycle(intvec V, list L, intvec P, int n, int ld, module M) |
---|
| 245 | "USAGE: |
---|
[5e8ee4c] | 246 | RETURN: int, 1 if Ufn-graph contains a cycle, or 0 otherwise |
---|
[5e2dd1] | 247 | PURPOSE:Searching the Ufnarovskij graph for cycles |
---|
| 248 | " |
---|
| 249 | {int i,j,w,r;intvec Vt,Vt2; |
---|
[df9f881] | 250 | int it, it2; |
---|
| 251 | if (size(V) < ld) |
---|
| 252 | {for (i = 1; i <= n; i++) |
---|
[5e2dd1] | 253 | {Vt = V,i; w = 0; |
---|
[df9f881] | 254 | for (j = 1; j <= size(P); j++) |
---|
| 255 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 256 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 257 | if (isInMat(Vt2,L[j]) > 0) |
---|
| 258 | {w = 1; break;} |
---|
| 259 | } |
---|
[5e2dd1] | 260 | } |
---|
[df9f881] | 261 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
---|
| 262 | if (r == 1) {break;} |
---|
[5e2dd1] | 263 | } |
---|
| 264 | return(r); |
---|
| 265 | } |
---|
| 266 | else |
---|
[df9f881] | 267 | {j = size(M); |
---|
| 268 | if (j > 0) |
---|
| 269 | { |
---|
| 270 | intmat Mt[j][nrows(M)]; |
---|
| 271 | for (it = 1; it <= j; it++) |
---|
| 272 | { for(it2 = 1; it2 <= nrows(M);it2++) |
---|
[f2b431] | 273 | {Mt[it,it2] = int(leadcoef(M[it2,it]));} |
---|
[df9f881] | 274 | } |
---|
| 275 | Vt = V[(size(V)-ld+1)..size(V)]; |
---|
| 276 | //Mt; type(Mt);Vt;type(Vt); |
---|
| 277 | if (isInMat(Vt,Mt) > 0) {return(1);} |
---|
| 278 | else |
---|
| 279 | {vector Vtt; |
---|
[f2b431] | 280 | for (it =1; it <= size(Vt); it++) |
---|
| 281 | {Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 282 | M = M,Vtt; |
---|
| 283 | kill Vtt; |
---|
| 284 | for (i = 1; i <= n; i++) |
---|
| 285 | {Vt = V,i; w = 0; |
---|
| 286 | for (j = 1; j <= size(P); j++) |
---|
| 287 | {if (P[j] <= size(Vt)) |
---|
| 288 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 289 | //L[j]; type(L[j]);Vt2;type(Vt2); |
---|
| 290 | if (isInMat(Vt2,L[j]) > 0) |
---|
| 291 | {w = 1; break;} |
---|
| 292 | } |
---|
| 293 | } |
---|
| 294 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
---|
| 295 | if (r == 1) {break;} |
---|
| 296 | } |
---|
| 297 | return(r); |
---|
[5e2dd1] | 298 | } |
---|
| 299 | } |
---|
[df9f881] | 300 | else |
---|
| 301 | { Vt = V[(size(V)-ld+1)..size(V)]; |
---|
| 302 | vector Vtt; |
---|
| 303 | for (it = 1; it <= size(Vt); it++) |
---|
| 304 | {Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 305 | M = Vtt; |
---|
| 306 | kill Vtt; |
---|
| 307 | for (i = 1; i <= n; i++) |
---|
| 308 | {Vt = V,i; w = 0; |
---|
[f2b431] | 309 | for (j = 1; j <= size(P); j++) |
---|
| 310 | {if (P[j] <= size(Vt)) |
---|
| 311 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 312 | //L[j]; type(L[j]);Vt2;type(Vt2); |
---|
| 313 | if (isInMat(Vt2,L[j]) > 0) |
---|
| 314 | {w = 1; break;} |
---|
| 315 | } |
---|
| 316 | } |
---|
| 317 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
---|
| 318 | if (r == 1) {break;} |
---|
[df9f881] | 319 | } |
---|
| 320 | return(r); |
---|
| 321 | } |
---|
[5e2dd1] | 322 | } |
---|
| 323 | } |
---|
| 324 | |
---|
[f2b431] | 325 | |
---|
| 326 | static proc findCycleDFS(int i, intmat T, intvec V) |
---|
| 327 | " |
---|
| 328 | PURPOSE: |
---|
| 329 | this is a classical deep-first search for cycles contained in a graph given by an intmat |
---|
| 330 | " |
---|
| 331 | { |
---|
| 332 | intvec rV; |
---|
| 333 | int k,k1,t; |
---|
| 334 | int j = V[size(V)]; |
---|
| 335 | if (T[j,i] > 0) {return(V);} |
---|
| 336 | else |
---|
| 337 | { |
---|
| 338 | for (k = 1; k <= ncols(T); k++) |
---|
| 339 | { |
---|
| 340 | t = 0; |
---|
| 341 | if (T[j,k] > 0) |
---|
| 342 | { |
---|
| 343 | for (k1 = 1; k1 <= size(V); k1++) {if (V[k1] == k) {t = 1; break;}} |
---|
| 344 | if (t == 0) |
---|
| 345 | { |
---|
| 346 | rV = V; |
---|
| 347 | rV[size(rV)+1] = k; |
---|
| 348 | rV = findCycleDFS(i,T,rV); |
---|
| 349 | if (rV[1] > -1) {return(rV);} |
---|
| 350 | } |
---|
| 351 | } |
---|
| 352 | } |
---|
| 353 | } |
---|
| 354 | return(intvec(-1)); |
---|
| 355 | } |
---|
| 356 | |
---|
| 357 | |
---|
| 358 | |
---|
[5e2dd1] | 359 | static proc findHCoeff(intvec V,int n,list L,intvec P,intvec H,list #) |
---|
| 360 | "USAGE: findHCoeff(V,n,L,P,H,degbound); L a list of intmats, degbound an integer |
---|
[5e8ee4c] | 361 | RETURN: intvec |
---|
[5e2dd1] | 362 | PURPOSE:Computing the coefficient of the Hilbert series (upto degree degbound) |
---|
| 363 | NOTE: Starting with a part of the Hilbert series we change the coefficient |
---|
[df9f881] | 364 | @* depending on how many basis elements we found on the actual branch |
---|
[5e2dd1] | 365 | " |
---|
| 366 | {int degbound = 0; |
---|
[df9f881] | 367 | if (size(#) > 0){if (#[1] > 0){degbound = #[1];}} |
---|
| 368 | int i,w,j,it; |
---|
| 369 | int h1 = 0; |
---|
| 370 | intvec Vt,Vt2,H1; |
---|
| 371 | module M; |
---|
| 372 | if (degbound == 0) |
---|
| 373 | {for (i = 1; i <= n; i++) |
---|
| 374 | {Vt = V, i; w = 0; |
---|
| 375 | for (j = 1; j<= size(P); j++) |
---|
| 376 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 377 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 378 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 379 | } |
---|
[df9f881] | 380 | } |
---|
| 381 | if (w == 0) |
---|
| 382 | {vector Vtt; |
---|
[f2b431] | 383 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 384 | M = M,Vtt; |
---|
| 385 | kill Vtt; |
---|
[df9f881] | 386 | } |
---|
[5e2dd1] | 387 | } |
---|
[df9f881] | 388 | if (size(M) == 0) {return(H);} |
---|
| 389 | else |
---|
| 390 | {M = simplify(M,2); |
---|
| 391 | for (i = 1; i <= size(M); i++) |
---|
| 392 | {kill Vt; intvec Vt; |
---|
[f2b431] | 393 | for (j =1; j <= size(M[i]); j++) {Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 394 | h1 = h1 + 1; H1 = findHCoeff(Vt,n,L,P,H1); |
---|
[df9f881] | 395 | } |
---|
| 396 | if (size(H1) < (size(V)+2)) {H1[(size(V)+2)] = h1;} |
---|
| 397 | else {H1[(size(V)+2)] = H1[(size(V)+2)] + h1;} |
---|
| 398 | H1 = H1 + H; |
---|
| 399 | return(H1); |
---|
[5e2dd1] | 400 | } |
---|
| 401 | } |
---|
| 402 | else |
---|
[df9f881] | 403 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
| 404 | if (size(V) == degbound) {return(H);} |
---|
| 405 | for (i = 1; i <= n; i++) |
---|
| 406 | {Vt = V, i; w = 0; |
---|
| 407 | for (j = 1; j<= size(P); j++) |
---|
| 408 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 409 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 410 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 411 | } |
---|
[df9f881] | 412 | } |
---|
| 413 | if (w == 0) |
---|
| 414 | {vector Vtt; |
---|
[f2b431] | 415 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 416 | M = M,Vtt; |
---|
| 417 | kill Vtt; |
---|
[df9f881] | 418 | } |
---|
[5e2dd1] | 419 | } |
---|
[df9f881] | 420 | if (size(M) == 0) {return(H);} |
---|
| 421 | else |
---|
| 422 | {M = simplify(M,2); |
---|
| 423 | for (i = 1; i <= size(M); i++) |
---|
| 424 | {kill Vt; intvec Vt; |
---|
[f2b431] | 425 | for (j =1; j <= size(M[i]); j++) |
---|
| 426 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 427 | h1 = h1 + 1; H1 = findHCoeff(Vt,n,L,P,H1,degbound); |
---|
[df9f881] | 428 | } |
---|
| 429 | if (size(H1) < (size(V)+2)) { H1[(size(V)+2)] = h1;} |
---|
| 430 | else {H1[(size(V)+2)] = H1[(size(V)+2)] + h1;} |
---|
| 431 | H1 = H1 + H; |
---|
| 432 | return(H1); |
---|
[5e2dd1] | 433 | } |
---|
| 434 | } |
---|
| 435 | } |
---|
| 436 | |
---|
| 437 | static proc findHCoeffMis(intvec V, int n, list L, intvec P, list R,list #) |
---|
| 438 | "USAGE: findHCoeffMis(V,n,L,P,R,degbound); degbound an optional integer, L a |
---|
| 439 | @* list of Intmats, R |
---|
[5e8ee4c] | 440 | RETURN: list |
---|
[5e2dd1] | 441 | PURPOSE:Computing the coefficients of the Hilbert series and the Mistletoes all |
---|
| 442 | @* at once |
---|
| 443 | " |
---|
| 444 | {int degbound = 0; |
---|
[df9f881] | 445 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
| 446 | int i,w,j,h1; |
---|
| 447 | intvec Vt,Vt2,H1; int it; |
---|
| 448 | module M; |
---|
| 449 | if (degbound == 0) |
---|
| 450 | {for (i = 1; i <= n; i++) |
---|
| 451 | {Vt = V, i; w = 0; |
---|
| 452 | for (j = 1; j<= size(P); j++) |
---|
| 453 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 454 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 455 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 456 | } |
---|
[df9f881] | 457 | } |
---|
| 458 | if (w == 0) |
---|
| 459 | {vector Vtt; |
---|
[f2b431] | 460 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 461 | M = M,Vtt; |
---|
| 462 | kill Vtt; |
---|
[df9f881] | 463 | } |
---|
[5e2dd1] | 464 | } |
---|
[df9f881] | 465 | if (size(M) == 0) {if (size(R) < 2){R[2] = list(V);} else {R[2] = R[2] + list(V);} return(R);} |
---|
| 466 | else |
---|
| 467 | {M = simplify(M,2); |
---|
| 468 | for (i = 1; i <= size(M); i++) |
---|
| 469 | {kill Vt; intvec Vt; |
---|
[f2b431] | 470 | for (j =1; j <= size(M[i]); j++) |
---|
| 471 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 472 | if (size(R[1]) < (size(V)+2)) { R[1][(size(V)+2)] = 1;} |
---|
| 473 | else |
---|
| 474 | {R[1][(size(V)+2)] = R[1][(size(V)+2)] + 1;} |
---|
| 475 | R = findHCoeffMis(Vt,n,L,P,R); |
---|
[df9f881] | 476 | } |
---|
| 477 | return(R); |
---|
[5e2dd1] | 478 | } |
---|
| 479 | } |
---|
| 480 | else |
---|
[df9f881] | 481 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
| 482 | if (size(V) == degbound) |
---|
| 483 | {if (size(R) < 2){R[2] = list (V);} |
---|
| 484 | else{R[2] = R[2] + list (V);} |
---|
| 485 | return(R); |
---|
[5e2dd1] | 486 | } |
---|
[df9f881] | 487 | for (i = 1; i <= n; i++) |
---|
| 488 | {Vt = V, i; w = 0; |
---|
| 489 | for (j = 1; j<= size(P); j++) |
---|
| 490 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 491 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 492 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 493 | } |
---|
[df9f881] | 494 | } |
---|
| 495 | if (w == 0) |
---|
| 496 | {vector Vtt; |
---|
[f2b431] | 497 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 498 | M = M,Vtt; |
---|
| 499 | kill Vtt; |
---|
[df9f881] | 500 | } |
---|
[5e2dd1] | 501 | } |
---|
[df9f881] | 502 | if (size(M) == 0) {if (size(R) < 2){R[2] = list(V);} else {R[2] = R[2] + list(V);} return(R);} |
---|
| 503 | else |
---|
| 504 | {M = simplify(M,2); |
---|
| 505 | for (i = 1; i <= ncols(M); i++) |
---|
| 506 | {kill Vt; intvec Vt; |
---|
[f2b431] | 507 | for (j =1; j <= size(M[i]); j++) |
---|
| 508 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 509 | if (size(R[1]) < (size(V)+2)) { R[1][(size(V)+2)] = 1;} |
---|
| 510 | else |
---|
| 511 | {R[1][(size(V)+2)] = R[1][(size(V)+2)] + 1;} |
---|
| 512 | R = findHCoeffMis(Vt,n,L,P,R,degbound); |
---|
[df9f881] | 513 | } |
---|
| 514 | return(R); |
---|
[5e2dd1] | 515 | } |
---|
| 516 | } |
---|
| 517 | } |
---|
| 518 | |
---|
| 519 | |
---|
| 520 | static proc findMisDim(intvec V,int n,list L,intvec P,list R,list #) |
---|
| 521 | "USAGE: |
---|
[5e8ee4c] | 522 | RETURN: list |
---|
[5e2dd1] | 523 | PURPOSE:Computing the K-dimension and the Mistletoes all at once |
---|
| 524 | " |
---|
| 525 | {int degbound = 0; |
---|
[df9f881] | 526 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
| 527 | int dimen,i,j,w; |
---|
| 528 | intvec Vt,Vt2; int it; |
---|
| 529 | module M; |
---|
| 530 | if (degbound == 0) |
---|
| 531 | {for (i = 1; i <= n; i++) |
---|
| 532 | {Vt = V, i; w = 0; |
---|
| 533 | for (j = 1; j<= size(P); j++) |
---|
| 534 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 535 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 536 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 537 | } |
---|
[df9f881] | 538 | } |
---|
| 539 | if (w == 0) |
---|
| 540 | {vector Vtt; |
---|
[f2b431] | 541 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 542 | M = M,Vtt; |
---|
| 543 | kill Vtt; |
---|
[df9f881] | 544 | } |
---|
[5e2dd1] | 545 | } |
---|
[df9f881] | 546 | if (size(M) == 0) |
---|
| 547 | {if (size(R) < 2){R[2] = list (V);} |
---|
| 548 | else{R[2] = R[2] + list(V);} |
---|
| 549 | return(R); |
---|
| 550 | } |
---|
| 551 | else |
---|
| 552 | {M = simplify(M,2); |
---|
| 553 | for (i = 1; i <= size(M); i++) |
---|
| 554 | {kill Vt; intvec Vt; |
---|
[f2b431] | 555 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 556 | R[1] = R[1] + 1; R = findMisDim(Vt,n,L,P,R); |
---|
[df9f881] | 557 | } |
---|
| 558 | return(R); |
---|
[5e2dd1] | 559 | } |
---|
| 560 | } |
---|
| 561 | else |
---|
[df9f881] | 562 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
| 563 | if (size(V) == degbound) |
---|
| 564 | {if (size(R) < 2){R[2] = list (V);} |
---|
| 565 | else{R[2] = R[2] + list (V);} |
---|
| 566 | return(R); |
---|
| 567 | } |
---|
| 568 | for (i = 1; i <= n; i++) |
---|
| 569 | {Vt = V, i; w = 0; |
---|
| 570 | for (j = 1; j<= size(P); j++) |
---|
| 571 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 572 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 573 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 574 | } |
---|
[df9f881] | 575 | } |
---|
| 576 | if (w == 0) |
---|
| 577 | {vector Vtt; |
---|
[f2b431] | 578 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 579 | M = M,Vtt; |
---|
| 580 | kill Vtt; |
---|
[df9f881] | 581 | } |
---|
[5e2dd1] | 582 | } |
---|
[df9f881] | 583 | if (size(M) == 0) |
---|
| 584 | {if (size(R) < 2){R[2] = list (V);} |
---|
| 585 | else{R[2] = R[2] + list(V);} |
---|
| 586 | return(R); |
---|
[5e2dd1] | 587 | } |
---|
[df9f881] | 588 | else |
---|
| 589 | {M = simplify(M,2); |
---|
| 590 | for (i = 1; i <= size(M); i++) |
---|
| 591 | {kill Vt; intvec Vt; |
---|
[f2b431] | 592 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 593 | R[1] = R[1] + 1; R = findMisDim(Vt,n,L,P,R,degbound); |
---|
[df9f881] | 594 | } |
---|
| 595 | return(R); |
---|
| 596 | } |
---|
[5e2dd1] | 597 | } |
---|
| 598 | } |
---|
| 599 | |
---|
| 600 | |
---|
| 601 | static proc findmistletoes(intvec V, int n, list L, intvec P, list #) |
---|
| 602 | "USAGE: findmistletoes(V,n,L,P,degbound); V a normal word, n the number of |
---|
| 603 | @* variables, L the GB, P the occuring degrees, |
---|
| 604 | @* and degbound the (optional) degreebound |
---|
[5e8ee4c] | 605 | RETURN: list |
---|
[5e2dd1] | 606 | PURPOSE:Computing mistletoes starting in V |
---|
| 607 | NOTE: V has to be normal w.r.t. L, it will not be checked for being so |
---|
| 608 | " |
---|
| 609 | {int degbound = 0; |
---|
[df9f881] | 610 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
| 611 | list R; intvec Vt,Vt2; int it; |
---|
| 612 | int i,j; |
---|
| 613 | module M; |
---|
| 614 | if (degbound == 0) |
---|
| 615 | {int w; |
---|
| 616 | for (i = 1; i <= n; i++) |
---|
| 617 | {Vt = V,i; w = 0; |
---|
| 618 | for (j = 1; j <= size(P); j++) |
---|
| 619 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 620 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 621 | if (isInMat(Vt2,L[j]) > 0) |
---|
| 622 | {w = 1; break;} |
---|
| 623 | } |
---|
[df9f881] | 624 | } |
---|
| 625 | if (w == 0) |
---|
| 626 | {vector Vtt; |
---|
[f2b431] | 627 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 628 | M = M,Vtt; |
---|
| 629 | kill Vtt; |
---|
[df9f881] | 630 | } |
---|
[5e2dd1] | 631 | } |
---|
[df9f881] | 632 | if (size(M)==0) {R = V; return(R);} |
---|
| 633 | else |
---|
| 634 | {M = simplify(M,2); |
---|
| 635 | for (i = 1; i <= size(M); i++) |
---|
| 636 | {kill Vt; intvec Vt; |
---|
[f2b431] | 637 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 638 | R = R + findmistletoes(Vt,n,L,P); |
---|
[df9f881] | 639 | } |
---|
| 640 | return(R); |
---|
[5e2dd1] | 641 | } |
---|
| 642 | } |
---|
| 643 | else |
---|
[df9f881] | 644 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
| 645 | if (size(V) == degbound) {R = V; return(R);} |
---|
| 646 | int w; |
---|
| 647 | for (i = 1; i <= n; i++) |
---|
| 648 | {Vt = V,i; w = 0; |
---|
| 649 | for (j = 1; j <= size(P); j++) |
---|
| 650 | {if (P[j] <= size(Vt)) |
---|
[f2b431] | 651 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 652 | if (isInMat(Vt2,L[j]) > 0){w = 1; break;} |
---|
| 653 | } |
---|
[df9f881] | 654 | } |
---|
| 655 | if (w == 0) |
---|
| 656 | {vector Vtt; |
---|
[f2b431] | 657 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 658 | M = M,Vtt; |
---|
| 659 | kill Vtt; |
---|
[df9f881] | 660 | } |
---|
| 661 | } |
---|
| 662 | if (size(M) == 0) {R = V; return(R);} |
---|
| 663 | else |
---|
| 664 | {M = simplify(M,2); |
---|
| 665 | for (i = 1; i <= ncols(M); i++) |
---|
| 666 | {kill Vt; intvec Vt; |
---|
[f2b431] | 667 | for (j =1; j <= size(M[i]); j++) |
---|
| 668 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 669 | //Vt; typeof(Vt); size(Vt); |
---|
| 670 | R = R + findmistletoes(Vt,n,L,P,degbound); |
---|
[df9f881] | 671 | } |
---|
| 672 | return(R); |
---|
| 673 | } |
---|
[5e2dd1] | 674 | } |
---|
| 675 | } |
---|
| 676 | |
---|
[f2b431] | 677 | static proc growthAlg(intmat T, list #) |
---|
| 678 | " |
---|
| 679 | real algorithm for checking the growth of an algebra |
---|
| 680 | " |
---|
| 681 | { |
---|
| 682 | int s = 1; |
---|
| 683 | if (size(#) > 0) { s = #[1];} |
---|
| 684 | int j; |
---|
| 685 | int n = ncols(T); |
---|
| 686 | intvec NV,C; NV[n] = 0; int m,i; |
---|
| 687 | intmat T2[n][n] = T[1..n,1..n]; intmat N[n][n]; |
---|
| 688 | if (T2 == N) |
---|
| 689 | { |
---|
| 690 | for (i = 1; i <= n; i++) |
---|
| 691 | { |
---|
| 692 | if (m < T[n+1,i]) { m = T[n+1,i];} |
---|
| 693 | } |
---|
| 694 | return(m); |
---|
| 695 | } |
---|
| 696 | |
---|
| 697 | //first part: the diagonals |
---|
| 698 | for (i = s; i <= n; i++) |
---|
| 699 | { |
---|
| 700 | if (T[i,i] > 0) |
---|
| 701 | { |
---|
| 702 | if ((T[i,i] >= 1) && (T[n+1,i] > 0)) {return(-1);} |
---|
| 703 | if ((T[i,i] == 1) && (T[n+1,i] == 0)) |
---|
| 704 | { |
---|
| 705 | T[i,i] = 0; |
---|
| 706 | T[n+1,i] = 1; |
---|
| 707 | return(growthAlg(T)); |
---|
| 708 | } |
---|
| 709 | } |
---|
| 710 | } |
---|
| 711 | |
---|
| 712 | //second part: searching for the last but one vertices |
---|
| 713 | T2 = T2*T2; |
---|
| 714 | for (i = s; i <= n; i++) |
---|
| 715 | { |
---|
| 716 | if ((intvec(T[i,1..n]) <> intvec(0)) && (intvec(T2[i,1..n]) == intvec(0))) |
---|
| 717 | { |
---|
| 718 | for (j = 1; j <= n; j++) |
---|
| 719 | { |
---|
| 720 | if ((T[i,j] > 0) && (m < T[n+1,j])) {m = T[n+1,j];} |
---|
| 721 | } |
---|
| 722 | T[n+1,i] = T[n+1,i] + m; |
---|
| 723 | T[i,1..n] = NV; |
---|
| 724 | return(growthAlg(T)); |
---|
| 725 | } |
---|
| 726 | } |
---|
| 727 | m = 0; |
---|
| 728 | |
---|
| 729 | //third part: searching for circles |
---|
| 730 | for (i = s; i <= n; i++) |
---|
| 731 | { |
---|
| 732 | T2 = T[1..n,1..n]; |
---|
| 733 | C = findCycleDFS(i,T2, intvec(i)); |
---|
| 734 | if (C[1] > 0) |
---|
| 735 | { |
---|
| 736 | for (j = 2; j <= size(C); j++) |
---|
| 737 | { |
---|
| 738 | T[i,1..n] = T[i,1..n] + T[C[j],1..n]; |
---|
| 739 | T[C[j],1..n] = NV; |
---|
| 740 | } |
---|
| 741 | for (j = 2; j <= size(C); j++) |
---|
| 742 | { |
---|
| 743 | T[1..n,i] = T[1..n,i] + T[1..n,C[j]]; |
---|
| 744 | T[1..n,C[j]] = NV; |
---|
| 745 | } |
---|
| 746 | T[i,i] = T[i,i] - size(C) + 1; |
---|
| 747 | m = 0; |
---|
| 748 | for (j = 1; j <= size(C); j++) |
---|
| 749 | { |
---|
| 750 | m = m + T[n+1,C[j]]; |
---|
| 751 | } |
---|
| 752 | for (j = 1; j <= size(C); j++) |
---|
| 753 | { |
---|
| 754 | T[n+1,C[j]] = m; |
---|
| 755 | } |
---|
| 756 | return(growthAlg(T,i)); |
---|
| 757 | } |
---|
| 758 | else {ERROR("No Cycle found, something seems wrong! Please contact the authors.");} |
---|
| 759 | } |
---|
| 760 | |
---|
| 761 | m = 0; |
---|
| 762 | for (i = 1; i <= n; i++) |
---|
| 763 | { |
---|
| 764 | if (m < T[n+1,i]) |
---|
| 765 | { |
---|
| 766 | m = T[n+1,i]; |
---|
| 767 | } |
---|
| 768 | } |
---|
| 769 | return(m); |
---|
| 770 | } |
---|
| 771 | |
---|
| 772 | static proc GlDimSuffix(intvec v, intvec g) |
---|
| 773 | { |
---|
| 774 | //Computes the shortest r such that g is a suffix for vr |
---|
| 775 | //only valid for lex orderings? |
---|
| 776 | intvec r,gt,vt,lt,g2; |
---|
| 777 | int lg,lv,l,i,c,f; |
---|
| 778 | lg = size(g); lv = size(v); |
---|
| 779 | if (lg <= lv) |
---|
| 780 | { |
---|
| 781 | l = lv-lg; |
---|
| 782 | } |
---|
| 783 | else |
---|
| 784 | { |
---|
| 785 | l = 0; g2 = g[(lv+1)..lg]; |
---|
| 786 | g = g[1..lv]; lg = size(g); |
---|
| 787 | c = 1; |
---|
| 788 | } |
---|
| 789 | while (l < lv) |
---|
| 790 | { |
---|
| 791 | vt = v[(l+1)..lv]; |
---|
| 792 | gt = g[1..(lv-l)]; |
---|
| 793 | lt = size(gt); |
---|
| 794 | for (i = 1; i <= lt; i++) |
---|
| 795 | { |
---|
| 796 | if (vt[i]<>gt[i]) {l++; break;} |
---|
| 797 | } |
---|
| 798 | if (lt <=i ) { f = 1; break;} |
---|
| 799 | } |
---|
| 800 | if (f == 0) {return(g);} |
---|
| 801 | r = g[(lv-l+1)..lg]; |
---|
| 802 | if (c == 1) {r = r,g2;} |
---|
| 803 | return(r); |
---|
| 804 | } |
---|
| 805 | |
---|
| 806 | static proc isNormal(intvec V, list G) |
---|
| 807 | { |
---|
| 808 | int i,j,k,l; |
---|
| 809 | k = 0; |
---|
| 810 | for (i = 1; i <= size(G); i++) |
---|
| 811 | { |
---|
| 812 | if ( size(G[i]) <= size(V) ) |
---|
| 813 | { |
---|
| 814 | while ( size(G[i])+k <= size(V) ) |
---|
| 815 | { |
---|
| 816 | if ( G[i] == V[(1+k)..size(V)] ) {return(1);} |
---|
| 817 | } |
---|
| 818 | } |
---|
| 819 | } |
---|
| 820 | return(0); |
---|
| 821 | } |
---|
| 822 | |
---|
| 823 | static proc findDChain(list L) |
---|
| 824 | { |
---|
| 825 | list Li; int i,j; |
---|
| 826 | for (i = 1; i <= size(L); i++) {Li[i] = size(L[i]);} |
---|
| 827 | Li = sort(Li); Li = Li[1]; |
---|
| 828 | return(Li[size(Li)]); |
---|
| 829 | } |
---|
| 830 | |
---|
[5e2dd1] | 831 | static proc isInList(intvec V, list L) |
---|
| 832 | "USAGE: isInList(V,L); V an intvec, L a list of intvecs |
---|
[5e8ee4c] | 833 | RETURN: int |
---|
[5e2dd1] | 834 | PURPOSE:Finding the position of V in L, returns 0, if V is not in M |
---|
| 835 | " |
---|
| 836 | {int i,n; |
---|
[df9f881] | 837 | n = 0; |
---|
| 838 | for (i = 1; i <= size(L); i++) {if (L[i] == V) {n = i; break;}} |
---|
| 839 | return(n); |
---|
[5e2dd1] | 840 | } |
---|
| 841 | |
---|
| 842 | static proc isInMat(intvec V, intmat M) |
---|
| 843 | "USAGE: isInMat(V,M);V an intvec, M an intmat |
---|
[5e8ee4c] | 844 | RETURN: int |
---|
[5e2dd1] | 845 | PURPOSE:Finding the position of V in M, returns 0, if V is not in M |
---|
| 846 | " |
---|
| 847 | {if (size(V) <> ncols(M)) {return(0);} |
---|
[df9f881] | 848 | int i; |
---|
| 849 | intvec Vt; |
---|
| 850 | for (i = 1; i <= nrows(M); i++) |
---|
| 851 | {Vt = M[i,1..ncols(M)]; |
---|
| 852 | if ((V-Vt) == 0){return(i);} |
---|
| 853 | } |
---|
| 854 | return(0); |
---|
[5e2dd1] | 855 | } |
---|
| 856 | |
---|
| 857 | static proc isInVec(int v,intvec V) |
---|
| 858 | "USAGE: isInVec(v,V); v an integer,V an intvec |
---|
[5e8ee4c] | 859 | RETURN: int |
---|
[5e2dd1] | 860 | PURPOSE:Finding the position of v in V, returns 0, if v is not in V |
---|
| 861 | " |
---|
| 862 | {int i,n; |
---|
[df9f881] | 863 | n = 0; |
---|
| 864 | for (i = 1; i <= size(V); i++) {if (V[i] == v) {n = i; break;}} |
---|
| 865 | return(n); |
---|
[5e2dd1] | 866 | } |
---|
| 867 | |
---|
[f2b431] | 868 | |
---|
| 869 | static proc isPF(intvec P, intvec I) |
---|
| 870 | " |
---|
| 871 | PURPOSE: |
---|
| 872 | checks, if a word P is a proper praefix of another word I |
---|
| 873 | " |
---|
| 874 | { |
---|
| 875 | int n = size(P); |
---|
| 876 | if (n <= 0) {return(1);} |
---|
| 877 | if (size(I) < n) {return(0);} |
---|
| 878 | intvec IP = I[1..n]; |
---|
| 879 | if (IP == P) {return(1);} |
---|
| 880 | else {return(0);} |
---|
| 881 | } |
---|
| 882 | |
---|
| 883 | static proc isIF(intvec IF, intvec I) |
---|
| 884 | " |
---|
| 885 | PURPOSE: |
---|
| 886 | checks, if a word IF is an infix of another word I |
---|
| 887 | " |
---|
| 888 | { |
---|
| 889 | int n = size(IF); |
---|
| 890 | int m = size(I); |
---|
| 891 | |
---|
| 892 | if (n <= 0) { |
---|
| 893 | return(1); |
---|
| 894 | } |
---|
| 895 | if (m < n) { |
---|
| 896 | return(0); |
---|
| 897 | } |
---|
| 898 | |
---|
| 899 | for (int i = 0; (n + i) <= m; i++){ |
---|
| 900 | intvec IIF = I[(1 + i)..(n + i)]; |
---|
| 901 | if (IIF == IF) { |
---|
| 902 | return(1); |
---|
| 903 | } |
---|
| 904 | } |
---|
| 905 | return(0); |
---|
| 906 | } |
---|
| 907 | |
---|
[5e2dd1] | 908 | proc ivL2lpI(list L) |
---|
| 909 | "USAGE: ivL2lpI(L); L a list of intvecs |
---|
| 910 | RETURN: ideal |
---|
[f2b431] | 911 | PURPOSE:Transforming a list of intvecs into an ideal of Letterplace monomials |
---|
[5e2dd1] | 912 | ASSUME: - Intvec corresponds to a Letterplace monomial |
---|
| 913 | @* - basering has to be a Letterplace ring |
---|
| 914 | EXAMPLE: example ivL2lpI; shows examples |
---|
| 915 | " |
---|
| 916 | {checkAssumptions(0,L); |
---|
[df9f881] | 917 | int i; ideal G; |
---|
| 918 | poly p; |
---|
| 919 | for (i = 1; i <= size(L); i++) |
---|
| 920 | {p = iv2lp(L[i]); |
---|
| 921 | G[(size(G) + 1)] = p; |
---|
| 922 | } |
---|
| 923 | return(G); |
---|
[5e2dd1] | 924 | } |
---|
| 925 | example |
---|
| 926 | { |
---|
| 927 | "EXAMPLE:"; echo = 2; |
---|
| 928 | ring r = 0,(x,y,z),dp; |
---|
| 929 | def R = makeLetterplaceRing(5);// constructs a Letterplace ring |
---|
| 930 | setring R; //sets basering to Letterplace ring |
---|
| 931 | intvec u = 1,1,2; intvec v = 2,1,3; intvec w = 3,1,1; |
---|
| 932 | // u = x^2y, v = yxz, w = zx^2 in intvec representation |
---|
| 933 | list L = u,v,w; |
---|
| 934 | ivL2lpI(L);// invokes the procedure, returns the ideal containing u,v,w |
---|
| 935 | } |
---|
| 936 | |
---|
| 937 | proc iv2lp(intvec I) |
---|
| 938 | "USAGE: iv2lp(I); I an intvec |
---|
| 939 | RETURN: poly |
---|
| 940 | PURPOSE:Transforming an intvec into the corresponding Letterplace polynomial |
---|
| 941 | ASSUME: - Intvec corresponds to a Letterplace monomial |
---|
| 942 | @* - basering has to be a Letterplace ring |
---|
| 943 | NOTE: - Assumptions will not be checked! |
---|
| 944 | EXAMPLE: example iv2lp; shows examples |
---|
| 945 | " |
---|
| 946 | {if (I[1] == 0) {return(1);} |
---|
[df9f881] | 947 | int i = size(I); |
---|
| 948 | if (i > attrib(basering,"uptodeg")) {ERROR("polynomial exceeds degreebound");} |
---|
| 949 | int j; poly p = 1; |
---|
| 950 | for (j = 1; j <= i; j++) {if (I[j] > 0) { p = lpMult(p,var(I[j]));}} //ignore zeroes, because they correspond to 1 |
---|
| 951 | return(p); |
---|
[5e2dd1] | 952 | } |
---|
| 953 | example |
---|
| 954 | { |
---|
| 955 | "EXAMPLE:"; echo = 2; |
---|
| 956 | ring r = 0,(x,y,z),dp; |
---|
| 957 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 958 | setring R; //sets basering to Letterplace ring |
---|
| 959 | intvec u = 1,1,2; intvec v = 2,1,3; intvec w = 3,1,1; |
---|
| 960 | // u = x^2y, v = yxz, w = zx^2 in intvec representation |
---|
| 961 | iv2lp(u); // invokes the procedure and returns the corresponding poly |
---|
| 962 | iv2lp(v); |
---|
| 963 | iv2lp(w); |
---|
| 964 | } |
---|
| 965 | |
---|
| 966 | proc iv2lpList(list L) |
---|
| 967 | "USAGE: iv2lpList(L); L a list of intmats |
---|
| 968 | RETURN: ideal |
---|
| 969 | PURPOSE:Converting a list of intmats into an ideal of corresponding monomials |
---|
| 970 | ASSUME: - The rows of each intmat in L must correspond to a Letterplace monomial |
---|
| 971 | @* - basering has to be a Letterplace ring |
---|
| 972 | EXAMPLE: example iv2lpList; shows examples |
---|
| 973 | " |
---|
| 974 | {checkAssumptions(0,L); |
---|
[df9f881] | 975 | ideal G; |
---|
| 976 | int i; |
---|
| 977 | for (i = 1; i <= size(L); i++){G = G + iv2lpMat(L[i]);} |
---|
| 978 | return(G); |
---|
[5e2dd1] | 979 | } |
---|
| 980 | example |
---|
| 981 | { |
---|
| 982 | "EXAMPLE:"; echo = 2; |
---|
| 983 | ring r = 0,(x,y,z),dp; |
---|
| 984 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 985 | setring R; // sets basering to Letterplace ring |
---|
| 986 | intmat u[3][1] = 1,1,2; intmat v[1][3] = 2,1,3; intmat w[2][3] = 3,1,1,2,3,1; |
---|
| 987 | // defines intmats of different size containing intvec representations of |
---|
| 988 | // monomials as rows |
---|
| 989 | list L = u,v,w; |
---|
| 990 | print(u); print(v); print(w); // shows the intmats contained in L |
---|
| 991 | iv2lpList(L); // returns the corresponding monomials as an ideal |
---|
| 992 | } |
---|
| 993 | |
---|
| 994 | |
---|
| 995 | proc iv2lpMat(intmat M) |
---|
| 996 | "USAGE: iv2lpMat(M); M an intmat |
---|
| 997 | RETURN: ideal |
---|
[f2b431] | 998 | PURPOSE:Converting an intmat into an ideal of the corresponding monomials |
---|
[5e2dd1] | 999 | ASSUME: - The rows of M must correspond to Letterplace monomials |
---|
| 1000 | @* - basering has to be a Letterplace ring |
---|
| 1001 | EXAMPLE: example iv2lpMat; shows examples |
---|
| 1002 | " |
---|
| 1003 | {list L = M; |
---|
[df9f881] | 1004 | checkAssumptions(0,L); |
---|
| 1005 | kill L; |
---|
| 1006 | ideal G; poly p; |
---|
| 1007 | int i; intvec I; |
---|
| 1008 | for (i = 1; i <= nrows(M); i++) |
---|
[5e2dd1] | 1009 | { I = M[i,1..ncols(M)]; |
---|
| 1010 | p = iv2lp(I); |
---|
| 1011 | G[size(G)+1] = p; |
---|
| 1012 | } |
---|
[df9f881] | 1013 | return(G); |
---|
[5e2dd1] | 1014 | } |
---|
| 1015 | example |
---|
| 1016 | { |
---|
| 1017 | "EXAMPLE:"; echo = 2; |
---|
| 1018 | ring r = 0,(x,y,z),dp; |
---|
| 1019 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1020 | setring R; // sets basering to Letterplace ring |
---|
| 1021 | intmat u[3][1] = 1,1,2; intmat v[1][3] = 2,1,3; intmat w[2][3] = 3,1,1,2,3,1; |
---|
| 1022 | // defines intmats of different size containing intvec representations of |
---|
| 1023 | // monomials as rows |
---|
| 1024 | iv2lpMat(u); // returns the monomials contained in u |
---|
| 1025 | iv2lpMat(v); // returns the monomials contained in v |
---|
| 1026 | iv2lpMat(w); // returns the monomials contained in w |
---|
| 1027 | } |
---|
| 1028 | |
---|
| 1029 | proc lpId2ivLi(ideal G) |
---|
| 1030 | "USAGE: lpId2ivLi(G); G an ideal |
---|
| 1031 | RETURN: list |
---|
[f2b431] | 1032 | PURPOSE:Transforming an ideal into the corresponding list of intvecs |
---|
[5e2dd1] | 1033 | ASSUME: - basering has to be a Letterplace ring |
---|
| 1034 | EXAMPLE: example lpId2ivLi; shows examples |
---|
| 1035 | " |
---|
| 1036 | {int i,j,k; |
---|
[df9f881] | 1037 | list M; |
---|
| 1038 | checkAssumptions(0,M); |
---|
| 1039 | for (i = 1; i <= size(G); i++) {M[i] = lp2iv(G[i]);} |
---|
| 1040 | return(M); |
---|
[5e2dd1] | 1041 | } |
---|
| 1042 | example |
---|
| 1043 | { |
---|
| 1044 | "EXAMPLE:"; echo = 2; |
---|
[df9f881] | 1045 | ring r = 0,(x,y),dp; |
---|
| 1046 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1047 | setring R; // sets basering to Letterplace ring |
---|
| 1048 | ideal L = x(1)*x(2),y(1)*y(2),x(1)*y(2)*x(3); |
---|
| 1049 | lpId2ivLi(L); // returns the corresponding intvecs as a list |
---|
[5e2dd1] | 1050 | } |
---|
| 1051 | |
---|
| 1052 | proc lp2iv(poly p) |
---|
| 1053 | "USAGE: lp2iv(p); p a poly |
---|
| 1054 | RETURN: intvec |
---|
[f2b431] | 1055 | PURPOSE:Transforming a monomial into the corresponding intvec |
---|
[5e2dd1] | 1056 | ASSUME: - basering has to be a Letterplace ring |
---|
| 1057 | NOTE: - Assumptions will not be checked! |
---|
| 1058 | EXAMPLE: example lp2iv; shows examples |
---|
| 1059 | " |
---|
| 1060 | {p = normalize(lead(p)); |
---|
[df9f881] | 1061 | intvec I; |
---|
| 1062 | int i,j; |
---|
| 1063 | if (deg(p) > attrib(basering,"uptodeg")) {ERROR("Monomial exceeds degreebound");} |
---|
| 1064 | if (p == 1) {return(I);} |
---|
| 1065 | if (p == 0) {ERROR("Monomial is not allowed to equal zero");} |
---|
| 1066 | intvec lep = leadexp(p); |
---|
| 1067 | for ( i = 1; i <= attrib(basering,"lV"); i++) {if (lep[i] == 1) {I = i; break;}} |
---|
| 1068 | for (i = (attrib(basering,"lV")+1); i <= size(lep); i++) |
---|
| 1069 | {if (lep[i] == 1) |
---|
[5e2dd1] | 1070 | { j = (i mod attrib(basering,"lV")); |
---|
| 1071 | if (j == 0) {I = I,attrib(basering,"lV");} |
---|
| 1072 | else {I = I,j;} |
---|
| 1073 | } |
---|
| 1074 | else { if (lep[i] > 1) {ERROR("monomial has a not allowed multidegree");}} |
---|
[df9f881] | 1075 | } |
---|
| 1076 | if (I[1] == 0) {ERROR("monomial has a not allowed multidegree");} |
---|
[5e2dd1] | 1077 | |
---|
[df9f881] | 1078 | return(I); |
---|
[5e2dd1] | 1079 | } |
---|
| 1080 | example |
---|
| 1081 | { |
---|
| 1082 | "EXAMPLE:"; echo = 2; |
---|
| 1083 | ring r = 0,(x,y,z),dp; |
---|
| 1084 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1085 | setring R; // sets basering to Letterplace ring |
---|
| 1086 | poly p = x(1)*x(2)*z(3); poly q = y(1)*y(2)*x(3)*x(4); |
---|
| 1087 | poly w= z(1)*y(2)*x(3)*z(4)*z(5); |
---|
| 1088 | // p,q,w are some polynomials we want to transform into their |
---|
| 1089 | // intvec representation |
---|
| 1090 | lp2iv(p); lp2iv(q); lp2iv(w); |
---|
| 1091 | } |
---|
| 1092 | |
---|
| 1093 | proc lp2ivId(ideal G) |
---|
| 1094 | "USAGE: lp2ivId(G); G an ideal |
---|
| 1095 | RETURN: list |
---|
| 1096 | PURPOSE:Converting an ideal into an list of intmats, |
---|
[f2b431] | 1097 | @* the corresponding intvecs forming the rows |
---|
[5e2dd1] | 1098 | ASSUME: - basering has to be a Letterplace ring |
---|
| 1099 | EXAMPLE: example lp2ivId; shows examples |
---|
| 1100 | " |
---|
| 1101 | {G = normalize(lead(G)); |
---|
[df9f881] | 1102 | intvec I; list L; |
---|
| 1103 | checkAssumptions(0,L); |
---|
| 1104 | int i,md; |
---|
| 1105 | for (i = 1; i <= size(G); i++) { if (md <= deg(G[i])) {md = deg(G[i]);}} |
---|
| 1106 | while (size(G) > 0) |
---|
| 1107 | {ideal Gt; |
---|
| 1108 | for (i = 1; i <= ncols(G); i++) {if (md == deg(G[i])) {Gt = Gt + G[i]; G[i] = 0;}} |
---|
| 1109 | if (size(Gt) > 0) |
---|
| 1110 | {G = simplify(G,2); |
---|
| 1111 | intmat M [size(Gt)][md]; |
---|
| 1112 | for (i = 1; i <= size(Gt); i++) {M[i,1..md] = lp2iv(Gt[i]);} |
---|
| 1113 | L = insert(L,M); |
---|
| 1114 | kill M; kill Gt; |
---|
| 1115 | md = md - 1; |
---|
| 1116 | } |
---|
| 1117 | else {kill Gt; md = md - 1;} |
---|
[5e2dd1] | 1118 | } |
---|
[df9f881] | 1119 | return(L); |
---|
[5e2dd1] | 1120 | } |
---|
| 1121 | example |
---|
| 1122 | { |
---|
| 1123 | "EXAMPLE:"; echo = 2; |
---|
| 1124 | ring r = 0,(x,y,z),dp; |
---|
| 1125 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1126 | setring R; // sets basering to Letterplace ring |
---|
| 1127 | poly p = x(1)*x(2)*z(3); poly q = y(1)*y(2)*x(3)*x(4); |
---|
| 1128 | poly w = z(1)*y(2)*x(3)*z(4); |
---|
| 1129 | // p,q,w are some polynomials we want to transform into their |
---|
| 1130 | // intvec representation |
---|
| 1131 | ideal G = p,q,w; |
---|
| 1132 | // define the ideal containing p,q and w |
---|
| 1133 | lp2ivId(G); // and return the list of intmats for this ideal |
---|
| 1134 | } |
---|
| 1135 | |
---|
| 1136 | // -----------------main procedures---------------------- |
---|
| 1137 | |
---|
[f2b431] | 1138 | proc lpUfGkDim(ideal G) |
---|
| 1139 | { |
---|
[437cdbc] | 1140 | G = lead(G); |
---|
| 1141 | |
---|
| 1142 | int l = 0; |
---|
| 1143 | for (int i = 1; i <= size(G); i++) { |
---|
| 1144 | // find the max degree in G |
---|
| 1145 | int d = deg(G[i]); |
---|
| 1146 | if (d > l) { |
---|
| 1147 | l = d; |
---|
| 1148 | } |
---|
| 1149 | |
---|
| 1150 | // also if G is the whole ring return minus infinity |
---|
| 1151 | if (leadmonom(G[i]) == 1) { |
---|
| 1152 | return(-2); // minus infinity |
---|
| 1153 | } |
---|
| 1154 | } |
---|
| 1155 | // if longet word has length 1 we handle it as a special case |
---|
| 1156 | if (l == 1) { |
---|
| 1157 | int n = attrib(basering, "lV"); // variable count |
---|
| 1158 | int k = size(G); |
---|
| 1159 | if (k == n) { |
---|
| 1160 | return(0); |
---|
| 1161 | } |
---|
| 1162 | if (k == n-1) { |
---|
| 1163 | return(1); |
---|
| 1164 | } |
---|
| 1165 | if (k <= n-2) { |
---|
| 1166 | return(-1); |
---|
| 1167 | } |
---|
| 1168 | } |
---|
| 1169 | |
---|
| 1170 | // otherwise count cycles in the Ufnarovskij Graph |
---|
[f2b431] | 1171 | return(UfGraphGrowth(lpUfGraph(G))); |
---|
| 1172 | } |
---|
| 1173 | |
---|
| 1174 | proc UfGraphGrowth(intmat UG) |
---|
| 1175 | { |
---|
| 1176 | int n = ncols(UG); // number of vertices |
---|
| 1177 | // iterate through all vertices |
---|
| 1178 | |
---|
| 1179 | intvec visited; |
---|
| 1180 | visited[n] = 0; |
---|
| 1181 | |
---|
| 1182 | intvec cyclic; |
---|
| 1183 | cyclic[n] = 0; |
---|
| 1184 | |
---|
| 1185 | int maxCycleCount = 0; |
---|
| 1186 | for (int v = 1; v <= n; v++) { |
---|
| 1187 | int cycleCount = Cycles(UG, v, visited, cyclic, 0); |
---|
| 1188 | if (cycleCount == -1) { |
---|
| 1189 | return(-1); |
---|
| 1190 | } |
---|
| 1191 | if (cycleCount > maxCycleCount) { |
---|
| 1192 | maxCycleCount = cycleCount; |
---|
| 1193 | } |
---|
| 1194 | } |
---|
| 1195 | return(maxCycleCount); |
---|
| 1196 | } |
---|
| 1197 | |
---|
| 1198 | proc Cycles(intmat G, int v, intvec visited, intvec cyclic, intvec path) |
---|
| 1199 | { |
---|
| 1200 | // Mark the current vertex as visited |
---|
| 1201 | visited[v] = 1; |
---|
| 1202 | |
---|
| 1203 | // Store the current vertex in path |
---|
| 1204 | if (path[1] == 0) { |
---|
| 1205 | path[1] = v; |
---|
| 1206 | } else { |
---|
| 1207 | path[size(path) + 1] = v; |
---|
| 1208 | } |
---|
| 1209 | |
---|
| 1210 | int cycles = 0; |
---|
| 1211 | for (int w = 1; w <= ncols(G); w++) { |
---|
| 1212 | if (G[v,w] == 1) { |
---|
| 1213 | if (visited[w] == 1) { // neuer zykel gefunden |
---|
| 1214 | // 1. alle Knoten in path bis w ÃŒberprÃŒfen ob in cyclic |
---|
| 1215 | for (int j = size(path); j >= 1; j--) { |
---|
| 1216 | if(cyclic[path[j]] == 1) { |
---|
| 1217 | // 1.1 falls ja return -1 |
---|
| 1218 | return (-1); |
---|
| 1219 | } |
---|
| 1220 | if (path[j] == w) { |
---|
| 1221 | break; |
---|
| 1222 | } |
---|
| 1223 | } |
---|
| 1224 | |
---|
| 1225 | // 2. ansonsten cycles++ |
---|
| 1226 | for (int j = size(path); j >= 1; j--) { |
---|
| 1227 | // 2.2 Kanten in diesem Zykel entfernen; Knoten cyclic |
---|
| 1228 | if (j == size(path)) { // Sonderfall bei der ersten Iteration |
---|
| 1229 | cyclic[v] = 1; |
---|
| 1230 | G[v, w] = 0; |
---|
| 1231 | } else { |
---|
| 1232 | cyclic[path[j]] = 1; |
---|
| 1233 | G[path[j], path[j+1]] = 0; |
---|
| 1234 | } |
---|
| 1235 | if (path[j] == w) { |
---|
| 1236 | break; |
---|
| 1237 | } |
---|
| 1238 | } |
---|
| 1239 | |
---|
| 1240 | // 3. auf jedem dieser Knoten Cycles() aufrufen |
---|
| 1241 | int maxCycleCount = 0; |
---|
| 1242 | for (int j = size(path); j >= 1; j--) { |
---|
| 1243 | int cycleCount = Cycles(G, path[j], visited, cyclic, path); |
---|
| 1244 | if(cycleCount == -1) { |
---|
| 1245 | return (-1); |
---|
| 1246 | } |
---|
| 1247 | if (cycleCount > maxCycleCount) { |
---|
| 1248 | maxCycleCount = cycleCount; |
---|
| 1249 | } |
---|
| 1250 | if (path[j] == w) { |
---|
| 1251 | break; |
---|
| 1252 | } |
---|
| 1253 | } |
---|
| 1254 | if (maxCycleCount >= cycles) { |
---|
| 1255 | cycles = maxCycleCount + 1; |
---|
| 1256 | } |
---|
| 1257 | } else { |
---|
| 1258 | int cycleCount = Cycles(G, w, visited, cyclic, path); |
---|
| 1259 | if (cycleCount == -1) { |
---|
| 1260 | return(-1); |
---|
| 1261 | } |
---|
| 1262 | if (cycleCount > cycles) { |
---|
| 1263 | cycles = cycleCount; |
---|
| 1264 | } |
---|
| 1265 | } |
---|
| 1266 | } |
---|
| 1267 | } |
---|
| 1268 | //printf("Path: %s Cycles: %s", path, cycles); |
---|
| 1269 | return(cycles); |
---|
| 1270 | } |
---|
| 1271 | |
---|
| 1272 | proc lpUfGraph(ideal G) |
---|
| 1273 | "USUAGE: lpUfGraph(G); G a set of monomials in a letterplace ring |
---|
| 1274 | RETURN: intmat |
---|
| 1275 | PURPOSE: Constructs the Ufnarovskij graph induced by G |
---|
| 1276 | @* the adjacency matrix of the Ufnarovskij graph induced by G |
---|
| 1277 | ASSUME: - basering is a Letterplace ring |
---|
| 1278 | - G are the leading monomials of a Groebner basis |
---|
| 1279 | " |
---|
| 1280 | { |
---|
| 1281 | int l = 0; // length of longest word (monomial) in G |
---|
| 1282 | for (int i = 1; i <= size(G); i++) { // find the max degree in G |
---|
| 1283 | int d = deg(G[i]); |
---|
| 1284 | if (d > l) { |
---|
| 1285 | l = d; |
---|
| 1286 | } |
---|
| 1287 | } |
---|
| 1288 | ideal V = lpStandardWords(G, l - 1); // vertices |
---|
| 1289 | int n = size(V); |
---|
| 1290 | intmat UG[n][n]; // Ufnarovskij graph |
---|
| 1291 | for (int i = 1; i <= n; i++) { |
---|
| 1292 | for (int j = 1; j <= n; j++) { |
---|
| 1293 | // V[i] = v, V[j] = w [Studzinski page 76] |
---|
| 1294 | intvec v = lp2iv(V[i]); |
---|
| 1295 | intvec w = lp2iv(V[j]); |
---|
| 1296 | intvec v_overlap; |
---|
| 1297 | intvec w_overlap; |
---|
| 1298 | //TODO how should the graph look like when l - 1 = 0 ? |
---|
| 1299 | if (l - 1 > 1) { |
---|
| 1300 | v_overlap = v[2 .. l-1]; |
---|
| 1301 | w_overlap = w[1 .. l-2]; |
---|
| 1302 | } |
---|
| 1303 | intvec vw = v; |
---|
| 1304 | vw[l] = w[l-1]; |
---|
| 1305 | if (v_overlap == w_overlap && !divides(G, vw)) { |
---|
| 1306 | UG[i,j] = 1; |
---|
| 1307 | } |
---|
| 1308 | } |
---|
| 1309 | } |
---|
| 1310 | return (UG); |
---|
| 1311 | } |
---|
| 1312 | |
---|
| 1313 | proc lpStandardWords(ideal G, int length) { |
---|
| 1314 | if (length == 0) { |
---|
| 1315 | return (1); |
---|
| 1316 | } |
---|
| 1317 | int lV = attrib(basering, "lV"); // variable count |
---|
| 1318 | // recursion could cause problems |
---|
| 1319 | ideal prevWords = lpStandardWords(G, length - 1); |
---|
| 1320 | ideal words; |
---|
| 1321 | for (int i = 1; i <= lV; i++) { |
---|
| 1322 | for (int j = 1; j <= size(prevWords); j++) { |
---|
| 1323 | int placeShift = (length - 1) * lV; // letterplace |
---|
| 1324 | // multiply every previous word with every variable |
---|
| 1325 | poly word = prevWords[j] * var(i + placeShift); |
---|
| 1326 | // assumes that G is simplified! |
---|
| 1327 | if (!divides(G, lp2iv(word))) { |
---|
| 1328 | words = words, word; |
---|
| 1329 | } |
---|
| 1330 | } |
---|
| 1331 | } |
---|
| 1332 | words = simplify(words, 2); // remove zeroes |
---|
| 1333 | return (words); |
---|
| 1334 | } |
---|
| 1335 | |
---|
| 1336 | proc divides(ideal G, intvec iv) { |
---|
| 1337 | for (int k = 1; k <= size(G); k++) { |
---|
| 1338 | if (isIF(lp2iv(G[k]), iv)) { |
---|
| 1339 | return (1); |
---|
| 1340 | } else { |
---|
| 1341 | if (k == size(G)) { |
---|
| 1342 | return (0); |
---|
| 1343 | } |
---|
| 1344 | } |
---|
| 1345 | } |
---|
| 1346 | } |
---|
| 1347 | |
---|
| 1348 | proc lpGraphOfNormalWords(ideal G) |
---|
| 1349 | "USUAGE: lpGraphOfNormalWords(G); G a set of monomials in a letterplace ring |
---|
| 1350 | RETURN: intmat |
---|
| 1351 | PURPOSE: Constructs the graph of normal words induced by G |
---|
| 1352 | @* the adjacency matrix of the graph of normal words induced by G |
---|
| 1353 | ASSUME: - basering is a Letterplace ring |
---|
| 1354 | - G are the leading monomials of a Groebner basis |
---|
| 1355 | " |
---|
| 1356 | { |
---|
| 1357 | // construct the Graph of normal words [Studzinski page 78] |
---|
| 1358 | // construct set of vertices |
---|
| 1359 | int v = attrib(basering,"lV"); int d = attrib(basering,"uptodeg"); |
---|
| 1360 | ideal V; poly p,q,w; |
---|
| 1361 | ideal LG = lead(G); |
---|
| 1362 | int i,j,k,b; intvec E,Et; |
---|
| 1363 | for (i = 1; i <= v; i++){V = V, var(i);} |
---|
| 1364 | for (i = 1; i <= size(LG); i++) |
---|
| 1365 | { |
---|
| 1366 | E = leadexp(LG[i]); |
---|
| 1367 | if (E == intvec(0)) {V = V,monomial(intvec(0));} |
---|
| 1368 | else |
---|
| 1369 | { |
---|
| 1370 | for (j = 1; j < d; j++) |
---|
| 1371 | { |
---|
| 1372 | Et = E[(j*v+1)..(d*v)]; |
---|
| 1373 | if (Et == intvec(0)) {break;} |
---|
| 1374 | else {V = V, monomial(Et);} |
---|
| 1375 | } |
---|
| 1376 | } |
---|
| 1377 | } |
---|
| 1378 | V = simplify(V,2+4); |
---|
| 1379 | printf("V = %p", V); |
---|
| 1380 | |
---|
| 1381 | |
---|
| 1382 | // construct incidence matrix |
---|
| 1383 | |
---|
| 1384 | list LV = lpId2ivLi(V); |
---|
| 1385 | intvec Ip,Iw; |
---|
| 1386 | int n = size(V); |
---|
| 1387 | intmat T[n+1][n]; |
---|
| 1388 | for (i = 1; i <= n; i++) |
---|
| 1389 | { |
---|
| 1390 | // printf("for1 (i=%p, n=%p)", i, n); |
---|
| 1391 | p = V[i]; Ip = lp2iv(p); |
---|
| 1392 | for (j = 1; j <= n; j++) |
---|
| 1393 | { |
---|
| 1394 | // printf("for2 (j=%p, n=%p)", j, n); |
---|
| 1395 | k = 1; b = 1; |
---|
| 1396 | q = V[j]; |
---|
| 1397 | w = lpNF(lpMult(p,q),LG); |
---|
| 1398 | if (w <> 0) |
---|
| 1399 | { |
---|
| 1400 | Iw = lp2iv(w); |
---|
| 1401 | while (k <= n) |
---|
| 1402 | { |
---|
| 1403 | // printf("while (k=%p, n=%p)", k, n); |
---|
| 1404 | if (isPF(LV[k],Iw) > 0) |
---|
| 1405 | {if (isPF(LV[k],Ip) == 0) {b = 0; k = n+1;} else {k++;} |
---|
| 1406 | } |
---|
| 1407 | else {k++;} |
---|
| 1408 | } |
---|
| 1409 | T[i,j] = b; |
---|
| 1410 | // print("Incidence Matrix:"); |
---|
| 1411 | // print(T); |
---|
| 1412 | } |
---|
| 1413 | } |
---|
| 1414 | } |
---|
| 1415 | return(T); |
---|
| 1416 | } |
---|
| 1417 | |
---|
| 1418 | proc lpGkDim(ideal G) |
---|
| 1419 | "USUAGE: lpGkDim(G); G an ideal in a letterplace ring |
---|
| 1420 | RETURN: int |
---|
| 1421 | PURPOSE: Determines the Gelfand Kirillov dimension of A/<G> |
---|
| 1422 | @*: -1 means it is infinite |
---|
| 1423 | ASSUME: - basering is a Letterplace ring |
---|
| 1424 | - G is a Groebner basis |
---|
| 1425 | " |
---|
| 1426 | { |
---|
| 1427 | return(growthAlg(lpGraphOfNormalWords(G))); |
---|
| 1428 | } |
---|
| 1429 | example |
---|
| 1430 | { |
---|
| 1431 | "EXAMPLE:"; echo = 2; |
---|
| 1432 | ring r = 0,(x,y,z),dp; |
---|
| 1433 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1434 | R; |
---|
| 1435 | setring R; // sets basering to Letterplace ring |
---|
| 1436 | ideal I = z(1);//an example of infinite GK dimension |
---|
| 1437 | lpGkDim(I); |
---|
| 1438 | I = x(1),y(1),z(1); // gkDim = 0 |
---|
| 1439 | lpGkDim(I); |
---|
| 1440 | I = x(1)*y(2), x(1)*z(2), z(1)*y(2), z(1)*z(2);//gkDim = 2 |
---|
| 1441 | lpGkDim(I); |
---|
| 1442 | } |
---|
| 1443 | |
---|
| 1444 | |
---|
[5e2dd1] | 1445 | proc ivDHilbert(list L, int n, list #) |
---|
| 1446 | "USAGE: ivDHilbert(L,n[,degbound]); L a list of intmats, n an integer, |
---|
| 1447 | @* degbound an optional integer |
---|
| 1448 | RETURN: list |
---|
| 1449 | PURPOSE:Computing the K-dimension and the Hilbert series |
---|
| 1450 | ASSUME: - basering is a Letterplace ring |
---|
| 1451 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1452 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 1453 | @* degbound <= attrib(basering,uptodeg) holds |
---|
[5e2dd1] | 1454 | NOTE: - If L is the list returned, then L[1] is an integer corresponding to the |
---|
| 1455 | @* dimension, L[2] is an intvec which contains the coefficients of the |
---|
| 1456 | @* Hilbert series |
---|
| 1457 | @* - If degbound is set, there will be a degree bound added. By default there |
---|
| 1458 | @* is no degree bound |
---|
| 1459 | @* - n is the number of variables |
---|
| 1460 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th coefficient of |
---|
| 1461 | @* the Hilbert series. |
---|
| 1462 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1463 | EXAMPLE: example ivDHilbert; shows examples |
---|
| 1464 | " |
---|
| 1465 | {int degbound = 0; |
---|
[df9f881] | 1466 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
| 1467 | checkAssumptions(degbound,L); |
---|
| 1468 | intvec H; int i,dimen; |
---|
| 1469 | H = ivHilbert(L,n,degbound); |
---|
| 1470 | for (i = 1; i <= size(H); i++){dimen = dimen + H[i];} |
---|
| 1471 | L = dimen,H; |
---|
| 1472 | return(L); |
---|
[5e2dd1] | 1473 | } |
---|
| 1474 | example |
---|
| 1475 | { |
---|
| 1476 | "EXAMPLE:"; echo = 2; |
---|
| 1477 | ring r = 0,(x,y),dp; |
---|
| 1478 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
[df9f881] | 1479 | R; |
---|
[5e2dd1] | 1480 | setring R; // sets basering to Letterplace ring |
---|
| 1481 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1482 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1483 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1484 | print(I1); |
---|
| 1485 | print(I2); |
---|
| 1486 | print(J1); |
---|
| 1487 | print(J2); |
---|
| 1488 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
| 1489 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1490 | //the procedure without a degree bound |
---|
| 1491 | ivDHilbert(G,2); |
---|
| 1492 | // the procedure with degree bound 5 |
---|
| 1493 | ivDHilbert(I,2,5); |
---|
| 1494 | } |
---|
| 1495 | |
---|
| 1496 | proc ivDHilbertSickle(list L, int n, list #) |
---|
| 1497 | "USAGE: ivDHilbertSickle(L,n[,degbound]); L a list of intmats, n an integer, |
---|
| 1498 | @* degbound an optional integer |
---|
| 1499 | RETURN: list |
---|
| 1500 | PURPOSE:Computing K-dimension, Hilbert series and mistletoes |
---|
| 1501 | ASSUME: - basering is a Letterplace ring. |
---|
| 1502 | @* - All rows of each intmat correspond to a Letterplace monomial. |
---|
| 1503 | @* - If you specify a different degree bound degbound, |
---|
[f2b431] | 1504 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[df9f881] | 1505 | NOTE: - If L is the list returned, then L[1] is an integer, L[2] is an intvec |
---|
| 1506 | @* which contains the coefficients of the Hilbert series and L[3] |
---|
[5e2dd1] | 1507 | @* is a list, containing the mistletoes as intvecs. |
---|
| 1508 | @* - If degbound is set, a degree bound will be added. By default there |
---|
| 1509 | @* is no degree bound. |
---|
| 1510 | @* - n is the number of variables. |
---|
[df9f881] | 1511 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th |
---|
[5e2dd1] | 1512 | @* coefficient of the Hilbert series. |
---|
| 1513 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1514 | EXAMPLE: example ivDHilbertSickle; shows examples |
---|
| 1515 | " |
---|
| 1516 | {int degbound = 0; |
---|
[df9f881] | 1517 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
| 1518 | checkAssumptions(degbound,L); |
---|
| 1519 | int i,dimen; list R; |
---|
| 1520 | R = ivSickleHil(L,n,degbound); |
---|
| 1521 | for (i = 1; i <= size(R[1]); i++){dimen = dimen + R[1][i];} |
---|
| 1522 | R[3] = R[2]; R[2] = R[1]; R[1] = dimen; |
---|
| 1523 | return(R); |
---|
[5e2dd1] | 1524 | } |
---|
| 1525 | example |
---|
| 1526 | { |
---|
| 1527 | "EXAMPLE:"; echo = 2; |
---|
| 1528 | ring r = 0,(x,y),dp; |
---|
| 1529 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1530 | R; |
---|
| 1531 | setring R; // sets basering to Letterplace ring |
---|
| 1532 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1533 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1534 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1535 | print(I1); |
---|
| 1536 | print(I2); |
---|
| 1537 | print(J1); |
---|
| 1538 | print(J2); |
---|
| 1539 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
| 1540 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1541 | ivDHilbertSickle(G,2); // invokes the procedure without a degree bound |
---|
| 1542 | ivDHilbertSickle(I,2,3); // invokes the procedure with degree bound 3 |
---|
| 1543 | } |
---|
| 1544 | |
---|
| 1545 | proc ivDimCheck(list L, int n) |
---|
| 1546 | "USAGE: ivDimCheck(L,n); L a list of intmats, n an integer |
---|
| 1547 | RETURN: int, 0 if the dimension is finite, or 1 otherwise |
---|
| 1548 | PURPOSE:Decides, whether the K-dimension is finite or not |
---|
[f2b431] | 1549 | ASSUME: - basering is a Letterplace ring. |
---|
| 1550 | @* - All rows of each intmat correspond to a Letterplace monomial. |
---|
| 1551 | NOTE: - n is the number of variables. |
---|
[5e2dd1] | 1552 | EXAMPLE: example ivDimCheck; shows examples |
---|
| 1553 | " |
---|
| 1554 | {checkAssumptions(0,L); |
---|
[df9f881] | 1555 | int i,r; |
---|
| 1556 | intvec P,H; |
---|
| 1557 | for (i = 1; i <= size(L); i++) |
---|
| 1558 | {P[i] = ncols(L[i]); |
---|
| 1559 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
| 1560 | } |
---|
| 1561 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1562 | kill H; |
---|
| 1563 | intmat S; int sd,ld; intvec V; |
---|
| 1564 | sd = P[1]; ld = P[1]; |
---|
| 1565 | for (i = 2; i <= size(P); i++) |
---|
| 1566 | {if (P[i] < sd) {sd = P[i];} |
---|
| 1567 | if (P[i] > ld) {ld = P[i];} |
---|
| 1568 | } |
---|
| 1569 | sd = (sd - 1); ld = ld - 1; |
---|
| 1570 | if (ld == 0) { return(allVars(L,P,n));} |
---|
| 1571 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1572 | else {S = createStartMat(sd,n);} |
---|
| 1573 | module M; |
---|
| 1574 | for (i = 1; i <= nrows(S); i++) |
---|
| 1575 | {V = S[i,1..ncols(S)]; |
---|
| 1576 | if (findCycle(V,L,P,n,ld,M)) {r = 1; break;} |
---|
| 1577 | } |
---|
| 1578 | return(r); |
---|
[5e2dd1] | 1579 | } |
---|
| 1580 | example |
---|
| 1581 | { |
---|
| 1582 | "EXAMPLE:"; echo = 2; |
---|
| 1583 | ring r = 0,(x,y),dp; |
---|
| 1584 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1585 | R; |
---|
| 1586 | setring R; // sets basering to Letterplace ring |
---|
| 1587 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1588 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1589 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1590 | print(I1); |
---|
| 1591 | print(I2); |
---|
| 1592 | print(J1); |
---|
| 1593 | print(J2); |
---|
| 1594 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
| 1595 | list I = J1,J2; // ideal, which is already a Groebner basis and which |
---|
| 1596 | ivDimCheck(G,2); // invokes the procedure, factor is of finite K-dimension |
---|
| 1597 | ivDimCheck(I,2); // invokes the procedure, factor is not of finite K-dimension |
---|
| 1598 | } |
---|
| 1599 | |
---|
| 1600 | proc ivHilbert(list L, int n, list #) |
---|
| 1601 | "USAGE: ivHilbert(L,n[,degbound]); L a list of intmats, n an integer, |
---|
| 1602 | @* degbound an optional integer |
---|
[5e8ee4c] | 1603 | RETURN: intvec, containing the coefficients of the Hilbert series |
---|
[5e2dd1] | 1604 | PURPOSE:Computing the Hilbert series |
---|
| 1605 | ASSUME: - basering is a Letterplace ring. |
---|
| 1606 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1607 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 1608 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 1609 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
| 1610 | @* is no degree bound. |
---|
| 1611 | @* - n is the number of variables. |
---|
| 1612 | @* - If I is returned, then I[k] is the (k-1)-th coefficient of the Hilbert |
---|
| 1613 | @* series. |
---|
| 1614 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1615 | EXAMPLE: example ivHilbert; shows examples |
---|
| 1616 | " |
---|
| 1617 | {int degbound = 0; |
---|
[df9f881] | 1618 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
| 1619 | intvec P,H; int i; |
---|
| 1620 | for (i = 1; i <= size(L); i++) |
---|
| 1621 | {P[i] = ncols(L[i]); |
---|
| 1622 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
[5e2dd1] | 1623 | } |
---|
[df9f881] | 1624 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1625 | H[1] = 1; |
---|
| 1626 | checkAssumptions(degbound,L); |
---|
| 1627 | if (degbound == 0) |
---|
| 1628 | {int sd; |
---|
| 1629 | intmat S; |
---|
| 1630 | sd = P[1]; |
---|
| 1631 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1632 | sd = (sd - 1); |
---|
| 1633 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1634 | else {S = createStartMat(sd,n);} |
---|
| 1635 | if (intvec(S) == 0) {return(H);} |
---|
| 1636 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
| 1637 | for (i = 1; i <= nrows(S); i++) |
---|
| 1638 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1639 | H = findHCoeff(St,n,L,P,H); |
---|
| 1640 | kill St; |
---|
| 1641 | } |
---|
| 1642 | return(H); |
---|
| 1643 | } |
---|
| 1644 | else |
---|
| 1645 | {for (i = 1; i <= size(P); i++) |
---|
| 1646 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
| 1647 | int sd; |
---|
| 1648 | intmat S; |
---|
| 1649 | sd = P[1]; |
---|
| 1650 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1651 | sd = (sd - 1); |
---|
| 1652 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1653 | else {S = createStartMat(sd,n);} |
---|
| 1654 | if (intvec(S) == 0) {return(H);} |
---|
| 1655 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
| 1656 | for (i = 1; i <= nrows(S); i++) |
---|
| 1657 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1658 | H = findHCoeff(St,n,L,P,H,degbound); |
---|
| 1659 | kill St; |
---|
| 1660 | } |
---|
| 1661 | return(H); |
---|
[5e2dd1] | 1662 | } |
---|
| 1663 | } |
---|
| 1664 | example |
---|
| 1665 | { |
---|
| 1666 | "EXAMPLE:"; echo = 2; |
---|
| 1667 | ring r = 0,(x,y),dp; |
---|
| 1668 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1669 | R; |
---|
| 1670 | setring R; // sets basering to Letterplace ring |
---|
| 1671 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1672 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1673 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1674 | print(I1); |
---|
| 1675 | print(I2); |
---|
| 1676 | print(J1); |
---|
| 1677 | print(J2); |
---|
| 1678 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
| 1679 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1680 | ivHilbert(G,2); // invokes the procedure without any degree bound |
---|
| 1681 | ivHilbert(I,2,5); // invokes the procedure with degree bound 5 |
---|
| 1682 | } |
---|
| 1683 | |
---|
| 1684 | |
---|
| 1685 | proc ivKDim(list L, int n, list #) |
---|
| 1686 | "USAGE: ivKDim(L,n[,degbound]); L a list of intmats, |
---|
| 1687 | @* n an integer, degbound an optional integer |
---|
[5e8ee4c] | 1688 | RETURN: int, the K-dimension of A/<L> |
---|
[5e2dd1] | 1689 | PURPOSE:Computing the K-dimension of A/<L> |
---|
| 1690 | ASSUME: - basering is a Letterplace ring. |
---|
| 1691 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1692 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 1693 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 1694 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
| 1695 | @* is no degree bound. |
---|
| 1696 | @* - n is the number of variables. |
---|
| 1697 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1698 | EXAMPLE: example ivKDim; shows examples |
---|
| 1699 | " |
---|
| 1700 | {int degbound = 0; |
---|
[df9f881] | 1701 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
| 1702 | intvec P,H; int i; |
---|
| 1703 | for (i = 1; i <= size(L); i++) |
---|
| 1704 | {P[i] = ncols(L[i]); |
---|
| 1705 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
[5e2dd1] | 1706 | } |
---|
[df9f881] | 1707 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1708 | kill H; |
---|
| 1709 | checkAssumptions(degbound,L); |
---|
| 1710 | if (degbound == 0) |
---|
| 1711 | {int sd; int dimen = 1; |
---|
| 1712 | intmat S; |
---|
| 1713 | sd = P[1]; |
---|
| 1714 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1715 | sd = (sd - 1); |
---|
| 1716 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1717 | else {S = createStartMat(sd,n);} |
---|
| 1718 | if (intvec(S) == 0) {return(dimen);} |
---|
| 1719 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
| 1720 | for (i = 1; i <= nrows(S); i++) |
---|
| 1721 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1722 | dimen = dimen + findDimen(St,n,L,P); |
---|
| 1723 | kill St; |
---|
| 1724 | } |
---|
| 1725 | return(dimen); |
---|
| 1726 | } |
---|
| 1727 | else |
---|
| 1728 | {for (i = 1; i <= size(P); i++) |
---|
| 1729 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
| 1730 | int sd; int dimen = 1; |
---|
| 1731 | intmat S; |
---|
| 1732 | sd = P[1]; |
---|
| 1733 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1734 | sd = (sd - 1); |
---|
| 1735 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1736 | else {S = createStartMat(sd,n);} |
---|
| 1737 | if (intvec(S) == 0) {return(dimen);} |
---|
| 1738 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
| 1739 | for (i = 1; i <= nrows(S); i++) |
---|
| 1740 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1741 | dimen = dimen + findDimen(St,n,L,P, degbound); |
---|
| 1742 | kill St; |
---|
| 1743 | } |
---|
| 1744 | return(dimen); |
---|
[5e2dd1] | 1745 | } |
---|
| 1746 | } |
---|
| 1747 | example |
---|
| 1748 | { |
---|
| 1749 | "EXAMPLE:"; echo = 2; |
---|
| 1750 | ring r = 0,(x,y),dp; |
---|
| 1751 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1752 | R; |
---|
| 1753 | setring R; // sets basering to Letterplace ring |
---|
| 1754 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1755 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1756 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1757 | print(I1); |
---|
| 1758 | print(I2); |
---|
| 1759 | print(J1); |
---|
| 1760 | print(J2); |
---|
| 1761 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
| 1762 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1763 | ivKDim(G,2); // invokes the procedure without any degree bound |
---|
| 1764 | ivKDim(I,2,5); // invokes the procedure with degree bound 5 |
---|
| 1765 | } |
---|
| 1766 | |
---|
| 1767 | proc ivMis2Dim(list M) |
---|
| 1768 | "USAGE: ivMis2Dim(M); M a list of intvecs |
---|
[5e8ee4c] | 1769 | RETURN: int, the K-dimension of the given algebra |
---|
[5e2dd1] | 1770 | PURPOSE:Computing the K-dimension out of given mistletoes |
---|
| 1771 | ASSUME: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
| 1772 | @* Otherwise the returned value may differ from the K-dimension. |
---|
| 1773 | @* - basering is a Letterplace ring. |
---|
| 1774 | EXAMPLE: example ivMis2Dim; shows examples |
---|
| 1775 | " |
---|
| 1776 | {checkAssumptions(0,M); |
---|
[df9f881] | 1777 | intvec L; |
---|
| 1778 | if (size(M) == 0){ERROR("There are no mistletoes, so it appears your dimension is infinite!");} |
---|
| 1779 | if (isInList(L,M) > 0) {print("1 is a mistletoe, therefore dim = 1"); return(1);} |
---|
| 1780 | int i,j,d,s; |
---|
[f2b431] | 1781 | j = 1; |
---|
[df9f881] | 1782 | d = 1 + size(M[1]); |
---|
| 1783 | for (i = 1; i < size(M); i++) |
---|
[f2b431] | 1784 | {s = size(M[i]); if (s > size(M[i+1])){s = size(M[i+1]);} |
---|
| 1785 | while ((M[i][j] == M[i+1][j]) && (j <= s)){j = j + 1;} |
---|
| 1786 | d = d + size(M[i+1])- j + 1; |
---|
[5e2dd1] | 1787 | } |
---|
[df9f881] | 1788 | return(d); |
---|
[5e2dd1] | 1789 | } |
---|
| 1790 | example |
---|
| 1791 | { |
---|
| 1792 | "EXAMPLE:"; echo = 2; |
---|
[df9f881] | 1793 | ring r = 0,(x,y),dp; |
---|
| 1794 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1795 | R; |
---|
| 1796 | setring R; // sets basering to Letterplace ring |
---|
| 1797 | intvec i1 = 1,2; intvec i2 = 2,1,2; |
---|
| 1798 | // the mistletoes are xy and yxy, which are already ordered lexicographically |
---|
| 1799 | list L = i1,i2; |
---|
| 1800 | ivMis2Dim(L); // returns the dimension of the factor algebra |
---|
[5e2dd1] | 1801 | } |
---|
| 1802 | |
---|
| 1803 | proc ivOrdMisLex(list M) |
---|
| 1804 | "USAGE: ivOrdMisLex(M); M a list of intvecs |
---|
[5e8ee4c] | 1805 | RETURN: list, containing the ordered intvecs of M |
---|
[5e2dd1] | 1806 | PURPOSE:Orders a given set of mistletoes lexicographically |
---|
| 1807 | ASSUME: - basering is a Letterplace ring. |
---|
[f2b431] | 1808 | - intvecs correspond to monomials |
---|
[5e2dd1] | 1809 | NOTE: - This is preprocessing, it's not needed if the mistletoes are returned |
---|
| 1810 | @* from the sickle algorithm. |
---|
| 1811 | @* - Each entry of the list returned is an intvec. |
---|
| 1812 | EXAMPLE: example ivOrdMisLex; shows examples |
---|
| 1813 | " |
---|
| 1814 | {checkAssumptions(0,M); |
---|
[df9f881] | 1815 | return(sort(M)[1]); |
---|
[5e2dd1] | 1816 | } |
---|
| 1817 | example |
---|
| 1818 | { |
---|
| 1819 | "EXAMPLE:"; echo = 2; |
---|
[df9f881] | 1820 | ring r = 0,(x,y),dp; |
---|
| 1821 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1822 | setring R; // sets basering to Letterplace ring |
---|
| 1823 | intvec i1 = 1,2,1; intvec i2 = 2,2,1; intvec i3 = 1,1; intvec i4 = 2,1,1,1; |
---|
| 1824 | // the corresponding monomials are xyx,y^2x,x^2,yx^3 |
---|
| 1825 | list M = i1,i2,i3,i4; |
---|
| 1826 | M; |
---|
| 1827 | ivOrdMisLex(M);// orders the list of monomials |
---|
[5e2dd1] | 1828 | } |
---|
| 1829 | |
---|
| 1830 | proc ivSickle(list L, int n, list #) |
---|
| 1831 | "USAGE: ivSickle(L,n,[degbound]); L a list of intmats, n an int, degbound an |
---|
| 1832 | @* optional integer |
---|
[5e8ee4c] | 1833 | RETURN: list, containing intvecs, the mistletoes of A/<L> |
---|
[f2b431] | 1834 | PURPOSE:Computing the mistletoes for a given Groebner basis L |
---|
[5e2dd1] | 1835 | ASSUME: - basering is a Letterplace ring. |
---|
| 1836 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1837 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 1838 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 1839 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
| 1840 | @* is no degree bound. |
---|
| 1841 | @* - n is the number of variables. |
---|
| 1842 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1843 | EXAMPLE: example ivSickle; shows examples |
---|
| 1844 | " |
---|
| 1845 | {list M; |
---|
[df9f881] | 1846 | int degbound = 0; |
---|
| 1847 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
| 1848 | int i; |
---|
| 1849 | intvec P,H; |
---|
| 1850 | for (i = 1; i <= size(L); i++) |
---|
| 1851 | {P[i] = ncols(L[i]); |
---|
| 1852 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
[5e2dd1] | 1853 | } |
---|
[df9f881] | 1854 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1855 | kill H; |
---|
| 1856 | checkAssumptions(degbound,L); |
---|
| 1857 | if (degbound == 0) |
---|
| 1858 | {intmat S; int sd; |
---|
| 1859 | sd = P[1]; |
---|
| 1860 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1861 | sd = (sd - 1); |
---|
| 1862 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1863 | else {S = createStartMat(sd,n);} |
---|
| 1864 | if (intvec(S) == 0) {return(list (intvec(0)));} |
---|
| 1865 | for (i = 1; i <= nrows(S); i++) |
---|
| 1866 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1867 | M = M + findmistletoes(St,n,L,P); |
---|
| 1868 | kill St; |
---|
| 1869 | } |
---|
| 1870 | return(M); |
---|
| 1871 | } |
---|
| 1872 | else |
---|
| 1873 | {for (i = 1; i <= size(P); i++) |
---|
| 1874 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
| 1875 | intmat S; int sd; |
---|
| 1876 | sd = P[1]; |
---|
| 1877 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1878 | sd = (sd - 1); |
---|
| 1879 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1880 | else {S = createStartMat(sd,n);} |
---|
| 1881 | if (intvec(S) == 0) {return(list (intvec(0)));} |
---|
| 1882 | for (i = 1; i <= nrows(S); i++) |
---|
| 1883 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1884 | M = M + findmistletoes(St,n,L,P,degbound); |
---|
| 1885 | kill St; |
---|
| 1886 | } |
---|
| 1887 | return(M); |
---|
[5e2dd1] | 1888 | } |
---|
| 1889 | } |
---|
| 1890 | example |
---|
| 1891 | { |
---|
| 1892 | "EXAMPLE:"; echo = 2; |
---|
| 1893 | ring r = 0,(x,y),dp; |
---|
| 1894 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1895 | setring R; // sets basering to Letterplace ring |
---|
| 1896 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1897 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1898 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1899 | print(I1); |
---|
| 1900 | print(I2); |
---|
| 1901 | print(J1); |
---|
| 1902 | print(J2); |
---|
| 1903 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
| 1904 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1905 | ivSickle(G,2); // invokes the procedure without any degree bound |
---|
| 1906 | ivSickle(I,2,5); // invokes the procedure with degree bound 5 |
---|
| 1907 | } |
---|
| 1908 | |
---|
| 1909 | proc ivSickleDim(list L, int n, list #) |
---|
| 1910 | "USAGE: ivSickleDim(L,n[,degbound]); L a list of intmats, n an integer, degbound |
---|
| 1911 | @* an optional integer |
---|
[5e8ee4c] | 1912 | RETURN: list |
---|
[5e2dd1] | 1913 | PURPOSE:Computing mistletoes and the K-dimension |
---|
| 1914 | ASSUME: - basering is a Letterplace ring. |
---|
| 1915 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1916 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 1917 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 1918 | NOTE: - If L is the list returned, then L[1] is an integer, L[2] is a list, |
---|
| 1919 | @* containing the mistletoes as intvecs. |
---|
| 1920 | @* - If degbound is set, a degree bound will be added. By default there |
---|
| 1921 | @* is no degree bound. |
---|
| 1922 | @* - n is the number of variables. |
---|
| 1923 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1924 | EXAMPLE: example ivSickleDim; shows examples |
---|
| 1925 | " |
---|
| 1926 | {list M; |
---|
[df9f881] | 1927 | int degbound = 0; |
---|
| 1928 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
| 1929 | int i,dimen; list R; |
---|
| 1930 | intvec P,H; |
---|
| 1931 | for (i = 1; i <= size(L); i++) |
---|
| 1932 | {P[i] = ncols(L[i]); |
---|
| 1933 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial, dimension equals zero");}} |
---|
[5e2dd1] | 1934 | } |
---|
[df9f881] | 1935 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1936 | kill H; |
---|
| 1937 | checkAssumptions(degbound,L); |
---|
| 1938 | if (degbound == 0) |
---|
| 1939 | {int sd; dimen = 1; |
---|
| 1940 | intmat S; |
---|
| 1941 | sd = P[1]; |
---|
| 1942 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1943 | sd = (sd - 1); |
---|
| 1944 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1945 | else {S = createStartMat(sd,n);} |
---|
| 1946 | if (intvec(S) == 0) {return(list(dimen,list(intvec(0))));} |
---|
| 1947 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
| 1948 | R[1] = dimen; |
---|
| 1949 | for (i = 1; i <= nrows(S); i++) |
---|
| 1950 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1951 | R = findMisDim(St,n,L,P,R); |
---|
| 1952 | kill St; |
---|
| 1953 | } |
---|
| 1954 | return(R); |
---|
| 1955 | } |
---|
| 1956 | else |
---|
| 1957 | {for (i = 1; i <= size(P); i++) |
---|
| 1958 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
| 1959 | int sd; dimen = 1; |
---|
| 1960 | intmat S; |
---|
| 1961 | sd = P[1]; |
---|
| 1962 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1963 | sd = (sd - 1); |
---|
| 1964 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1965 | else {S = createStartMat(sd,n);} |
---|
| 1966 | if (intvec(S) == 0) {return(list(dimen,list(intvec(0))));} |
---|
| 1967 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
| 1968 | R[1] = dimen; |
---|
| 1969 | for (i = 1; i <= nrows(S); i++) |
---|
| 1970 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1971 | R = findMisDim(St,n,L,P,R,degbound); |
---|
| 1972 | kill St; |
---|
| 1973 | } |
---|
| 1974 | return(R); |
---|
[5e2dd1] | 1975 | } |
---|
| 1976 | } |
---|
| 1977 | example |
---|
| 1978 | { |
---|
| 1979 | "EXAMPLE:"; echo = 2; |
---|
| 1980 | ring r = 0,(x,y),dp; |
---|
| 1981 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1982 | setring R; // sets basering to Letterplace ring |
---|
| 1983 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1984 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1985 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1986 | print(I1); |
---|
| 1987 | print(I2); |
---|
| 1988 | print(J1); |
---|
| 1989 | print(J2); |
---|
| 1990 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
| 1991 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1992 | ivSickleDim(G,2); // invokes the procedure without any degree bound |
---|
| 1993 | ivSickleDim(I,2,5); // invokes the procedure with degree bound 5 |
---|
| 1994 | } |
---|
| 1995 | |
---|
| 1996 | proc ivSickleHil(list L, int n, list #) |
---|
| 1997 | "USAGE:ivSickleHil(L,n[,degbound]); L a list of intmats, n an integer, |
---|
| 1998 | @* degbound an optional integer |
---|
[5e8ee4c] | 1999 | RETURN: list |
---|
[5e2dd1] | 2000 | PURPOSE:Computing the mistletoes and the Hilbert series |
---|
| 2001 | ASSUME: - basering is a Letterplace ring. |
---|
| 2002 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 2003 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 2004 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 2005 | NOTE: - If L is the list returned, then L[1] is an intvec, L[2] is a list, |
---|
| 2006 | @* containing the mistletoes as intvecs. |
---|
| 2007 | @* - If degbound is set, a degree bound will be added. By default there |
---|
| 2008 | @* is no degree bound. |
---|
| 2009 | @* - n is the number of variables. |
---|
| 2010 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
| 2011 | @* coefficient of the Hilbert series. |
---|
| 2012 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2013 | EXAMPLE: example ivSickleHil; shows examples |
---|
| 2014 | " |
---|
| 2015 | {int degbound = 0; |
---|
[df9f881] | 2016 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
| 2017 | intvec P,H; int i; list R; |
---|
| 2018 | for (i = 1; i <= size(L); i++) |
---|
| 2019 | {P[i] = ncols(L[i]); |
---|
| 2020 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
[5e2dd1] | 2021 | } |
---|
[df9f881] | 2022 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 2023 | H[1] = 1; |
---|
| 2024 | checkAssumptions(degbound,L); |
---|
| 2025 | if (degbound == 0) |
---|
| 2026 | {int sd; |
---|
| 2027 | intmat S; |
---|
| 2028 | sd = P[1]; |
---|
| 2029 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 2030 | sd = (sd - 1); |
---|
| 2031 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 2032 | else {S = createStartMat(sd,n);} |
---|
| 2033 | if (intvec(S) == 0) {return(list(H,list(intvec (0))));} |
---|
| 2034 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
| 2035 | R[1] = H; kill H; |
---|
| 2036 | for (i = 1; i <= nrows(S); i++) |
---|
| 2037 | {intvec St = S[i,1..ncols(S)]; |
---|
| 2038 | R = findHCoeffMis(St,n,L,P,R); |
---|
| 2039 | kill St; |
---|
| 2040 | } |
---|
| 2041 | return(R); |
---|
| 2042 | } |
---|
| 2043 | else |
---|
| 2044 | {for (i = 1; i <= size(P); i++) |
---|
| 2045 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
| 2046 | int sd; |
---|
| 2047 | intmat S; |
---|
| 2048 | sd = P[1]; |
---|
| 2049 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 2050 | sd = (sd - 1); |
---|
| 2051 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 2052 | else {S = createStartMat(sd,n);} |
---|
| 2053 | if (intvec(S) == 0) {return(list(H,list(intvec(0))));} |
---|
| 2054 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
| 2055 | R[1] = H; kill H; |
---|
| 2056 | for (i = 1; i <= nrows(S); i++) |
---|
| 2057 | {intvec St = S[i,1..ncols(S)]; |
---|
| 2058 | R = findHCoeffMis(St,n,L,P,R,degbound); |
---|
| 2059 | kill St; |
---|
| 2060 | } |
---|
| 2061 | return(R); |
---|
[5e2dd1] | 2062 | } |
---|
| 2063 | } |
---|
| 2064 | example |
---|
| 2065 | { |
---|
| 2066 | "EXAMPLE:"; echo = 2; |
---|
| 2067 | ring r = 0,(x,y),dp; |
---|
| 2068 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2069 | setring R; // sets basering to Letterplace ring |
---|
| 2070 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 2071 | intmat I1[2][2] = 1,1,2,2; intmat I2[1][3] = 1,2,1; |
---|
| 2072 | intmat J1[1][2] = 1,1; intmat J2[2][3] = 2,1,2,1,2,1; |
---|
| 2073 | print(I1); |
---|
| 2074 | print(I2); |
---|
| 2075 | print(J1); |
---|
| 2076 | print(J2); |
---|
| 2077 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
| 2078 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 2079 | ivSickleHil(G,2); // invokes the procedure without any degree bound |
---|
| 2080 | ivSickleHil(I,2,5); // invokes the procedure with degree bound 5 |
---|
| 2081 | } |
---|
| 2082 | |
---|
| 2083 | proc lpDHilbert(ideal G, list #) |
---|
| 2084 | "USAGE: lpDHilbert(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
[5e8ee4c] | 2085 | RETURN: list |
---|
[5e2dd1] | 2086 | PURPOSE:Computing K-dimension and Hilbert series, starting with a lp-ideal |
---|
[f2b431] | 2087 | ASSUME: - basering is a Letterplace ring. |
---|
[5e2dd1] | 2088 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 2089 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e8ee4c] | 2090 | NOTE: - If L is the list returned, then L[1] is an integer corresponding to the |
---|
| 2091 | @* dimension, L[2] is an intvec which contains the coefficients of the |
---|
| 2092 | @* Hilbert series |
---|
[5e2dd1] | 2093 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2094 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2095 | @* - n can be set to a different number of variables. |
---|
| 2096 | @* Default: n = attrib(basering, lV). |
---|
| 2097 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th |
---|
| 2098 | @* coefficient of the Hilbert series. |
---|
| 2099 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2100 | EXAMPLE: example lpDHilbert; shows examples |
---|
| 2101 | " |
---|
| 2102 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
[df9f881] | 2103 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 2104 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 2105 | list L; |
---|
| 2106 | L = lp2ivId(normalize(lead(G))); |
---|
| 2107 | return(ivDHilbert(L,n,degbound)); |
---|
[5e2dd1] | 2108 | } |
---|
| 2109 | example |
---|
| 2110 | { |
---|
| 2111 | "EXAMPLE:"; echo = 2; |
---|
| 2112 | ring r = 0,(x,y),dp; |
---|
| 2113 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2114 | setring R; // sets basering to Letterplace ring |
---|
| 2115 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
| 2116 | //Groebner basis |
---|
| 2117 | lpDHilbert(G,5,2); // invokes procedure with degree bound 5 and 2 variables |
---|
| 2118 | // note that the optional parameters are not necessary, due to the finiteness |
---|
| 2119 | // of the K-dimension of the factor algebra |
---|
| 2120 | lpDHilbert(G); // procedure with ring parameters |
---|
| 2121 | lpDHilbert(G,0); // procedure without degreebound |
---|
| 2122 | } |
---|
| 2123 | |
---|
| 2124 | proc lpDHilbertSickle(ideal G, list #) |
---|
| 2125 | "USAGE: lpDHilbertSickle(G[,degbound,n]); G an ideal, degbound, n optional |
---|
| 2126 | @* integers |
---|
[5e8ee4c] | 2127 | RETURN: list |
---|
[5e2dd1] | 2128 | PURPOSE:Computing K-dimension, Hilbert series and mistletoes at once |
---|
[f2b431] | 2129 | ASSUME: - basering is a Letterplace ring. |
---|
[5e2dd1] | 2130 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 2131 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[df9f881] | 2132 | NOTE: - If L is the list returned, then L[1] is an integer, the K-dimension, |
---|
[f7d745] | 2133 | @* L[2] is an intvec, the Hilbert series and L[3] is an ideal, |
---|
[df9f881] | 2134 | @* the mistletoes |
---|
[5e2dd1] | 2135 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2136 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2137 | @* - n can be set to a different number of variables. |
---|
| 2138 | @* Default: n = attrib(basering, lV). |
---|
| 2139 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
| 2140 | @* coefficient of the Hilbert series. |
---|
| 2141 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2142 | EXAMPLE: example lpDHilbertSickle; shows examples |
---|
| 2143 | " |
---|
| 2144 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
[df9f881] | 2145 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 2146 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 2147 | list L; |
---|
| 2148 | L = lp2ivId(normalize(lead(G))); |
---|
| 2149 | L = ivDHilbertSickle(L,n,degbound); |
---|
| 2150 | L[3] = ivL2lpI(L[3]); |
---|
| 2151 | return(L); |
---|
[5e2dd1] | 2152 | } |
---|
| 2153 | example |
---|
| 2154 | { |
---|
| 2155 | "EXAMPLE:"; echo = 2; |
---|
| 2156 | ring r = 0,(x,y),dp; |
---|
| 2157 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2158 | setring R; // sets basering to Letterplace ring |
---|
| 2159 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
| 2160 | //Groebner basis |
---|
| 2161 | lpDHilbertSickle(G,5,2); //invokes procedure with degree bound 5 and 2 variables |
---|
| 2162 | // note that the optional parameters are not necessary, due to the finiteness |
---|
| 2163 | // of the K-dimension of the factor algebra |
---|
[df9f881] | 2164 | lpDHilbertSickle(G); // procedure with ring parameters |
---|
| 2165 | lpDHilbertSickle(G,0); // procedure without degreebound |
---|
[5e2dd1] | 2166 | } |
---|
| 2167 | |
---|
[f2b431] | 2168 | proc lpGlDimBound (ideal I) |
---|
| 2169 | "USAGE: lpGlDimBound(I); I an ideal |
---|
| 2170 | RETURN: int, an upper bound for the global dimension, -1 means infinity |
---|
| 2171 | PURPOSE: computing an upper bound for the global dimension |
---|
| 2172 | ASSUME: - basering is a Letterplace ring. |
---|
| 2173 | NOTE: -1 is also returned when degree bound is reached |
---|
| 2174 | EXAMPLE: example lpGlDimBound; shows example |
---|
| 2175 | " |
---|
| 2176 | { |
---|
| 2177 | ideal G = lead(I); |
---|
| 2178 | list L = lpId2ivLi(G); list RL; |
---|
| 2179 | int n = attrib(basering,"lV"); |
---|
| 2180 | int d = attrib(basering,"uptodeg"); |
---|
| 2181 | int i,j,r; list V; intvec g,v,s,s1,vs; |
---|
| 2182 | for (i = 1; i <= n; i++) {L[i] = i;} |
---|
| 2183 | for (j = 1; j <= size(L); j++) |
---|
| 2184 | { |
---|
| 2185 | i=1; |
---|
| 2186 | while (i <= size(V)) |
---|
| 2187 | { |
---|
| 2188 | v = V[i], g = L[i]; s = GLDimSuffix(v,g); |
---|
| 2189 | if (size(s)>1) {s1 = s[1..(size(s)-1)];} |
---|
| 2190 | else {s1 = s[1];} |
---|
| 2191 | vs = v,s1; |
---|
| 2192 | if (isNormal(vs,L)) |
---|
| 2193 | { |
---|
| 2194 | if (ContainedIn(s1,V)==0) |
---|
| 2195 | { |
---|
| 2196 | V = insert(V,s1,size(V)); |
---|
| 2197 | } |
---|
| 2198 | } |
---|
| 2199 | } |
---|
| 2200 | } |
---|
| 2201 | r=findDChain(L); |
---|
| 2202 | if (r == d) {return(-1);} |
---|
| 2203 | else { return(r);} |
---|
| 2204 | } |
---|
| 2205 | example |
---|
| 2206 | { |
---|
| 2207 | "EXAMPLE:"; echo = 2; |
---|
| 2208 | ring r = 0,(x,y),dp; |
---|
| 2209 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2210 | setring R; // sets basering to Letterplace ring |
---|
| 2211 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
| 2212 | //Groebner basis |
---|
| 2213 | lpGlDimBound(G); // invokes procedure with Groebner basis G |
---|
| 2214 | } |
---|
| 2215 | |
---|
[5e2dd1] | 2216 | proc lpHilbert(ideal G, list #) |
---|
| 2217 | "USAGE: lpHilbert(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
[5e8ee4c] | 2218 | RETURN: intvec, containing the coefficients of the Hilbert series |
---|
[5e2dd1] | 2219 | PURPOSE:Computing the Hilbert series |
---|
[f2b431] | 2220 | ASSUME: - basering is a Letterplace ring. |
---|
[5e2dd1] | 2221 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 2222 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 2223 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2224 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2225 | @* - n is the number of variables, which can be set to a different number. |
---|
| 2226 | @* Default: attrib(basering, lV). |
---|
| 2227 | @* - If I is returned, then I[k] is the (k-1)-th coefficient of the Hilbert |
---|
| 2228 | @* series. |
---|
| 2229 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2230 | EXAMPLE: example lpHilbert; shows examples |
---|
| 2231 | " |
---|
| 2232 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
[df9f881] | 2233 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 2234 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 2235 | list L; |
---|
| 2236 | L = lp2ivId(normalize(lead(G))); |
---|
| 2237 | return(ivHilbert(L,n,degbound)); |
---|
[5e2dd1] | 2238 | } |
---|
| 2239 | example |
---|
| 2240 | { |
---|
| 2241 | "EXAMPLE:"; echo = 2; |
---|
| 2242 | ring r = 0,(x,y),dp; |
---|
| 2243 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2244 | setring R; // sets basering to Letterplace ring |
---|
| 2245 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
| 2246 | //Groebner basis |
---|
| 2247 | lpHilbert(G,5,2); // invokes procedure with degree bound 5 and 2 variables |
---|
| 2248 | // note that the optional parameters are not necessary, due to the finiteness |
---|
| 2249 | // of the K-dimension of the factor algebra |
---|
| 2250 | lpDHilbert(G); // procedure with ring parameters |
---|
| 2251 | lpDHilbert(G,0); // procedure without degreebound |
---|
| 2252 | } |
---|
| 2253 | |
---|
| 2254 | proc lpDimCheck(ideal G) |
---|
| 2255 | "USAGE: lpDimCheck(G); |
---|
[5e8ee4c] | 2256 | RETURN: int, 1 if K-dimension of the factor algebra is infinite, 0 otherwise |
---|
[5e2dd1] | 2257 | PURPOSE:Checking a factor algebra for finiteness of the K-dimension |
---|
[f2b431] | 2258 | ASSUME: - basering is a Letterplace ring. |
---|
[5e2dd1] | 2259 | EXAMPLE: example lpDimCheck; shows examples |
---|
| 2260 | " |
---|
| 2261 | {int n = attrib(basering,"lV"); |
---|
[df9f881] | 2262 | list L; |
---|
| 2263 | ideal R; |
---|
| 2264 | R = normalize(lead(G)); |
---|
| 2265 | L = lp2ivId(R); |
---|
| 2266 | return(ivDimCheck(L,n)); |
---|
[5e2dd1] | 2267 | } |
---|
| 2268 | example |
---|
| 2269 | { |
---|
| 2270 | "EXAMPLE:"; echo = 2; |
---|
| 2271 | ring r = 0,(x,y),dp; |
---|
| 2272 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2273 | setring R; // sets basering to Letterplace ring |
---|
| 2274 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); |
---|
| 2275 | // Groebner basis |
---|
| 2276 | ideal I = x(1)*x(2), y(1)*x(2)*y(3), x(1)*y(2)*x(3); |
---|
| 2277 | // Groebner basis |
---|
| 2278 | lpDimCheck(G); // invokes procedure, factor algebra is of finite K-dimension |
---|
| 2279 | lpDimCheck(I); // invokes procedure, factor algebra is of infinite Kdimension |
---|
| 2280 | } |
---|
| 2281 | |
---|
| 2282 | proc lpKDim(ideal G, list #) |
---|
| 2283 | "USAGE: lpKDim(G[,degbound, n]); G an ideal, degbound, n optional integers |
---|
[5e8ee4c] | 2284 | RETURN: int, the K-dimension of the factor algebra |
---|
[5e2dd1] | 2285 | PURPOSE:Computing the K-dimension of a factor algebra, given via an ideal |
---|
[f2b431] | 2286 | ASSUME: - basering is a Letterplace ring |
---|
[5e2dd1] | 2287 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 2288 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 2289 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2290 | @* degree bound. Default: attrib(basering, uptodeg). |
---|
| 2291 | @* - n is the number of variables, which can be set to a different number. |
---|
| 2292 | @* Default: attrib(basering, lV). |
---|
| 2293 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2294 | EXAMPLE: example lpKDim; shows examples |
---|
| 2295 | " |
---|
| 2296 | {int degbound = attrib(basering, "uptodeg");int n = attrib(basering, "lV"); |
---|
[df9f881] | 2297 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 2298 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 2299 | list L; |
---|
| 2300 | L = lp2ivId(normalize(lead(G))); |
---|
| 2301 | return(ivKDim(L,n,degbound)); |
---|
[5e2dd1] | 2302 | } |
---|
| 2303 | example |
---|
| 2304 | { |
---|
| 2305 | "EXAMPLE:"; echo = 2; |
---|
| 2306 | ring r = 0,(x,y),dp; |
---|
| 2307 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2308 | setring R; // sets basering to Letterplace ring |
---|
| 2309 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); |
---|
| 2310 | // ideal G contains a Groebner basis |
---|
| 2311 | lpKDim(G); //procedure invoked with ring parameters |
---|
| 2312 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
| 2313 | // ring is not necessary |
---|
| 2314 | lpKDim(G,0); // procedure without any degree bound |
---|
| 2315 | } |
---|
| 2316 | |
---|
| 2317 | proc lpMis2Dim(ideal M) |
---|
| 2318 | "USAGE: lpMis2Dim(M); M an ideal |
---|
[5e8ee4c] | 2319 | RETURN: int, the K-dimension of the factor algebra |
---|
[5e2dd1] | 2320 | PURPOSE:Computing the K-dimension out of given mistletoes |
---|
[f2b431] | 2321 | ASSUME: - basering is a Letterplace ring. |
---|
[5e2dd1] | 2322 | @* - M contains only monomials |
---|
| 2323 | NOTE: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
| 2324 | EXAMPLE: example lpMis2Dim; shows examples |
---|
| 2325 | " |
---|
| 2326 | {list L; |
---|
[df9f881] | 2327 | L = lpId2ivLi(M); |
---|
| 2328 | return(ivMis2Dim(L)); |
---|
[5e2dd1] | 2329 | } |
---|
| 2330 | example |
---|
| 2331 | { |
---|
| 2332 | "EXAMPLE:"; echo = 2; |
---|
[df9f881] | 2333 | ring r = 0,(x,y),dp; |
---|
| 2334 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
[5e2dd1] | 2335 | setring R; // sets basering to Letterplace ring |
---|
[df9f881] | 2336 | ideal L = x(1)*y(2),y(1)*x(2)*y(3); |
---|
| 2337 | // ideal containing the mistletoes |
---|
| 2338 | lpMis2Dim(L); // returns the K-dimension of the factor algebra |
---|
[5e2dd1] | 2339 | } |
---|
| 2340 | |
---|
| 2341 | proc lpOrdMisLex(ideal M) |
---|
| 2342 | "USAGE: lpOrdMisLex(M); M an ideal of mistletoes |
---|
[5e8ee4c] | 2343 | RETURN: ideal, containing the mistletoes, ordered lexicographically |
---|
[5e2dd1] | 2344 | PURPOSE:A given set of mistletoes is ordered lexicographically |
---|
[f2b431] | 2345 | ASSUME: - basering is a Letterplace ring. |
---|
[5e2dd1] | 2346 | NOTE: This is preprocessing, it is not needed if the mistletoes are returned |
---|
| 2347 | @* from the sickle algorithm. |
---|
| 2348 | EXAMPLE: example lpOrdMisLex; shows examples |
---|
| 2349 | " |
---|
| 2350 | {return(ivL2lpI(sort(lpId2ivLi(M))[1]));} |
---|
| 2351 | example |
---|
| 2352 | { |
---|
| 2353 | "EXAMPLE:"; echo = 2; |
---|
[df9f881] | 2354 | ring r = 0,(x,y),dp; |
---|
| 2355 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2356 | setring R; // sets basering to Letterplace ring |
---|
| 2357 | 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); |
---|
| 2358 | // some monomials |
---|
| 2359 | lpOrdMisLex(M); // orders the monomials lexicographically |
---|
[5e2dd1] | 2360 | } |
---|
| 2361 | |
---|
| 2362 | proc lpSickle(ideal G, list #) |
---|
| 2363 | "USAGE: lpSickle(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
[5e8ee4c] | 2364 | RETURN: ideal |
---|
[5e2dd1] | 2365 | PURPOSE:Computing the mistletoes of K[X]/<G> |
---|
[f2b431] | 2366 | ASSUME: - basering is a Letterplace ring. |
---|
[5e2dd1] | 2367 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 2368 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 2369 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2370 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2371 | @* - n is the number of variables, which can be set to a different number. |
---|
| 2372 | @* Default: attrib(basering, lV). |
---|
| 2373 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2374 | EXAMPLE: example lpSickle; shows examples |
---|
| 2375 | " |
---|
| 2376 | {int degbound = attrib(basering,"uptodeg"); int n = attrib(basering, "lV"); |
---|
[df9f881] | 2377 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 2378 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 2379 | list L; ideal R; |
---|
| 2380 | R = normalize(lead(G)); |
---|
| 2381 | L = lp2ivId(R); |
---|
| 2382 | L = ivSickle(L,n,degbound); |
---|
| 2383 | R = ivL2lpI(L); |
---|
| 2384 | return(R); |
---|
[5e2dd1] | 2385 | } |
---|
| 2386 | example |
---|
| 2387 | { |
---|
| 2388 | "EXAMPLE:"; echo = 2; |
---|
| 2389 | ring r = 0,(x,y),dp; |
---|
| 2390 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2391 | setring R; // sets basering to Letterplace ring |
---|
| 2392 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
| 2393 | //Groebner basis |
---|
| 2394 | lpSickle(G); //invokes the procedure with ring parameters |
---|
| 2395 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
| 2396 | // ring is not necessary |
---|
| 2397 | lpSickle(G,0); // procedure without any degree bound |
---|
| 2398 | } |
---|
| 2399 | |
---|
| 2400 | proc lpSickleDim(ideal G, list #) |
---|
| 2401 | "USAGE: lpSickleDim(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
[5e8ee4c] | 2402 | RETURN: list |
---|
[5e2dd1] | 2403 | PURPOSE:Computing the K-dimension and the mistletoes |
---|
[f2b431] | 2404 | ASSUME: - basering is a Letterplace ring. |
---|
[5e2dd1] | 2405 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 2406 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 2407 | NOTE: - If L is the list returned, then L[1] is an integer, the K-dimension, |
---|
| 2408 | @* L[2] is an ideal, the mistletoes. |
---|
| 2409 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2410 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2411 | @* - n is the number of variables, which can be set to a different number. |
---|
| 2412 | @* Default: attrib(basering, lV). |
---|
| 2413 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2414 | EXAMPLE: example lpSickleDim; shows examples |
---|
| 2415 | " |
---|
| 2416 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
[df9f881] | 2417 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 2418 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 2419 | list L; |
---|
| 2420 | L = lp2ivId(normalize(lead(G))); |
---|
| 2421 | L = ivSickleDim(L,n,degbound); |
---|
| 2422 | L[2] = ivL2lpI(L[2]); |
---|
| 2423 | return(L); |
---|
[5e2dd1] | 2424 | } |
---|
| 2425 | example |
---|
| 2426 | { |
---|
| 2427 | "EXAMPLE:"; echo = 2; |
---|
| 2428 | ring r = 0,(x,y),dp; |
---|
| 2429 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2430 | setring R; // sets basering to Letterplace ring |
---|
| 2431 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
| 2432 | //Groebner basis |
---|
| 2433 | lpSickleDim(G); // invokes the procedure with ring parameters |
---|
| 2434 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
| 2435 | // ring is not necessary |
---|
| 2436 | lpSickleDim(G,0); // procedure without any degree bound |
---|
| 2437 | } |
---|
| 2438 | |
---|
| 2439 | proc lpSickleHil(ideal G, list #) |
---|
| 2440 | "USAGE: lpSickleHil(G); |
---|
[5e8ee4c] | 2441 | RETURN: list |
---|
[5e2dd1] | 2442 | PURPOSE:Computing the Hilbert series and the mistletoes |
---|
[f2b431] | 2443 | ASSUME: - basering is a Letterplace ring. |
---|
[5e2dd1] | 2444 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 2445 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 2446 | NOTE: - If L is the list returned, then L[1] is an intvec, corresponding to the |
---|
| 2447 | @* Hilbert series, L[2] is an ideal, the mistletoes. |
---|
| 2448 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2449 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2450 | @* - n is the number of variables, which can be set to a different number. |
---|
| 2451 | @* Default: attrib(basering, lV). |
---|
| 2452 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
| 2453 | @* coefficient of the Hilbert series. |
---|
| 2454 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2455 | EXAMPLE: example lpSickleHil; shows examples |
---|
| 2456 | " |
---|
| 2457 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
[df9f881] | 2458 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 2459 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 2460 | list L; |
---|
| 2461 | L = lp2ivId(normalize(lead(G))); |
---|
| 2462 | L = ivSickleHil(L,n,degbound); |
---|
| 2463 | L[2] = ivL2lpI(L[2]); |
---|
| 2464 | return(L); |
---|
[5e2dd1] | 2465 | } |
---|
| 2466 | example |
---|
| 2467 | { |
---|
| 2468 | "EXAMPLE:"; echo = 2; |
---|
| 2469 | ring r = 0,(x,y),dp; |
---|
| 2470 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2471 | setring R; // sets basering to Letterplace ring |
---|
| 2472 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
| 2473 | //Groebner basis |
---|
| 2474 | lpSickleHil(G); // invokes the procedure with ring parameters |
---|
| 2475 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
| 2476 | // ring is not necessary |
---|
| 2477 | lpSickleHil(G,0); // procedure without any degree bound |
---|
| 2478 | } |
---|
| 2479 | |
---|
| 2480 | proc sickle(ideal G, list #) |
---|
| 2481 | "USAGE: sickle(G[,m, d, h, degbound]); G an ideal; m,d,h,degbound optional |
---|
| 2482 | @* integers |
---|
[5e8ee4c] | 2483 | RETURN: list |
---|
[5e2dd1] | 2484 | PURPOSE:Allowing the user to access all procs with one command |
---|
[f2b431] | 2485 | ASSUME: - basering is a Letterplace ring. |
---|
[5e2dd1] | 2486 | @* - if you specify a different degree bound degbound, |
---|
[f2b431] | 2487 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
[5e2dd1] | 2488 | NOTE: The returned object will always be a list, but the entries of the |
---|
| 2489 | @* returned list may be very different |
---|
| 2490 | @* case m=1,d=1,h=1: see lpDHilbertSickle |
---|
| 2491 | @* case m=1,d=1,h=0: see lpSickleDim |
---|
| 2492 | @* case m=1,d=0,h=1: see lpSickleHil |
---|
| 2493 | @* case m=1,d=0,h=0: see lpSickle (this is the default case) |
---|
| 2494 | @* case m=0,d=1,h=1: see lpDHilbert |
---|
| 2495 | @* case m=0,d=1,h=0: see lpKDim |
---|
| 2496 | @* case m=0,d=0,h=1: see lpHilbert |
---|
| 2497 | @* case m=0,d=0,h=0: returns an error |
---|
| 2498 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2499 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2500 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2501 | EXAMPLE: example sickle; shows examples |
---|
| 2502 | " |
---|
| 2503 | {int m,d,h,degbound; |
---|
[df9f881] | 2504 | m = 1; d = 0; h = 0; degbound = attrib(basering,"uptodeg"); |
---|
| 2505 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] < 1) {m = 0;}}} |
---|
| 2506 | if (size(#) > 1) {if (typeof(#[1])=="int"){if (#[2] > 0) {d = 1;}}} |
---|
| 2507 | if (size(#) > 2) {if (typeof(#[1])=="int"){if (#[3] > 0) {h = 1;}}} |
---|
| 2508 | if (size(#) > 3) {if (typeof(#[1])=="int"){if (#[4] >= 0) {degbound = #[4];}}} |
---|
| 2509 | if (m == 1) |
---|
| 2510 | {if (d == 0) |
---|
| 2511 | {if (h == 0) {return(lpSickle(G,degbound,attrib(basering,"lV")));} |
---|
| 2512 | else {return(lpSickleHil(G,degbound,attrib(basering,"lV")));} |
---|
| 2513 | } |
---|
| 2514 | else |
---|
| 2515 | {if (h == 0) {return(lpSickleDim(G,degbound,attrib(basering,"lV")));} |
---|
| 2516 | else {return(lpDHilbertSickle(G,degbound,attrib(basering,"lV")));} |
---|
| 2517 | } |
---|
[5e2dd1] | 2518 | } |
---|
| 2519 | else |
---|
[df9f881] | 2520 | {if (d == 0) |
---|
| 2521 | {if (h == 0) {ERROR("You request to do nothing, so relax and do so");} |
---|
| 2522 | else {return(lpHilbert(G,degbound,attrib(basering,"lV")));} |
---|
| 2523 | } |
---|
| 2524 | else |
---|
| 2525 | {if (h == 0) {return(lpKDim(G,degbound,attrib(basering,"lV")));} |
---|
| 2526 | else {return(lpDHilbert(G,degbound,attrib(basering,"lV")));} |
---|
| 2527 | } |
---|
[5e2dd1] | 2528 | } |
---|
| 2529 | } |
---|
| 2530 | example |
---|
| 2531 | { |
---|
| 2532 | "EXAMPLE:"; echo = 2; |
---|
| 2533 | ring r = 0,(x,y),dp; |
---|
| 2534 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2535 | setring R; // sets basering to Letterplace ring |
---|
| 2536 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); |
---|
| 2537 | // G contains a Groebner basis |
---|
| 2538 | sickle(G,1,1,1); // computes mistletoes, K-dimension and the Hilbert series |
---|
| 2539 | sickle(G,1,0,0); // computes mistletoes only |
---|
| 2540 | sickle(G,0,1,0); // computes K-dimension only |
---|
| 2541 | sickle(G,0,0,1); // computes Hilbert series only |
---|
| 2542 | } |
---|
| 2543 | |
---|
[f2b431] | 2544 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2545 | /* vl: new stuff for conversion to Magma and to SD |
---|
| 2546 | todo: doc, example |
---|
| 2547 | */ |
---|
| 2548 | proc extractVars(r) |
---|
| 2549 | { |
---|
| 2550 | int i = 1; |
---|
| 2551 | int j = 1; |
---|
| 2552 | string candidate; |
---|
| 2553 | list result = list(); |
---|
| 2554 | for (i = 1; i<=nvars(r);i++) |
---|
| 2555 | { |
---|
| 2556 | candidate = string(var(i))[1,find(string(var(i)),"(")-1]; |
---|
| 2557 | if (!inList(result, candidate)) |
---|
| 2558 | { |
---|
| 2559 | result = insert(result,candidate,size(result)); |
---|
| 2560 | } |
---|
| 2561 | } |
---|
| 2562 | return(result); |
---|
| 2563 | } |
---|
| 2564 | |
---|
| 2565 | proc letterPlacePoly2MagmaString(poly h) |
---|
| 2566 | { |
---|
| 2567 | int pos; |
---|
| 2568 | string s = string(h); |
---|
| 2569 | while(find(s,"(")) |
---|
| 2570 | { |
---|
| 2571 | pos = find(s,"("); |
---|
| 2572 | while(s[pos]!=")") |
---|
| 2573 | { |
---|
| 2574 | s = s[1,pos-1]+s[pos+1,size(s)-pos]; |
---|
| 2575 | } |
---|
| 2576 | if (size(s)!=pos) |
---|
| 2577 | { |
---|
| 2578 | s = s[1,pos-1]+s[pos+1,size(s)-pos]; // The last (")") |
---|
| 2579 | } |
---|
| 2580 | else |
---|
| 2581 | { |
---|
| 2582 | s = s[1,pos-1]; |
---|
| 2583 | } |
---|
| 2584 | } |
---|
| 2585 | return(s); |
---|
| 2586 | } |
---|
| 2587 | |
---|
| 2588 | proc letterPlaceIdeal2SD(ideal I, int upToDeg) |
---|
| 2589 | { |
---|
| 2590 | int i; |
---|
| 2591 | print("Don't forget to fill in the formal Data in the file"); |
---|
| 2592 | string result = "<?xml version=\"1.0\"?>"+newline+"<FREEALGEBRA createdAt=\"\" createdBy=\"Singular\" id=\"FREEALGEBRA/\">"+newline; |
---|
| 2593 | result = result + "<vars>"+string(extractVars(basering))+"</vars>"+newline; |
---|
| 2594 | result = result + "<basis>"+newline; |
---|
| 2595 | for (i = 1;i<=size(I);i++) |
---|
| 2596 | { |
---|
| 2597 | result = result + "<poly>"+letterPlacePoly2MagmaString(I[i])+"</poly>"+newline; |
---|
| 2598 | } |
---|
| 2599 | result = result + "</basis>"+newline; |
---|
| 2600 | result = result + "<uptoDeg>"+ string(upToDeg)+"</uptoDeg>"+newline; |
---|
| 2601 | result = result + "<Comment></Comment>"+newline; |
---|
| 2602 | result = result + "<Version></Version>"+newline; |
---|
| 2603 | result = result + "</FREEALGEBRA>"; |
---|
| 2604 | return(result); |
---|
| 2605 | } |
---|
| 2606 | |
---|
| 2607 | |
---|
[5e2dd1] | 2608 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2609 | |
---|
| 2610 | |
---|
[6dce0a] | 2611 | proc tst_fpadim() |
---|
| 2612 | { |
---|
| 2613 | example ivDHilbert; |
---|
| 2614 | example ivDHilbertSickle; |
---|
| 2615 | example ivDimCheck; |
---|
| 2616 | example ivHilbert; |
---|
| 2617 | example ivKDim; |
---|
| 2618 | example ivMis2Dim; |
---|
| 2619 | example ivOrdMisLex; |
---|
| 2620 | example ivSickle; |
---|
| 2621 | example ivSickleHil; |
---|
| 2622 | example ivSickleDim; |
---|
| 2623 | example lpDHilbert; |
---|
| 2624 | example lpDHilbertSickle; |
---|
| 2625 | example lpHilbert; |
---|
| 2626 | example lpDimCheck; |
---|
| 2627 | example lpKDim; |
---|
| 2628 | example lpMis2Dim; |
---|
| 2629 | example lpOrdMisLex; |
---|
| 2630 | example lpSickle; |
---|
| 2631 | example lpSickleHil; |
---|
| 2632 | example lpSickleDim; |
---|
| 2633 | example sickle; |
---|
| 2634 | example ivL2lpI; |
---|
| 2635 | example iv2lp; |
---|
| 2636 | example iv2lpList; |
---|
| 2637 | example iv2lpMat; |
---|
| 2638 | example lp2iv; |
---|
| 2639 | example lp2ivId; |
---|
| 2640 | example lpId2ivLi; |
---|
| 2641 | } |
---|
| 2642 | |
---|
| 2643 | |
---|
| 2644 | |
---|
| 2645 | |
---|
| 2646 | |
---|
[5e2dd1] | 2647 | /* |
---|
[df9f881] | 2648 | Here are some examples one may try. Just copy them into your console. |
---|
| 2649 | These are relations for braid groups, up to degree d: |
---|
| 2650 | |
---|
| 2651 | |
---|
| 2652 | LIB "fpadim.lib"; |
---|
| 2653 | ring r = 0,(x,y,z),dp; |
---|
| 2654 | int d =10; // degree |
---|
| 2655 | def R = makeLetterplaceRing(d); |
---|
| 2656 | setring R; |
---|
| 2657 | 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), |
---|
| 2658 | z(1)*x(2)*z(3) - y(1)*z(2)*x(3), x(1)*x(2)*x(3) + y(1)*y(2)*y(3) + |
---|
| 2659 | z(1)*z(2)*z(3) + x(1)*y(2)*z(3); |
---|
| 2660 | option(prot); |
---|
| 2661 | option(redSB);option(redTail);option(mem); |
---|
| 2662 | ideal J = system("freegb",I,d,3); |
---|
| 2663 | lpDimCheck(J); |
---|
| 2664 | sickle(J,1,1,1,d);//Computes mistletoes, K-dimension and the Hilbert series |
---|
| 2665 | |
---|
| 2666 | |
---|
| 2667 | |
---|
| 2668 | LIB "fpadim.lib"; |
---|
| 2669 | ring r = 0,(x,y,z),dp; |
---|
| 2670 | int d =11; // degree |
---|
| 2671 | def R = makeLetterplaceRing(d); |
---|
| 2672 | setring R; |
---|
| 2673 | 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), |
---|
| 2674 | z(1)*x(2)*z(3) - y(1)*z(2)*x(3), x(1)*x(2)*x(3) + y(1)*y(2)*y(3) + |
---|
| 2675 | z(1)*z(2)*z(3) + x(1)*y(2)*z(3); |
---|
| 2676 | option(prot); |
---|
| 2677 | option(redSB);option(redTail);option(mem); |
---|
| 2678 | ideal J = system("freegb",I,d,3); |
---|
| 2679 | lpDimCheck(J); |
---|
| 2680 | sickle(J,1,1,1,d); |
---|
| 2681 | |
---|
| 2682 | |
---|
| 2683 | |
---|
| 2684 | LIB "fpadim.lib"; |
---|
| 2685 | ring r = 0,(x,y,z),dp; |
---|
| 2686 | int d = 6; // degree |
---|
| 2687 | def R = makeLetterplaceRing(d); |
---|
| 2688 | setring R; |
---|
| 2689 | 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), |
---|
| 2690 | 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); |
---|
| 2691 | option(prot); |
---|
| 2692 | option(redSB);option(redTail);option(mem); |
---|
| 2693 | ideal J = system("freegb",I,d,3); |
---|
| 2694 | lpDimCheck(J); |
---|
| 2695 | sickle(J,1,1,1,d); |
---|
[5e2dd1] | 2696 | */ |
---|
| 2697 | |
---|
| 2698 | /* |
---|
[df9f881] | 2699 | Here are some examples, which can also be found in [studzins]: |
---|
| 2700 | |
---|
| 2701 | // takes up to 880Mb of memory |
---|
| 2702 | LIB "fpadim.lib"; |
---|
| 2703 | ring r = 0,(x,y,z),dp; |
---|
| 2704 | int d =10; // degree |
---|
| 2705 | def R = makeLetterplaceRing(d); |
---|
| 2706 | setring R; |
---|
| 2707 | ideal I = |
---|
| 2708 | 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); |
---|
| 2709 | option(prot); |
---|
| 2710 | option(redSB);option(redTail);option(mem); |
---|
| 2711 | ideal J = system("freegb",I,d,nvars(r)); |
---|
| 2712 | lpDimCheck(J); |
---|
| 2713 | sickle(J,1,1,1,d); // dimension is 24872 |
---|
| 2714 | |
---|
| 2715 | |
---|
| 2716 | LIB "fpadim.lib"; |
---|
| 2717 | ring r = 0,(x,y,z),dp; |
---|
| 2718 | int d =10; // degree |
---|
| 2719 | def R = makeLetterplaceRing(d); |
---|
| 2720 | setring R; |
---|
| 2721 | 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); |
---|
| 2722 | option(prot); |
---|
| 2723 | option(redSB);option(redTail);option(mem); |
---|
| 2724 | ideal J = system("freegb",I,d,3); |
---|
| 2725 | lpDimCheck(J); |
---|
| 2726 | sickle(J,1,1,1,d); |
---|
[5e2dd1] | 2727 | */ |
---|
[f2b431] | 2728 | |
---|
| 2729 | |
---|
| 2730 | /* |
---|
| 2731 | Example for computing GK dimension: |
---|
| 2732 | returns a ring which contains an ideal I |
---|
| 2733 | run gkDim(I) inside this ring and it should return 2n (the GK dimension |
---|
| 2734 | of n-th Weyl algebra including evaluation operators). |
---|
| 2735 | |
---|
| 2736 | proc createWeylEx(int n, int d) |
---|
| 2737 | " |
---|
| 2738 | " |
---|
| 2739 | { |
---|
| 2740 | int baseringdef; |
---|
| 2741 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
| 2742 | { |
---|
| 2743 | def save = basering; |
---|
| 2744 | baseringdef = 1; |
---|
| 2745 | } |
---|
| 2746 | ring r = 0,(d(1..n),x(1..n),e(1..n)),dp; |
---|
| 2747 | def R = makeLetterplaceRing(d); |
---|
| 2748 | setring R; |
---|
| 2749 | ideal I; int i,j; |
---|
| 2750 | |
---|
| 2751 | for (i = 1; i <= n; i++) |
---|
| 2752 | { |
---|
| 2753 | for (j = i+1; j<= n; j++) |
---|
| 2754 | { |
---|
| 2755 | I[size(I)+1] = lpMult(var(i),var(j)); |
---|
| 2756 | } |
---|
| 2757 | } |
---|
| 2758 | |
---|
| 2759 | for (i = 1; i <= n; i++) |
---|
| 2760 | { |
---|
| 2761 | for (j = i+1; j<= n; j++) |
---|
| 2762 | { |
---|
| 2763 | I[size(I)+1] = lpMult(var(n+i),var(n+j)); |
---|
| 2764 | } |
---|
| 2765 | } |
---|
| 2766 | for (i = 1; i <= n; i++) |
---|
| 2767 | { |
---|
| 2768 | for (j = 1; j<= n; j++) |
---|
| 2769 | { |
---|
| 2770 | I[size(I)+1] = lpMult(var(i),var(n+j)); |
---|
| 2771 | } |
---|
| 2772 | } |
---|
| 2773 | for (i = 1; i <= n; i++) |
---|
| 2774 | { |
---|
| 2775 | for (j = 1; j<= n; j++) |
---|
| 2776 | { |
---|
| 2777 | I[size(I)+1] = lpMult(var(i),var(2*n+j)); |
---|
| 2778 | } |
---|
| 2779 | } |
---|
| 2780 | for (i = 1; i <= n; i++) |
---|
| 2781 | { |
---|
| 2782 | for (j = 1; j<= n; j++) |
---|
| 2783 | { |
---|
| 2784 | I[size(I)+1] = lpMult(var(2*n+i),var(n+j)); |
---|
| 2785 | } |
---|
| 2786 | } |
---|
| 2787 | for (i = 1; i <= n; i++) |
---|
| 2788 | { |
---|
| 2789 | for (j = 1; j<= n; j++) |
---|
| 2790 | { |
---|
| 2791 | I[size(I)+1] = lpMult(var(2*n+i),var(2*n+j)); |
---|
| 2792 | } |
---|
| 2793 | } |
---|
| 2794 | I = simplify(I,2+4); |
---|
| 2795 | I = letplaceGBasis(I); |
---|
| 2796 | export(I); |
---|
| 2797 | if (baseringdef == 1) {setring save;} |
---|
| 2798 | return(R); |
---|
| 2799 | } |
---|
| 2800 | |
---|
| 2801 | proc TestGKAuslander3() |
---|
| 2802 | { |
---|
| 2803 | ring r = (0,q),(z,x,y),(dp(1),dp(2)); |
---|
| 2804 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2805 | R; setring R; // sets basering to Letterplace ring |
---|
| 2806 | ideal I; |
---|
| 2807 | I = q*x(1)*y(2) - y(1)*x(2), z(1)*y(2) - y(1)*z(2), z(1)*x(2) - x(1)*z(2); |
---|
| 2808 | I = letplaceGBasis(I); |
---|
| 2809 | lpGkDim(I); // must be 3 |
---|
| 2810 | I = x(1)*y(2)*z(3) - y(1)*x(2), z(1)*y(2) - y(1)*z(2), z(1)*x(2) - x(1)*z(2);//gkDim = 2 |
---|
| 2811 | I = letplaceGBasis(I); // not finite BUT contains a poly in x,y only |
---|
| 2812 | lpGkDim(I); // must be 4 |
---|
| 2813 | |
---|
| 2814 | ring r = 0,(y,x,z),dp; |
---|
| 2815 | def R = makeLetterplaceRing(10); // constructs a Letterplace ring |
---|
| 2816 | R; setring R; // sets basering to Letterplace ring |
---|
| 2817 | ideal I; |
---|
| 2818 | I = x(1)*y(2)*z(3) - y(1)*x(2), z(1)*y(2) - y(1)*z(2), z(1)*x(2) - x(1)*z(2);//gkDim = 2 |
---|
| 2819 | I = letplaceGBasis(I); // computed as it would be homogenized; infinite |
---|
| 2820 | poly p = x(1)*y(2)*y(3)*x(4)-y(1)*x(2)*x(3)*y(4); |
---|
| 2821 | lpNF(p, I); // 0 as expected |
---|
| 2822 | |
---|
| 2823 | // with inverse of z |
---|
| 2824 | ring r = 0,(iz,z,x,y),dp; |
---|
| 2825 | def R = makeLetterplaceRing(11); // constructs a Letterplace ring |
---|
| 2826 | R; setring R; // sets basering to Letterplace ring |
---|
| 2827 | ideal I; |
---|
| 2828 | I = x(1)*y(2)*z(3) - y(1)*x(2), z(1)*y(2) - y(1)*z(2), z(1)*x(2) - x(1)*z(2), |
---|
| 2829 | iz(1)*y(2) - y(1)*iz(2), iz(1)*x(2) - x(1)*iz(2), iz(1)*z(2)-1, z(1)*iz(2) -1; |
---|
| 2830 | I = letplaceGBasis(I); // |
---|
| 2831 | setring r; |
---|
| 2832 | def R2 = makeLetterplaceRing(23); // constructs a Letterplace ring |
---|
| 2833 | setring R2; // sets basering to Letterplace ring |
---|
| 2834 | ideal I = imap(R,I); |
---|
| 2835 | lpGkDim(I); |
---|
| 2836 | |
---|
| 2837 | |
---|
| 2838 | ring r = 0,(t,z,x,y),(dp(2),dp(2)); |
---|
| 2839 | def R = makeLetterplaceRing(20); // constructs a Letterplace ring |
---|
| 2840 | R; setring R; // sets basering to Letterplace ring |
---|
| 2841 | ideal I; |
---|
| 2842 | I = x(1)*y(2)*z(3) - y(1)*x(2)*t(3), z(1)*y(2) - y(1)*z(2), z(1)*x(2) - x(1)*z(2), |
---|
| 2843 | t(1)*y(2) - y(1)*t(2), t(1)*x(2) - x(1)*t(2), t(1)*z(2) - z(1)*t(2);//gkDim = 2 |
---|
| 2844 | I = letplaceGBasis(I); // computed as it would be homogenized; infinite |
---|
| 2845 | LIB "elim.lib"; |
---|
| 2846 | ideal Inoz = nselect(I,intvec(2,6,10,14,18,22,26,30)); |
---|
| 2847 | for(int i=1; i<=20; i++) |
---|
| 2848 | { |
---|
| 2849 | Inoz=subst(Inoz,t(i),1); |
---|
| 2850 | } |
---|
| 2851 | ideal J = x(1)*y(2)*y(3)*x(4)-y(1)*x(2)*x(3)*y(4); |
---|
| 2852 | J = letplaceGBasis(J); |
---|
| 2853 | |
---|
| 2854 | poly p = x(1)*y(2)*y(3)*x(4)-y(1)*x(2)*x(3)*y(4); |
---|
| 2855 | lpNF(p, I); // 0 as expected |
---|
| 2856 | |
---|
| 2857 | ring r2 = 0,(x,y),dp; |
---|
| 2858 | def R2 = makeLetterplaceRing(50); // constructs a Letterplace ring |
---|
| 2859 | setring R2; |
---|
| 2860 | ideal J = x(1)*y(2)*y(3)*x(4)-y(1)*x(2)*x(3)*y(4); |
---|
| 2861 | J = letplaceGBasis(J); |
---|
| 2862 | } |
---|
| 2863 | |
---|
| 2864 | */ |
---|
| 2865 | |
---|
| 2866 | |
---|
| 2867 | /* actual work: |
---|
| 2868 | // downup algebra A |
---|
| 2869 | LIB "fpadim.lib"; |
---|
| 2870 | ring r = (0,a,b,g),(x,y),Dp; |
---|
| 2871 | def R = makeLetterplaceRing(6); // constructs a Letterplace ring |
---|
| 2872 | setring R; |
---|
| 2873 | poly F1 = g*x(1); |
---|
| 2874 | poly F2 = g*y(1); |
---|
| 2875 | ideal J = x(1)*x(2)*y(3)-a*x(1)*y(2)*x(3) - b*y(1)*x(2)*x(3) - F1, |
---|
| 2876 | x(1)*y(2)*y(3)-a*y(1)*x(2)*y(3) - b*y(1)*y(2)*x(3) - F2; |
---|
| 2877 | J = letplaceGBasis(J); |
---|
| 2878 | lpGkDim(J); // 3 == correct |
---|
| 2879 | |
---|
| 2880 | // downup algebra B |
---|
| 2881 | LIB "fpadim.lib"; |
---|
| 2882 | ring r = (0,a,b,g, p(1..7),q(1..7)),(x,y),Dp; |
---|
| 2883 | def R = makeLetterplaceRing(6); // constructs a Letterplace ring |
---|
| 2884 | setring R; |
---|
| 2885 | ideal imn = 1, y(1)*y(2)*y(3), x(1)*y(2), y(1)*x(2), x(1)*x(2), y(1)*y(2), x(1), y(1); |
---|
| 2886 | int i; |
---|
| 2887 | poly F1, F2; |
---|
| 2888 | for(i=1;i<=7;i++) |
---|
| 2889 | { |
---|
| 2890 | F1 = F1 + p(i)*imn[i]; |
---|
| 2891 | F2 = F2 + q(i)*imn[i]; |
---|
| 2892 | } |
---|
| 2893 | ideal J = x(1)*x(2)*y(3)-a*x(1)*y(2)*x(3) - b*y(1)*x(2)*x(3) - F1, |
---|
| 2894 | x(1)*y(2)*y(3)-a*y(1)*x(2)*y(3) - b*y(1)*y(2)*x(3) - F2; |
---|
| 2895 | J = letplaceGBasis(J); |
---|
| 2896 | lpGkDim(J); // 3 == correct |
---|
| 2897 | |
---|
| 2898 | */ |
---|