[6fe3a0] | 1 | // Last change 12.02.2001 (Eric Westenberger) |
---|
[4e461ff] | 2 | /////////////////////////////////////////////////////////////////////////////// |
---|
[6fe3a0] | 3 | version="$Id: zeroset.lib,v 1.11 2005-04-25 10:13:07 Singular Exp $"; |
---|
[fd3fb7] | 4 | category="Symbolic-numerical solving"; |
---|
[4e461ff] | 5 | info=" |
---|
[d12655] | 6 | LIBRARY: zeroset.lib Procedures For Roots and Factorization |
---|
[6fe3a0] | 7 | AUTHOR: Thomas Bayer, email: tbayer@mathematik.uni-kl.de |
---|
| 8 | http://wwwmayr.informatik.tu-muenchen.de/personen/bayert/ |
---|
[d12655] | 9 | Current Adress: Institut fuer Informatik, TU Muenchen |
---|
| 10 | |
---|
[b9b906] | 11 | OVERVIEW: |
---|
[9173792] | 12 | Algorithms for finding the zero-set of a zero-dim. ideal in Q(a)[x_1,..,x_n], |
---|
[d12655] | 13 | Roots and Factorization of univariate polynomials over Q(a)[t] |
---|
[34b0314] | 14 | where a is an algebraic number. Written in the frame of the |
---|
[b9b906] | 15 | diploma thesis (advisor: Prof. Gert-Martin Greuel) 'Computations of moduli |
---|
[6fe3a0] | 16 | spaces of semiquasihomogeneous singularities and an implementation in Singular'. |
---|
[d12655] | 17 | This library is meant as a preliminary extension of the functionality |
---|
| 18 | of Singular for univariate factorization of polynomials over simple algebraic |
---|
| 19 | extensions in characteristic 0. |
---|
| 20 | Subprocedures with postfix 'Main' require that the ring contains a variable |
---|
| 21 | 'a' and no parameters, and the ideal 'mpoly', where 'minpoly' from the |
---|
| 22 | basering is stored. |
---|
[4e461ff] | 23 | |
---|
| 24 | PROCEDURES: |
---|
[6fe3a0] | 25 | EGCD(f, g) gcd over an algebraic extension field of Q |
---|
| 26 | Factor(f) factorization of f over an algebraic extension field |
---|
| 27 | Quotient(f, g) quotient q of f w.r.t. g (in f = q*g + remainder) |
---|
| 28 | Remainder(f,g) remainder of the division of f by g |
---|
| 29 | Roots(f) computes all roots of f in an extension field of Q |
---|
| 30 | SQFRNorm(f) norm of f (f must be squarefree) |
---|
| 31 | ZeroSet(I) zero-set of the 0-dim. ideal I |
---|
[4e461ff] | 32 | |
---|
[d12655] | 33 | AUXILIARY PROCEDURES: |
---|
[6fe3a0] | 34 | EGCDMain(f, g) gcd over an algebraic extension field of Q |
---|
| 35 | FactorMain(f) factorization of f over an algebraic extension field |
---|
[9173792] | 36 | InvertNumberMain(c) inverts an element of an algebraic extension field |
---|
[6fe3a0] | 37 | QuotientMain(f, g) quotient of f w.r.t. g |
---|
| 38 | RemainderMain(f,g) remainder of the division of f by g |
---|
| 39 | RootsMain(f) computes all roots of f, might extend the ground field |
---|
| 40 | SQFRNormMain(f) norm of f (f must be squarefree) |
---|
[4e461ff] | 41 | ContainedQ(data, f) f in data ? |
---|
[6fe3a0] | 42 | SameQ(a, b) a == b (list a,b) |
---|
[4e461ff] | 43 | "; |
---|
| 44 | |
---|
[d12655] | 45 | LIB "primitiv.lib"; |
---|
| 46 | LIB "primdec.lib"; |
---|
[4e461ff] | 47 | |
---|
| 48 | // note : return a ring : ring need not be exported !!! |
---|
[6fe3a0] | 49 | |
---|
| 50 | // Artihmetic in Q(a)[x] without built-in procedures |
---|
[4e461ff] | 51 | // assume basering = Q[x,a] and minpoly is represented by mpoly(a). |
---|
[6fe3a0] | 52 | // the algorithms are taken from "Polynomial Algorithms in Computer Algebra", |
---|
[4e461ff] | 53 | // F. Winkler, Springer Verlag Wien, 1996. |
---|
| 54 | |
---|
| 55 | |
---|
| 56 | // To do : |
---|
[d12655] | 57 | // squarefree factorization |
---|
[4e461ff] | 58 | // multiplicities |
---|
| 59 | |
---|
| 60 | // Improvement : |
---|
| 61 | // a main problem is the growth of the coefficients. Try Roots(x7 - 1) |
---|
[9173792] | 62 | // return ideal mpoly ! |
---|
[4e461ff] | 63 | // mpoly is not monic, comes from primitive_extra |
---|
| 64 | |
---|
[d12655] | 65 | // IMPLEMENTATION |
---|
[4e461ff] | 66 | // |
---|
| 67 | // In procedures with name 'proc-name'Main a polynomial ring over a simple |
---|
| 68 | // extension field is represented as Q[x...,a] together with the ideal |
---|
| 69 | // 'mpoly' (attribute "isSB"). The arithmetic in the extension field is |
---|
[6fe3a0] | 70 | // implemented in the procedures in the procedures 'MultPolys' (multiplication) |
---|
[d12655] | 71 | // and 'InvertNumber' (inversion). After addition and substraction one should |
---|
[4e461ff] | 72 | // apply 'SimplifyPoly' to the result to reduce the result w.r.t. 'mpoly'. |
---|
[d12655] | 73 | // This is done by reducing each coefficient seperately, which is more |
---|
[4e461ff] | 74 | // efficient for polynomials with many terms. |
---|
| 75 | |
---|
| 76 | |
---|
| 77 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 78 | |
---|
| 79 | proc Roots(poly f) |
---|
[9173792] | 80 | "USAGE: Roots(f); where f is a polynomial |
---|
| 81 | PURPOSE: compute all roots of f in a finite extension of the ground field |
---|
[4e461ff] | 82 | without multiplicities. |
---|
| 83 | RETURN: ring, a polynomial ring over an extension field of the ground field, |
---|
[9173792] | 84 | containing a list 'roots' and polynomials 'newA' and 'f': |
---|
| 85 | @format |
---|
| 86 | - 'roots' is the list of roots of the polynomial f (no multiplicities) |
---|
[34b0314] | 87 | - if the ground field is Q(a') and the extension field is Q(a), then |
---|
| 88 | 'newA' is the representation of a' in Q(a). |
---|
[9173792] | 89 | If the basering contains a parameter 'a' and the minpoly remains unchanged |
---|
| 90 | then 'newA' = 'a'. |
---|
| 91 | If the basering does not contain a parameter then 'newA' = 'a' (default). |
---|
| 92 | - 'f' is the polynomial f in Q(a) (a' being substituted by 'newA') |
---|
| 93 | @end format |
---|
[4e461ff] | 94 | ASSUME: ground field to be Q or a simple extension of Q given by a minpoly |
---|
| 95 | EXAMPLE: example Roots; shows an example |
---|
| 96 | " |
---|
| 97 | { |
---|
| 98 | int dbPrt = printlevel-voice+3; |
---|
| 99 | |
---|
| 100 | // create a new ring where par(1) is replaced by the variable |
---|
| 101 | // with the same name or, if basering does not contain a parameter, |
---|
| 102 | // with a new variable 'a'. |
---|
| 103 | |
---|
| 104 | def ROB = basering; |
---|
| 105 | def ROR = TransferRing(basering); |
---|
| 106 | setring ROR; |
---|
[6fe3a0] | 107 | export(ROR); |
---|
[4e461ff] | 108 | |
---|
| 109 | // get the polynomial f and find the roots |
---|
| 110 | |
---|
| 111 | poly f = imap(ROB, f); |
---|
| 112 | list result = RootsMain(f); // find roots of f |
---|
| 113 | |
---|
| 114 | // store the roots and the the new representation of 'a' and transform |
---|
| 115 | // the coefficients of f. |
---|
| 116 | |
---|
| 117 | list roots = result[1]; |
---|
| 118 | poly newA = result[2]; |
---|
| 119 | map F = basering, maxideal(1); |
---|
| 120 | F[nvars(basering)] = newA; |
---|
| 121 | poly fn = SimplifyPoly(F(f)); |
---|
| 122 | |
---|
| 123 | // create a new ring with minploy = mpoly[1] (from ROR) |
---|
| 124 | |
---|
| 125 | def RON = NewBaseRing(); |
---|
| 126 | setring(RON); |
---|
| 127 | list roots = imap(ROR, roots); |
---|
| 128 | poly newA = imap(ROR, newA); |
---|
| 129 | poly f = imap(ROR, fn); |
---|
| 130 | kill(ROR); |
---|
| 131 | export(roots); |
---|
| 132 | export(newA); |
---|
| 133 | export(f); dbprint(dbPrt," |
---|
| 134 | // 'Roots' created a new ring which contains the list 'roots' and |
---|
| 135 | // the polynomials 'f' and 'newA' |
---|
| 136 | // To access the roots, newA and the new representation of f, type |
---|
| 137 | def R = Roots(f); setring R; roots; newA; f; |
---|
| 138 | "); |
---|
| 139 | return(RON); |
---|
| 140 | } |
---|
| 141 | example |
---|
| 142 | {"EXAMPLE:"; echo = 2; |
---|
| 143 | ring R = (0,a), x, lp; |
---|
| 144 | minpoly = a2+1; |
---|
| 145 | poly f = x3 - a; |
---|
| 146 | def R1 = Roots(f); |
---|
| 147 | setring R1; |
---|
| 148 | minpoly; |
---|
| 149 | newA; |
---|
| 150 | f; |
---|
| 151 | roots; |
---|
| 152 | map F; |
---|
| 153 | F[1] = roots[1]; |
---|
| 154 | F(f); |
---|
| 155 | } |
---|
| 156 | |
---|
| 157 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 158 | |
---|
| 159 | proc RootsMain(poly f) |
---|
[9173792] | 160 | "USAGE: RootsMain(f); where f is a polynomial |
---|
| 161 | PURPOSE: compute all roots of f in a finite extension of the ground field |
---|
[4e461ff] | 162 | without multiplicities. |
---|
| 163 | RETURN: list, all entries are polynomials |
---|
[9173792] | 164 | @format |
---|
| 165 | _[1] = roots of f, each entry is a polynomial |
---|
[34b0314] | 166 | _[2] = 'newA' - if the ground field is Q(a') and the extension field |
---|
[9173792] | 167 | is Q(a), then 'newA' is the representation of a' in Q(a) |
---|
[34b0314] | 168 | _[3] = minpoly of the algebraic extension of the ground field |
---|
[9173792] | 169 | @end format |
---|
| 170 | ASSUME: basering = Q[x,a] ideal mpoly must be defined, it might be 0! |
---|
[35f23d] | 171 | NOTE: might change the ideal mpoly !! |
---|
[4e461ff] | 172 | EXAMPLE: example Roots; shows an example |
---|
| 173 | " |
---|
| 174 | { |
---|
| 175 | int i, linFactors, nlinFactors, dbPrt; |
---|
| 176 | intvec wt = 1,0; // deg(a) = 0 |
---|
| 177 | list factorList, nlFactors, nlMult, roots, result; |
---|
| 178 | poly fa, lc; |
---|
| 179 | |
---|
| 180 | dbPrt = printlevel-voice+3; |
---|
| 181 | |
---|
| 182 | // factor f in Q(a)[t] to obtain the roots lying in Q(a) |
---|
| 183 | // firstly, find roots of the linear factors, |
---|
| 184 | // nonlinear factors are processed later |
---|
| 185 | |
---|
| 186 | dbprint(dbPrt, "Roots of " + string(f) + ", minimal polynomial = " + string(mpoly[1])); |
---|
| 187 | factorList = FactorMain(f); // Factorize f |
---|
| 188 | dbprint(dbPrt, (" prime factors of f are : " + string(factorList[1]))); |
---|
| 189 | |
---|
| 190 | linFactors = 0; |
---|
| 191 | nlinFactors = 0; |
---|
| 192 | for(i = 2; i <= size(factorList[1]); i = i + 1) { // find linear and nonlinear factors |
---|
| 193 | fa = factorList[1][i]; |
---|
| 194 | if(deg(fa, wt) == 1) { |
---|
| 195 | linFactors++; // get the root from the linear factor |
---|
| 196 | lc = LeadTerm(fa, 1)[3]; |
---|
| 197 | fa = MultPolys(InvertNumberMain(lc), fa); // make factor monic |
---|
| 198 | roots[linFactors] = var(1) - fa; // fa is monic !! |
---|
| 199 | } |
---|
| 200 | else { // ignore nonlinear factors |
---|
| 201 | nlinFactors++; |
---|
| 202 | nlFactors[nlinFactors] = factorList[1][i]; |
---|
| 203 | nlMult[nlinFactors] = factorList[2][i]; |
---|
| 204 | } |
---|
| 205 | } |
---|
[34b0314] | 206 | if(linFactors == size(factorList[1]) - 1) { // all roots of f are contained in the ground field |
---|
[4e461ff] | 207 | result[1] = roots; |
---|
| 208 | result[2] = var(2); |
---|
| 209 | result[3] = mpoly[1]; |
---|
| 210 | return(result); |
---|
| 211 | } |
---|
| 212 | |
---|
[34b0314] | 213 | // process the nonlinear factors, i.e., extend the ground field |
---|
[4e461ff] | 214 | // where a nonlinear factor (irreducible) is a minimal polynomial |
---|
| 215 | // compute the primitive element of this extension |
---|
| 216 | |
---|
| 217 | ideal primElem, minPolys, Fid; |
---|
| 218 | list partSol; |
---|
| 219 | map F, Xchange; |
---|
| 220 | poly f1, newA, mp, oldMinPoly; |
---|
| 221 | |
---|
| 222 | Fid = mpoly; |
---|
| 223 | F[1] = var(1); |
---|
| 224 | Xchange[1] = var(2); // the variables have to be exchanged |
---|
| 225 | Xchange[2] = var(1); // for the use of 'primitive' |
---|
| 226 | |
---|
| 227 | if(nlinFactors == 1) { // one nl factor |
---|
| 228 | |
---|
| 229 | // compute the roots of the nonlinear (irreducible, monic) factor f1 of f |
---|
| 230 | // by extending the basefield by a' with minimal polynomial f1 |
---|
| 231 | // Then call Roots(f1) to find the roots of f1 over the new base field |
---|
| 232 | |
---|
| 233 | f1 = nlFactors[1]; |
---|
| 234 | if(mpoly[1] != 0) { |
---|
| 235 | mp = mpoly[1]; |
---|
| 236 | minPolys = Xchange(mp), Xchange(f1); |
---|
| 237 | primElem = primitive_extra(minPolys); // no random coord. change |
---|
| 238 | mpoly = std(primElem[1]); |
---|
| 239 | F = basering, maxideal(1); |
---|
| 240 | F[2] = primElem[2]; // transfer all to the new representation |
---|
| 241 | newA = primElem[2]; // new representation of a |
---|
| 242 | f1 = SimplifyPoly(F(f1)); //reduce(F(f1), mpoly); |
---|
| 243 | if(size(roots) > 0) {roots = SimplifyData(F(roots));} |
---|
| 244 | } |
---|
| 245 | else { |
---|
| 246 | mpoly = std(Xchange(f1)); |
---|
| 247 | newA = var(2); |
---|
| 248 | } |
---|
| 249 | result[3] = mpoly[1]; |
---|
| 250 | oldMinPoly = mpoly[1]; |
---|
| 251 | partSol = RootsMain(f1); // find roots of f1 over extended field |
---|
| 252 | |
---|
| 253 | if(oldMinPoly != partSol[3]) { // minpoly has changed ? |
---|
| 254 | // all previously computed roots must be transformed |
---|
| 255 | // because the minpoly has changed |
---|
| 256 | result[3] = partSol[3]; // new minpoly |
---|
| 257 | F[2] = partSol[2]; // new representation of algebraic number |
---|
| 258 | if(size(roots) > 0) {roots = SimplifyData(F(roots)); } |
---|
| 259 | newA = SimplifyPoly(F(newA)); // F(newA); |
---|
| 260 | } |
---|
| 261 | roots = roots + partSol[1]; // add roots |
---|
| 262 | result[2] = newA; |
---|
| 263 | result[1] = roots; |
---|
| 264 | } |
---|
| 265 | else { // more than one nonlinear (irreducible) factor (f_1,...,f_r) |
---|
| 266 | // solve each of them by RootsMain(f_i), append their roots |
---|
| 267 | // change the minpoly and transform all previously computed |
---|
| 268 | // roots if necessary. |
---|
| 269 | // Note that the for-loop is more or less book-keeping |
---|
| 270 | |
---|
| 271 | newA = var(2); |
---|
| 272 | result[2] = newA; |
---|
| 273 | for(i = 1; i <= size(nlFactors); i = i + 1) { |
---|
| 274 | oldMinPoly = mpoly[1]; |
---|
| 275 | partSol = RootsMain(nlFactors[i]); // main work |
---|
| 276 | nlFactors[i] = 0; // delete factor |
---|
| 277 | result[3] = partSol[3]; // store minpoly |
---|
| 278 | |
---|
| 279 | // book-keeping starts here as in the case 1 nonlinear factor |
---|
| 280 | |
---|
| 281 | if(oldMinPoly != partSol[3]) { // minpoly has changed |
---|
| 282 | F = basering, maxideal(1); |
---|
| 283 | F[2] = partSol[2]; // transfer all to the new representation |
---|
| 284 | newA = SimplifyPoly(F(newA)); // F(newA); new representation of a |
---|
| 285 | result[2] = newA; |
---|
| 286 | if(i < size(nlFactors)) { |
---|
| 287 | nlFactors = SimplifyData(F(nlFactors)); |
---|
| 288 | } // transform remaining factors |
---|
| 289 | if(size(roots) > 0) {roots = SimplifyData(F(roots));} |
---|
| 290 | } |
---|
| 291 | roots = roots + partSol[1]; // transform roots |
---|
| 292 | result[1] = roots; |
---|
| 293 | } // end more than one nl factor |
---|
| 294 | |
---|
| 295 | } |
---|
| 296 | return(result); |
---|
| 297 | } |
---|
| 298 | |
---|
| 299 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 300 | |
---|
| 301 | proc ZeroSet(ideal I, list #) |
---|
[9173792] | 302 | "USAGE: ZeroSet(I [,opt] ); I=ideal, opt=integer |
---|
| 303 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a finite extension |
---|
[34b0314] | 304 | of the ground field. |
---|
[4e461ff] | 305 | RETURN: ring, a polynomial ring over an extension field of the ground field, |
---|
[9173792] | 306 | containing a list 'zeroset', a polynomial 'newA', and an |
---|
| 307 | ideal 'id': |
---|
| 308 | @format |
---|
| 309 | - 'zeroset' is the list of the zeros of the ideal I, each zero is an ideal. |
---|
[34b0314] | 310 | - if the ground field is Q(a') and the extension field is Q(a), then |
---|
[9173792] | 311 | 'newA' is the representation of a' in Q(a). |
---|
| 312 | If the basering contains a parameter 'a' and the minpoly remains unchanged |
---|
| 313 | then 'newA' = 'a'. |
---|
[34b0314] | 314 | If the basering does not contain a parameter then 'newA' = 'a' (default). |
---|
[9173792] | 315 | - 'id' is the ideal I in Q(a)[x_1,...] (a' substituted by 'newA') |
---|
| 316 | @end format |
---|
[d12655] | 317 | ASSUME: dim(I) = 0, and ground field to be Q or a simple extension of Q given |
---|
| 318 | by a minpoly. |
---|
[4e461ff] | 319 | OPTIONS: opt = 0 no primary decomposition (default) |
---|
[d12655] | 320 | opt > 0 primary decomposition |
---|
[b9b906] | 321 | NOTE: If I contains an algebraic number (parameter) then 'I' must be |
---|
[35f23d] | 322 | transformed w.r.t. 'newA' in the new ring. |
---|
[4e461ff] | 323 | EXAMPLE: example ZeroSet; shows an example |
---|
| 324 | " |
---|
| 325 | { |
---|
| 326 | int primaryDecQ, dbPrt; |
---|
| 327 | list rp; |
---|
| 328 | |
---|
| 329 | dbPrt = printlevel-voice+2; |
---|
| 330 | |
---|
| 331 | if(size(#) > 0) { primaryDecQ = #[1]; } |
---|
| 332 | else { primaryDecQ = 0; } |
---|
| 333 | |
---|
[6fe3a0] | 334 | // create a new ring 'ZSR' with one additional variable instead of the |
---|
| 335 | // parameter |
---|
| 336 | // if the basering does not contain a parameter then 'a' is used as the |
---|
| 337 | // additional variable. |
---|
[4e461ff] | 338 | |
---|
| 339 | def RZSB = basering; |
---|
[6fe3a0] | 340 | def ZSR = TransferRing(RZSB); |
---|
[4e461ff] | 341 | setring ZSR; |
---|
| 342 | |
---|
| 343 | // get ideal I and find the zero-set |
---|
| 344 | |
---|
| 345 | ideal id = std(imap(RZSB, I)); |
---|
[6fe3a0] | 346 | // print(dim(id)); |
---|
| 347 | if(dim(id) > 1) { // new variable adjoined to ZSR |
---|
| 348 | ERROR(" ideal not zerodimensional "); |
---|
| 349 | } |
---|
[4e461ff] | 350 | |
---|
| 351 | list result = ZeroSetMain(id, primaryDecQ); |
---|
| 352 | |
---|
| 353 | // store the zero-set, minimal polynomial and the new representative of 'a' |
---|
| 354 | |
---|
| 355 | list zeroset = result[1]; |
---|
| 356 | poly newA = result[2]; |
---|
| 357 | poly minPoly = result[3][1]; |
---|
| 358 | |
---|
[6fe3a0] | 359 | // transform the generators of the ideal I w.r.t. the new representation |
---|
| 360 | // of 'a' |
---|
[4e461ff] | 361 | |
---|
| 362 | map F = basering, maxideal(1); |
---|
| 363 | F[nvars(basering)] = newA; |
---|
| 364 | id = SimplifyData(F(id)); |
---|
| 365 | |
---|
| 366 | // create a new ring with minpoly = minPoly |
---|
| 367 | |
---|
| 368 | def RZBN = NewBaseRing(); |
---|
| 369 | setring RZBN; |
---|
| 370 | |
---|
| 371 | list zeroset = imap(ZSR, zeroset); |
---|
| 372 | poly newA = imap(ZSR, newA); |
---|
| 373 | ideal id = imap(ZSR, id); |
---|
| 374 | kill(ZSR); |
---|
| 375 | |
---|
| 376 | export(id); |
---|
| 377 | export(zeroset); |
---|
| 378 | export(newA); |
---|
| 379 | dbprint(dbPrt," |
---|
| 380 | // 'ZeroSet' created a new ring which contains the list 'zeroset', the ideal |
---|
| 381 | // 'id' and the polynomial 'newA'. 'id' is the ideal of the input transformed |
---|
| 382 | // w.r.t. 'newA'. |
---|
| 383 | // To access the zero-set, 'newA' and the new representation of the ideal, type |
---|
| 384 | def R = ZeroSet(I); setring R; zeroset; newA; id; |
---|
| 385 | "); |
---|
[6fe3a0] | 386 | setring RZSB; |
---|
[4e461ff] | 387 | return(RZBN); |
---|
| 388 | } |
---|
| 389 | example |
---|
| 390 | {"EXAMPLE:"; echo = 2; |
---|
| 391 | ring R = (0,a), (x,y,z), lp; |
---|
| 392 | minpoly = a2 + 1; |
---|
| 393 | ideal I = x2 - 1/2, a*z - 1, y - 2; |
---|
| 394 | def T = ZeroSet(I); |
---|
| 395 | setring T; |
---|
| 396 | minpoly; |
---|
| 397 | newA; |
---|
| 398 | id; |
---|
| 399 | zeroset; |
---|
| 400 | map F1 = basering, zeroset[1]; |
---|
| 401 | map F2 = basering, zeroset[2]; |
---|
| 402 | F1(id); |
---|
| 403 | F2(id); |
---|
| 404 | } |
---|
| 405 | |
---|
| 406 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 407 | |
---|
| 408 | proc InvertNumberMain(poly f) |
---|
[9173792] | 409 | "USAGE: InvertNumberMain(f); where f is a polynomial |
---|
[4e461ff] | 410 | PURPOSE: compute 1/f if f is a number in Q(a) i.e., f is represented by a |
---|
| 411 | polynomial in Q[a]. |
---|
| 412 | RETURN: poly 1/f |
---|
[9173792] | 413 | ASSUME: basering = Q[x_1,...,x_n,a], ideal mpoly must be defined and != 0 ! |
---|
[4e461ff] | 414 | " |
---|
| 415 | { |
---|
| 416 | if(diff(f, var(1)) != 0) { ERROR("number must not contain variable !");} |
---|
| 417 | |
---|
| 418 | int n = nvars(basering); |
---|
| 419 | def RINB = basering; |
---|
| 420 | string ringSTR = "ring RINR = 0, " + string(var(n)) + ", dp;"; |
---|
| 421 | execute(ringSTR); // new ring = Q[a] |
---|
| 422 | |
---|
| 423 | list gcdList; |
---|
| 424 | poly f, g, inv; |
---|
| 425 | |
---|
| 426 | f = imap(RINB, f); |
---|
| 427 | g = imap(RINB, mpoly)[1]; |
---|
| 428 | |
---|
| 429 | if(diff(f, var(1)) != 0) { inv = extgcd(f, g)[2]; } // f contains var(1) |
---|
| 430 | else { inv = 1/f;} // f element in Q |
---|
| 431 | |
---|
| 432 | setring(RINB); |
---|
| 433 | return(imap(RINR, inv)); |
---|
| 434 | } |
---|
| 435 | |
---|
| 436 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 437 | |
---|
| 438 | proc MultPolys(poly f, poly g) |
---|
| 439 | "USAGE: MultPolys(f, g); poly f,g |
---|
| 440 | PURPOSE: multiply the polynomials f and g and reduce them w.r.t. mpoly |
---|
| 441 | RETURN: poly f*g |
---|
| 442 | ASSUME: basering = Q[x,a], ideal mpoly must be defined, it might be 0 ! |
---|
| 443 | " |
---|
| 444 | { |
---|
| 445 | return(SimplifyPoly(f * g)); |
---|
| 446 | } |
---|
| 447 | |
---|
| 448 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 449 | |
---|
| 450 | proc LeadTerm(poly f, int i) |
---|
| 451 | "USAGE: LeadTerm(f); poly f, int i |
---|
[9173792] | 452 | PURPOSE: compute the leading coef and term of f w.r.t var(i), where the last |
---|
[4e461ff] | 453 | ring variable is treated as a parameter. |
---|
| 454 | RETURN: list of polynomials |
---|
| 455 | _[1] = leading term |
---|
| 456 | _[2] = leading monomial |
---|
| 457 | _[3] = leading coefficient |
---|
| 458 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
| 459 | " |
---|
| 460 | { |
---|
| 461 | list result; |
---|
| 462 | matrix co = coef(f, var(i)); |
---|
| 463 | result[1] = co[1, 1]*co[2, 1]; |
---|
| 464 | result[2] = co[1, 1]; |
---|
| 465 | result[3] = co[2, 1]; |
---|
| 466 | return(result); |
---|
| 467 | } |
---|
| 468 | |
---|
| 469 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 470 | |
---|
| 471 | proc Quotient(poly f, poly g) |
---|
[6fe3a0] | 472 | "USAGE: Quotient(f, g); where f,g are polynomials; |
---|
| 473 | PURPOSE: compute the quotient q and remainder r s.t. f = g*q + r, deg(r) < deg(g) |
---|
[4e461ff] | 474 | RETURN: list of polynomials |
---|
[6fe3a0] | 475 | @format |
---|
| 476 | _[1] = quotient q |
---|
| 477 | _[2] = remainder r |
---|
| 478 | @end format |
---|
[4e461ff] | 479 | ASSUME: basering = Q[x] or Q(a)[x] |
---|
| 480 | EXAMPLE: example Quotient; shows an example |
---|
| 481 | " |
---|
| 482 | { |
---|
[6fe3a0] | 483 | def QUOB = basering; |
---|
| 484 | def QUOR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
| 485 | setring QUOR; |
---|
| 486 | export(QUOR); |
---|
| 487 | poly f = imap(QUOB, f); |
---|
| 488 | poly g = imap(QUOB, g); |
---|
| 489 | list result = QuotientMain(f, g); |
---|
| 490 | |
---|
| 491 | setring(QUOB); |
---|
| 492 | list result = imap(QUOR, result); |
---|
| 493 | kill(QUOR); |
---|
| 494 | return(result); |
---|
[4e461ff] | 495 | } |
---|
| 496 | example |
---|
| 497 | {"EXAMPLE:"; echo = 2; |
---|
| 498 | ring R = (0,a), x, lp; |
---|
| 499 | minpoly = a2+1; |
---|
| 500 | poly f = x4 - 2; |
---|
| 501 | poly g = x - a; |
---|
| 502 | list qr = Quotient(f, g); |
---|
| 503 | qr; |
---|
| 504 | qr[1]*g + qr[2] - f; |
---|
| 505 | } |
---|
| 506 | |
---|
| 507 | proc QuotientMain(poly f, poly g) |
---|
[6fe3a0] | 508 | "USAGE: QuotientMain(f, g); where f,g are polynomials |
---|
| 509 | PURPOSE: compute the quotient q and remainder r s.t. f = g*q + r, deg(r) < deg(g) |
---|
[4e461ff] | 510 | RETURN: list of polynomials |
---|
[6fe3a0] | 511 | @format |
---|
| 512 | _[1] = quotient q |
---|
| 513 | _[2] = remainder r |
---|
| 514 | @end format |
---|
| 515 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
| 516 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
[4e461ff] | 517 | EXAMPLE: example Quotient; shows an example |
---|
| 518 | " |
---|
| 519 | { |
---|
[6fe3a0] | 520 | if(g == 0) { ERROR("Division by zero !");} |
---|
| 521 | |
---|
| 522 | def QMB = basering; |
---|
| 523 | def QMR = NewBaseRing(); |
---|
| 524 | setring QMR; |
---|
| 525 | poly f, g, h; |
---|
| 526 | h = imap(QMB, f) / imap(QMB, g); |
---|
| 527 | setring QMB; |
---|
| 528 | return(list(imap(QMR, h), 0)); |
---|
[4e461ff] | 529 | } |
---|
| 530 | |
---|
| 531 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 532 | |
---|
| 533 | proc Remainder(poly f, poly g) |
---|
[9173792] | 534 | "USAGE: Remainder(f, g); where f,g are polynomials |
---|
| 535 | PURPOSE: compute the remainder of the division of f by g, i.e. a polynomial r |
---|
| 536 | s.t. f = g*q + r, deg(r) < deg(g). |
---|
[4e461ff] | 537 | RETURN: poly |
---|
| 538 | ASSUME: basering = Q[x] or Q(a)[x] |
---|
| 539 | " |
---|
| 540 | { |
---|
| 541 | def REMB = basering; |
---|
| 542 | def REMR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
| 543 | setring(REMR); |
---|
[6fe3a0] | 544 | export(REMR); |
---|
[4e461ff] | 545 | poly f = imap(REMB, f); |
---|
| 546 | poly g = imap(REMB, g); |
---|
| 547 | poly h = RemainderMain(f, g); |
---|
| 548 | |
---|
| 549 | setring(REMB); |
---|
| 550 | poly r = imap(REMR, h); |
---|
| 551 | kill(REMR); |
---|
| 552 | return(r); |
---|
| 553 | } |
---|
| 554 | example |
---|
| 555 | {"EXAMPLE:"; echo = 2; |
---|
| 556 | ring R = (0,a), x, lp; |
---|
| 557 | minpoly = a2+1; |
---|
| 558 | poly f = x4 - 1; |
---|
| 559 | poly g = x3 - 1; |
---|
| 560 | Remainder(f, g); |
---|
| 561 | } |
---|
| 562 | |
---|
| 563 | proc RemainderMain(poly f, poly g) |
---|
[9173792] | 564 | "USAGE: RemainderMain(f, g); where f,g are polynomials |
---|
| 565 | PURPOSE: compute the remainder r s.t. f = g*q + r, deg(r) < deg(g) |
---|
[4e461ff] | 566 | RETURN: poly |
---|
[9173792] | 567 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
| 568 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
[4e461ff] | 569 | " |
---|
| 570 | { |
---|
| 571 | int dg; |
---|
| 572 | intvec wt = 1,0;; |
---|
| 573 | poly lc, g1, r; |
---|
| 574 | |
---|
| 575 | if(deg(g, wt) == 0) { return(0); } |
---|
| 576 | |
---|
| 577 | lc = LeadTerm(g, 1)[3]; |
---|
| 578 | g1 = MultPolys(InvertNumberMain(lc), g); // make g monic |
---|
| 579 | |
---|
| 580 | return(SimplifyPoly(reduce(f, std(g1)))); |
---|
| 581 | } |
---|
| 582 | |
---|
| 583 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 584 | |
---|
| 585 | proc EGCD(poly f, poly g) |
---|
[9173792] | 586 | "USAGE: EGCD(f, g); where f,g are polynomials |
---|
| 587 | PURPOSE: compute the polynomial gcd of f and g over Q(a)[x] |
---|
| 588 | RETURN: polynomial h s.t. h is a greatest common divisor of f and g (not nec. |
---|
| 589 | monic) |
---|
[4e461ff] | 590 | ASSUME: basering = Q(a)[t] |
---|
| 591 | EXAMPLE: example EGCD; shows an example |
---|
| 592 | " |
---|
| 593 | { |
---|
| 594 | def GCDB = basering; |
---|
| 595 | def GCDR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
| 596 | setring GCDR; |
---|
[6fe3a0] | 597 | export(GCDR); |
---|
[4e461ff] | 598 | poly f = imap(GCDB, f); |
---|
| 599 | poly g = imap(GCDB, g); |
---|
| 600 | poly h = EGCDMain(f, g); // squarefree norm of f |
---|
| 601 | |
---|
| 602 | setring(GCDB); |
---|
| 603 | poly h = imap(GCDR, h); |
---|
| 604 | kill(GCDR); |
---|
| 605 | return(h); |
---|
| 606 | } |
---|
| 607 | example |
---|
| 608 | {"EXAMPLE:"; echo = 2; |
---|
| 609 | ring R = (0,a), x, lp; |
---|
| 610 | minpoly = a2+1; |
---|
| 611 | poly f = x4 - 1; |
---|
| 612 | poly g = x2 - 2*a*x - 1; |
---|
| 613 | EGCD(f, g); |
---|
| 614 | } |
---|
| 615 | |
---|
| 616 | proc EGCDMain(poly f, poly g) |
---|
[9173792] | 617 | "USAGE: EGCDMain(f, g); where f,g are polynomials |
---|
| 618 | PURPOSE: compute the polynomial gcd of f and g over Q(a)[x] |
---|
[4e461ff] | 619 | RETURN: poly |
---|
[9173792] | 620 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
| 621 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
[4e461ff] | 622 | EXAMPLE: example EGCD; shows an example |
---|
| 623 | " |
---|
| 624 | // might be extended to return s1, s2 s.t. f*s1 + g*s2 = gcd |
---|
| 625 | { |
---|
| 626 | int i = 1; |
---|
| 627 | poly r1, r2, r; |
---|
| 628 | |
---|
| 629 | r1 = f; |
---|
| 630 | r2 = g; |
---|
| 631 | |
---|
| 632 | while(r2 != 0) { |
---|
| 633 | r = RemainderMain(r1, r2); |
---|
| 634 | r1 = r2; |
---|
| 635 | r2 = r; |
---|
| 636 | } |
---|
| 637 | return(r1); |
---|
| 638 | } |
---|
| 639 | |
---|
| 640 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 641 | |
---|
| 642 | proc MEGCD(poly f, poly g, int varIndex) |
---|
| 643 | "USAGE: MEGCD(f, g, i); poly f, g; int i |
---|
[9173792] | 644 | PURPOSE: compute the polynomial gcd of f and g in the i'th variable |
---|
[4e461ff] | 645 | RETURN: poly |
---|
[d12655] | 646 | ASSUME: f, g are polynomials in var(i), last variable is the algebraic number |
---|
[4e461ff] | 647 | EXAMPLE: example MEGCD; shows an example |
---|
| 648 | " |
---|
| 649 | // might be extended to return s1, s2 s.t. f*s1 + g*s2 = gc |
---|
| 650 | // not used ! |
---|
| 651 | { |
---|
| 652 | string @str, @sf, @sg, @mp, @parName; |
---|
| 653 | |
---|
| 654 | def @RGCDB = basering; |
---|
| 655 | |
---|
| 656 | @sf = string(f); |
---|
| 657 | @sg = string(g); |
---|
| 658 | @mp = string(minpoly); |
---|
| 659 | |
---|
| 660 | if(npars(basering) == 0) { @parName = "0";} |
---|
| 661 | else { @parName = "(0, " + parstr(basering) + ")"; } |
---|
| 662 | @str = "ring @RGCD = " + @parName + ", " + string(var(varIndex)) + ", dp;"; |
---|
| 663 | execute(@str); |
---|
| 664 | if(@mp != "0") { execute ("minpoly = " + @mp + ";"); } |
---|
| 665 | execute("poly @f = " + @sf + ";"); |
---|
| 666 | execute("poly @g = " + @sg + ";"); |
---|
[6fe3a0] | 667 | export(@RGCD); |
---|
[4e461ff] | 668 | poly @h = EGCD(@f, @g); |
---|
| 669 | setring(@RGCDB); |
---|
| 670 | poly h = imap(@RGCD, @h); |
---|
| 671 | kill(@RGCD); |
---|
| 672 | return(h); |
---|
| 673 | } |
---|
| 674 | |
---|
| 675 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 676 | |
---|
| 677 | proc SQFRNorm(poly f) |
---|
[9173792] | 678 | "USAGE: SQFRNorm(f); where f is a polynomial |
---|
| 679 | PURPOSE: compute the norm of the squarefree polynomial f in Q(a)[x]. |
---|
| 680 | RETURN: list with 3 entries |
---|
| 681 | @format |
---|
| 682 | _[1] = squarefree norm of g (poly) |
---|
| 683 | _[2] = g (= f(x - s*a)) (poly) |
---|
| 684 | _[3] = s (int) |
---|
| 685 | @end format |
---|
[4e461ff] | 686 | ASSUME: f must be squarefree, basering = Q(a)[x] and minpoly != 0. |
---|
| 687 | NOTE: the norm is an element of Q[x] |
---|
| 688 | EXAMPLE: example SQFRNorm; shows an example |
---|
| 689 | " |
---|
| 690 | { |
---|
| 691 | def SNB = basering; |
---|
[6fe3a0] | 692 | def SNR = TransferRing(SNB); // new ring with parameter 'a' |
---|
| 693 | // replaced by a variable |
---|
[4e461ff] | 694 | setring SNR; |
---|
| 695 | poly f = imap(SNB, f); |
---|
| 696 | list result = SQFRNormMain(f); // squarefree norm of f |
---|
| 697 | |
---|
[6fe3a0] | 698 | setring SNB; |
---|
[4e461ff] | 699 | list result = imap(SNR, result); |
---|
[6fe3a0] | 700 | kill SNR; |
---|
[4e461ff] | 701 | return(result); |
---|
| 702 | } |
---|
| 703 | example |
---|
| 704 | {"EXAMPLE:"; echo = 2; |
---|
| 705 | ring R = (0,a), x, lp; |
---|
| 706 | minpoly = a2+1; |
---|
| 707 | poly f = x4 - 2*x + 1; |
---|
| 708 | SQFRNorm(f); |
---|
| 709 | } |
---|
| 710 | |
---|
| 711 | proc SQFRNormMain(poly f) |
---|
[9173792] | 712 | "USAGE: SQFRNorm(f); where f is a polynomial |
---|
| 713 | PURPOSE: compute the norm of the squarefree polynomial f in Q(a)[x]. |
---|
| 714 | RETURN: list with 3 entries |
---|
| 715 | @format |
---|
| 716 | _[1] = squarefree norm of g (poly) |
---|
| 717 | _[2] = g (= f(x - s*a)) (poly) |
---|
| 718 | _[3] = s (int) |
---|
| 719 | @end format |
---|
| 720 | ASSUME: f must be squarefree, basering = Q[x,a] and ideal mpoly is equal to |
---|
[d12655] | 721 | 'minpoly',this represents the ring Q(a)[x] together with 'minpoly'. |
---|
[4e461ff] | 722 | NOTE: the norm is an element of Q[x] |
---|
| 723 | EXAMPLE: example SqfrNorm; shows an example |
---|
| 724 | " |
---|
| 725 | { |
---|
[6fe3a0] | 726 | def SNRMB = basering; |
---|
[4e461ff] | 727 | int s = 0; |
---|
| 728 | intvec wt = 1,0; |
---|
| 729 | ideal mapId; |
---|
| 730 | // list result; |
---|
| 731 | poly g, N, N1, h; |
---|
| 732 | string ringSTR; |
---|
| 733 | |
---|
| 734 | mapId[1] = var(1) - var(2); // linear transformation |
---|
| 735 | mapId[2] = var(2); |
---|
[6fe3a0] | 736 | map Fs = SNRMB, mapId; |
---|
[4e461ff] | 737 | |
---|
| 738 | N = resultant(f, mpoly[1], var(2)); // norm of f |
---|
| 739 | N1 = diff(N, var(1)); |
---|
| 740 | g = f; |
---|
| 741 | |
---|
| 742 | ringSTR = "ring SNRM1 = 0, " + string(var(1)) + ", dp;"; // univariate ring |
---|
| 743 | execute(ringSTR); |
---|
| 744 | poly N, N1, h; // N, N1 do not contain 'a', use built-in gcd |
---|
| 745 | h = gcd(imap(SNRMB, N), imap(SNRMB, N1)); |
---|
| 746 | setring(SNRMB); |
---|
| 747 | h = imap(SNRM1, h); |
---|
| 748 | while(deg(h, wt) != 0) { // while norm is not squarefree |
---|
| 749 | s = s + 1; |
---|
| 750 | g = reduce(Fs(g), mpoly); |
---|
| 751 | N = reduce(resultant(g, mpoly[1], var(2)), mpoly); // norm of g |
---|
| 752 | N1 = reduce(diff(N, var(1)), mpoly); |
---|
| 753 | setring(SNRM1); |
---|
| 754 | h = gcd(imap(SNRMB, N), imap(SNRMB, N1)); |
---|
| 755 | setring(SNRMB); |
---|
| 756 | h = imap(SNRM1, h); |
---|
| 757 | } |
---|
| 758 | return(list(N, g, s)); |
---|
| 759 | } |
---|
| 760 | |
---|
| 761 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 762 | |
---|
| 763 | proc Factor(poly f) |
---|
[9173792] | 764 | "USAGE: Factor(f); where f is a polynomial |
---|
| 765 | PURPOSE: compute the factorization of the squarefree poly f over Q(a)[t] |
---|
| 766 | RETURN: list with two entries |
---|
| 767 | @format |
---|
| 768 | _[1] = factors (monic), first entry is the leading coefficient |
---|
| 769 | _[2] = multiplicities (not yet implemented) |
---|
| 770 | @end format |
---|
| 771 | ASSUME: basering must be the univariate polynomial ring over a field, which |
---|
[35f23d] | 772 | is Q or a simple extension of Q given by a minpoly. |
---|
[9173792] | 773 | NOTE: if basering = Q[t] then this is the built-in @code{factorize} |
---|
[4e461ff] | 774 | EXAMPLE: example Factor; shows an example |
---|
| 775 | " |
---|
| 776 | { |
---|
| 777 | def FRB = basering; |
---|
| 778 | def FRR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
| 779 | setring FRR; |
---|
[6fe3a0] | 780 | export(FRR); |
---|
[4e461ff] | 781 | poly f = imap(FRB, f); |
---|
| 782 | list result = FactorMain(f); // squarefree norm of f |
---|
| 783 | |
---|
| 784 | setring(FRB); |
---|
| 785 | list result = imap(FRR, result); |
---|
| 786 | kill(FRR); |
---|
| 787 | return(result); |
---|
| 788 | } |
---|
| 789 | example |
---|
| 790 | {"EXAMPLE:"; echo = 2; |
---|
| 791 | ring R = (0,a), x, lp; |
---|
| 792 | minpoly = a2+1; |
---|
| 793 | poly f = x4 - 1; |
---|
| 794 | list fl = Factor(f); |
---|
| 795 | fl; |
---|
| 796 | fl[1][1]*fl[1][2]*fl[1][3]*fl[1][4]*fl[1][5] - f; |
---|
| 797 | } |
---|
| 798 | |
---|
| 799 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 800 | |
---|
| 801 | proc FactorMain(poly f) |
---|
[9173792] | 802 | "USAGE: FactorMain(f); where f is a polynomial |
---|
| 803 | PURPOSE: compute the factorization of the squarefree poly f over Q(a)[t], |
---|
[d12655] | 804 | minpoly = p(a). |
---|
[9173792] | 805 | RETURN: list with 2 entries |
---|
| 806 | @format |
---|
| 807 | _[1] = factors, first is a constant |
---|
| 808 | _[2] = multiplicities (not yet implemented) |
---|
| 809 | @end format |
---|
| 810 | ASSUME: basering = Q[x,a], representing Q(a)[x]. An ideal mpoly must |
---|
| 811 | be defined, representing the minimal polynomial (it might be 0!). |
---|
[4e461ff] | 812 | EXAMPLE: example Factor; shows an example |
---|
| 813 | " |
---|
| 814 | // extend this by a squarefree factorization !! |
---|
| 815 | // multiplicities are not valid !! |
---|
| 816 | { |
---|
| 817 | int i, s; |
---|
| 818 | list normList, factorList, quo_rem; |
---|
| 819 | poly f1, h, h1, H, g, leadCoef, invCoeff; |
---|
| 820 | ideal fac1, fac2; |
---|
| 821 | map F; |
---|
| 822 | |
---|
| 823 | // if no minimal polynomial is defined then use 'factorize' |
---|
| 824 | // FactorOverQ is wrapped around 'factorize' |
---|
| 825 | |
---|
| 826 | if(mpoly[1] == 0) { |
---|
| 827 | // print(" factorize : deg = " + string(deg(f, intvec(1,0)))); |
---|
| 828 | factorList = factorize(f); // FactorOverQ(f); |
---|
| 829 | return(factorList); |
---|
| 830 | } |
---|
| 831 | |
---|
| 832 | // if mpoly != 0 and f does not contain the algebraic number, a root of |
---|
| 833 | // f might be contained in Q(a). Hence one must not use 'factorize'. |
---|
| 834 | |
---|
| 835 | fac1[1] = 1; |
---|
| 836 | fac2[1] = 1; |
---|
| 837 | normList = SQFRNormMain(f); |
---|
| 838 | // print(" factorize : deg = " + string(deg(normList[1], intvec(1,0)))); |
---|
| 839 | factorList = factorize(normList[1]); // factor squarefree norm of f over Q[x] |
---|
| 840 | g = normList[2]; |
---|
| 841 | s = normList[3]; |
---|
| 842 | F[1] = var(1) + s*var(2); // inverse transformation |
---|
| 843 | F[2] = var(2); |
---|
| 844 | fac1[1] = factorList[1][1]; |
---|
| 845 | fac2[1] = factorList[2][1]; |
---|
| 846 | for(i = 2; i <= size(factorList[1]); i = i + 1) { |
---|
| 847 | H = factorList[1][i]; |
---|
| 848 | h = EGCDMain(H, g); |
---|
| 849 | quo_rem = QuotientMain(g, h); |
---|
| 850 | g = quo_rem[1]; |
---|
| 851 | fac1[i] = SimplifyPoly(F(h)); |
---|
| 852 | fac2[i] = 1; // to be changed later |
---|
| 853 | } |
---|
| 854 | return(list(fac1, fac2)); |
---|
| 855 | } |
---|
| 856 | |
---|
| 857 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 858 | |
---|
| 859 | proc ZeroSetMain(ideal I, int primDecQ) |
---|
| 860 | "USAGE: ZeroSetMain(ideal I, int opt); ideal I, int opt |
---|
[9173792] | 861 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a simple extension |
---|
[34b0314] | 862 | of the ground field. |
---|
[4e461ff] | 863 | RETURN: list |
---|
[d12655] | 864 | - 'f' is the polynomial f in Q(a) (a' being substituted by newA) |
---|
[4e461ff] | 865 | _[1] = zero-set (list), is the list of the zero-set of the ideal I, |
---|
| 866 | each entry is an ideal. |
---|
[34b0314] | 867 | _[2] = 'newA'; if the ground field is Q(a') and the extension field |
---|
[4e461ff] | 868 | is Q(a), then 'newA' is the representation of a' in Q(a). |
---|
| 869 | If the basering contains a parameter 'a' and the minpoly |
---|
| 870 | remains unchanged then 'newA' = 'a'. If the basering does not |
---|
| 871 | contain a parameter then 'newA' = 'a' (default). |
---|
[d12655] | 872 | _[3] = 'mpoly' (ideal), the minimal polynomial of the simple extension |
---|
| 873 | of the ground field. |
---|
| 874 | ASSUME: basering = K[x_1,x_2,...,x_n] where K = Q or a simple extension of Q |
---|
| 875 | given by a minpoly; dim(I) = 0. |
---|
[4e461ff] | 876 | NOTE: opt = 0 no primary decomposition |
---|
[d12655] | 877 | opt > 0 use a primary decomposition |
---|
[4e461ff] | 878 | EXAMPLE: example ZeroSet; shows an example |
---|
| 879 | " |
---|
| 880 | { |
---|
| 881 | // main work is done in ZeroSetMainWork, here the zero-set of each ideal from the |
---|
| 882 | // primary decompostion is coputed by menas of ZeroSetMainWork, and then the |
---|
| 883 | // minpoly and the parameter representing the algebraic extension are |
---|
| 884 | // transformed according to 'newA', i.e., only bookeeping is done. |
---|
| 885 | |
---|
[6fe3a0] | 886 | def altring=basering; |
---|
[4e461ff] | 887 | int i, j, n, noMP, dbPrt; |
---|
| 888 | intvec w; |
---|
| 889 | list currentSol, result, idealList, primDecList, zeroSet; |
---|
| 890 | ideal J; |
---|
| 891 | map Fa; |
---|
| 892 | poly newA, oldMinPoly; |
---|
| 893 | |
---|
| 894 | dbPrt = printlevel-voice+2; |
---|
| 895 | dbprint(dbPrt, "ZeroSet of " + string(I) + ", minpoly = " + string(minpoly)); |
---|
| 896 | |
---|
| 897 | n = nvars(basering) - 1; |
---|
| 898 | for(i = 1; i <= n; i++) { w[i] = 1;} |
---|
| 899 | w[n + 1] = 0; |
---|
| 900 | |
---|
| 901 | if(primDecQ == 0) { return(ZeroSetMainWork(I, w, 0)); } |
---|
| 902 | |
---|
| 903 | newA = var(n + 1); |
---|
| 904 | if(mpoly[1] == 0) { noMP = 1;} |
---|
| 905 | else {noMP = 0;} |
---|
| 906 | |
---|
| 907 | primDecList = primdecGTZ(I); // primary decomposition |
---|
| 908 | dbprint(dbPrt, "primary decomposition consists of " + string(size(primDecList)) + " primary ideals "); |
---|
| 909 | // idealList = PDSort(idealList); // high degrees first |
---|
| 910 | |
---|
| 911 | for(i = 1; i <= size(primDecList); i = i + 1) { |
---|
| 912 | idealList[i] = primDecList[i][2]; // use prime component |
---|
| 913 | dbprint(dbPrt, string(i) + " " + string(idealList[i])); |
---|
| 914 | } |
---|
| 915 | |
---|
| 916 | // compute the zero-set of each primary ideal and join them. |
---|
[34b0314] | 917 | // If necessary, change the ground field and transform the zero-set |
---|
[4e461ff] | 918 | |
---|
| 919 | dbprint(dbPrt, " |
---|
| 920 | find the zero-set of each primary ideal, form the union |
---|
| 921 | and keep track of the minimal polynomials "); |
---|
| 922 | |
---|
| 923 | for(i = 1; i <= size(idealList); i = i + 1) { |
---|
| 924 | J = idealList[i]; |
---|
| 925 | idealList[i] = 0; |
---|
| 926 | oldMinPoly = mpoly[1]; |
---|
| 927 | dbprint(dbPrt, " ideal#" + string(i) + " of " + string(size(idealList)) + " = " + string(J)); |
---|
| 928 | currentSol = ZeroSetMainWork(J, w, 0); |
---|
| 929 | |
---|
| 930 | if(oldMinPoly != currentSol[3]) { // change minpoly and transform solutions |
---|
| 931 | dbprint(dbPrt, " change minpoly to " + string(currentSol[3][1])); |
---|
| 932 | dbprint(dbPrt, " new representation of algebraic number = " + string(currentSol[2])); |
---|
| 933 | if(!noMP) { // transform the algebraic number a |
---|
| 934 | Fa = basering, maxideal(1); |
---|
| 935 | Fa[n + 1] = currentSol[2]; |
---|
| 936 | newA = SimplifyPoly(Fa(newA)); // new representation of a |
---|
| 937 | if(size(zeroSet) > 0) {zeroSet = SimplifyZeroset(Fa(zeroSet)); } |
---|
| 938 | if(i < size(idealList)) { idealList = SimplifyZeroset(Fa(idealList)); } |
---|
| 939 | } |
---|
| 940 | else { noMP = 0;} |
---|
| 941 | } |
---|
| 942 | zeroSet = zeroSet + currentSol[1]; // add new elements |
---|
| 943 | } |
---|
| 944 | return(list(zeroSet, newA, mpoly)); |
---|
| 945 | } |
---|
| 946 | |
---|
| 947 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 948 | |
---|
| 949 | proc ZeroSetMainWork(ideal id, intvec wt, int sVars) |
---|
| 950 | "USAGE: ZeroSetMainWork(I, wt, sVars); |
---|
[9173792] | 951 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a finite extension |
---|
[34b0314] | 952 | of the ground field (without multiplicities). |
---|
[4e461ff] | 953 | RETURN: list, all entries are polynomials |
---|
| 954 | _[1] = zeros, each entry is an ideal |
---|
[34b0314] | 955 | _[2] = newA; if the ground field is Q(a') this is the rep. of a' w.r.t. a |
---|
| 956 | _[3] = minpoly of the algebraic extension of the ground field (ideal) |
---|
[4e461ff] | 957 | _[4] = name of algebraic number (default = 'a') |
---|
| 958 | ASSUME: basering = Q[x_1,x_2,...,x_n,a] |
---|
[9173792] | 959 | ideal mpoly must be defined, it might be 0! |
---|
[4e461ff] | 960 | NOTE: might change 'mpoly' !! |
---|
| 961 | EXAMPLE: example IdealSolve; shows an example |
---|
| 962 | " |
---|
| 963 | { |
---|
[6fe3a0] | 964 | def altring=basering; |
---|
[4e461ff] | 965 | int i, j, k, nrSols, n, noMP; |
---|
| 966 | ideal I, generators, gens, solid, partsolid; |
---|
| 967 | list linSol, linearSolution, nLinSol, nonlinSolutions, partSol, sol, solutions, result; |
---|
| 968 | list linIndex, nlinIndex, index; |
---|
| 969 | map Fa, Fsubs; |
---|
| 970 | poly oldMinPoly, newA; |
---|
| 971 | |
---|
| 972 | if(mpoly[1] == 0) { noMP = 1;} |
---|
| 973 | else { noMP = 0;} |
---|
| 974 | n = nvars(basering) - 1; |
---|
| 975 | newA = var(n + 1); |
---|
| 976 | |
---|
| 977 | I = std(id); |
---|
| 978 | |
---|
| 979 | // find linear solutions of univariate generators |
---|
| 980 | |
---|
| 981 | linSol = LinearZeroSetMain(I, wt); |
---|
| 982 | generators = linSol[3]; // they are a standardbasis |
---|
| 983 | linIndex = linSol[2]; |
---|
| 984 | linearSolution = linSol[1]; |
---|
| 985 | if(size(linIndex) + sVars == n) { // all variables solved |
---|
| 986 | solid = SubsMapIdeal(linearSolution, linIndex, 0); |
---|
| 987 | result[1] = list(solid); |
---|
| 988 | result[2] = var(n + 1); |
---|
| 989 | result[3] = mpoly; |
---|
| 990 | return(result); |
---|
| 991 | } |
---|
| 992 | |
---|
| 993 | // find roots of the nonlinear univariate polynomials of generators |
---|
| 994 | // if necessary, transform linear solutions w.r.t. newA |
---|
| 995 | |
---|
| 996 | oldMinPoly = mpoly[1]; |
---|
| 997 | nLinSol = NonLinearZeroSetMain(generators, wt); // find solutions of univariate generators |
---|
| 998 | nonlinSolutions = nLinSol[1]; // store solutions |
---|
| 999 | nlinIndex = nLinSol[4]; // and index of solved variables |
---|
| 1000 | generators = nLinSol[5]; // new generators |
---|
| 1001 | |
---|
| 1002 | // change minpoly if necessary and transform the ideal and the partial solutions |
---|
| 1003 | |
---|
| 1004 | if(oldMinPoly != nLinSol[3]) { |
---|
| 1005 | newA = nLinSol[2]; |
---|
| 1006 | if(!noMP && size(linearSolution) > 0) { // transform the algebraic number a |
---|
| 1007 | Fa = basering, maxideal(1); |
---|
| 1008 | Fa[n + 1] = newA; |
---|
| 1009 | linearSolution = SimplifyData(Fa(linearSolution)); // ... |
---|
| 1010 | } |
---|
| 1011 | } |
---|
| 1012 | |
---|
| 1013 | // check if all variables are solved. |
---|
| 1014 | |
---|
| 1015 | if(size(linIndex) + size(nlinIndex) == n - sVars) { |
---|
| 1016 | solutions = MergeSolutions(linearSolution, linIndex, nonlinSolutions, nlinIndex, list(), n); |
---|
| 1017 | } |
---|
| 1018 | |
---|
| 1019 | else { |
---|
| 1020 | |
---|
| 1021 | // some variables are not solved. |
---|
| 1022 | // substitute each partial solution in generators and find the |
---|
| 1023 | // zero set of the resulting ideal by recursive application |
---|
| 1024 | // of ZeroSetMainWork ! |
---|
| 1025 | |
---|
| 1026 | index = linIndex + nlinIndex; |
---|
| 1027 | nrSols = 0; |
---|
| 1028 | for(i = 1; i <= size(nonlinSolutions); i = i + 1) { |
---|
| 1029 | sol = linearSolution + nonlinSolutions[i]; |
---|
| 1030 | solid = SubsMapIdeal(sol, index, 1); |
---|
| 1031 | Fsubs = basering, solid; |
---|
| 1032 | gens = std(SimplifyData(Fsubs(generators))); // substitute partial solution |
---|
| 1033 | oldMinPoly = mpoly[1]; |
---|
| 1034 | partSol = ZeroSetMainWork(gens, wt, size(index) + sVars); |
---|
| 1035 | |
---|
| 1036 | if(oldMinPoly != partSol[3]) { // minpoly has changed |
---|
| 1037 | Fa = basering, maxideal(1); |
---|
| 1038 | Fa[n + 1] = partSol[2]; // a -> p(a), representation of a w.r.t. new minpoly |
---|
| 1039 | newA = reduce(Fa(newA), mpoly); |
---|
| 1040 | generators = std(SimplifyData(Fa(generators))); |
---|
| 1041 | if(size(linearSolution) > 0) { linearSolution = SimplifyData(Fa(linearSolution));} |
---|
| 1042 | if(size(nonlinSolutions) > 0) { |
---|
| 1043 | nonlinSolutions = SimplifyZeroset(Fa(nonlinSolutions)); |
---|
| 1044 | } |
---|
| 1045 | sol = linearSolution + nonlinSolutions[i]; |
---|
| 1046 | } |
---|
| 1047 | |
---|
| 1048 | for(j = 1; j <= size(partSol[1]); j++) { // for all partial solutions |
---|
| 1049 | partsolid = partSol[1][j]; |
---|
| 1050 | for(k = 1; k <= size(index); k++) { |
---|
| 1051 | partsolid[index[k]] = sol[k]; |
---|
| 1052 | } |
---|
| 1053 | nrSols++; |
---|
| 1054 | solutions[nrSols] = partsolid; |
---|
| 1055 | } |
---|
| 1056 | } |
---|
| 1057 | |
---|
| 1058 | } // end else |
---|
| 1059 | return(list(solutions, newA, mpoly)); |
---|
| 1060 | } |
---|
| 1061 | |
---|
| 1062 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1063 | |
---|
| 1064 | proc LinearZeroSetMain(ideal I, intvec wt) |
---|
| 1065 | "USAGE: LinearZeroSetMain(I, wt) |
---|
| 1066 | PURPOSE: solve the univariate linear polys in I |
---|
| 1067 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
| 1068 | RETURN: list |
---|
| 1069 | _[1] = partial solution of I |
---|
| 1070 | _[2] = index of solved vars |
---|
| 1071 | _[3] = new generators (standardbasis) |
---|
| 1072 | " |
---|
| 1073 | { |
---|
[6fe3a0] | 1074 | def altring=basering; |
---|
[4e461ff] | 1075 | int i, ok, n, found, nrSols; |
---|
| 1076 | ideal generators, newGens; |
---|
| 1077 | list result, index, totalIndex, vars, sol, temp; |
---|
| 1078 | map F; |
---|
| 1079 | poly f; |
---|
| 1080 | |
---|
| 1081 | result[1] = index; // sol[1] should be the empty list |
---|
| 1082 | n = nvars(basering) - 1; |
---|
| 1083 | generators = I; // might be wrong, use index ! |
---|
| 1084 | ok = 1; |
---|
| 1085 | nrSols = 0; |
---|
| 1086 | while(ok) { |
---|
| 1087 | found = 0; |
---|
| 1088 | for(i = 1; i <= size(generators); i = i + 1) { |
---|
| 1089 | f = generators[i]; |
---|
| 1090 | vars = Variables(f, n); |
---|
| 1091 | if(size(vars) == 1 && deg(f, wt) == 1) { // univariate,linear |
---|
| 1092 | nrSols++; found++; |
---|
| 1093 | index[nrSols] = vars[1]; |
---|
| 1094 | sol[nrSols] = var(vars[1]) - MultPolys(InvertNumberMain(LeadTerm(f, vars[1])[3]), f); |
---|
| 1095 | } |
---|
| 1096 | } |
---|
| 1097 | if(found > 0) { |
---|
| 1098 | F = basering, SubsMapIdeal(sol, index, 1); |
---|
| 1099 | newGens = std(SimplifyData(F(generators))); // substitute, simplify alg. number |
---|
| 1100 | if(size(newGens) == 0) {ok = 0;} |
---|
| 1101 | generators = newGens; |
---|
| 1102 | } |
---|
| 1103 | else { |
---|
| 1104 | ok = 0; |
---|
| 1105 | } |
---|
| 1106 | } |
---|
| 1107 | if(nrSols > 0) { result[1] = sol;} |
---|
| 1108 | result[2] = index; |
---|
| 1109 | result[3] = generators; |
---|
| 1110 | return(result); |
---|
| 1111 | } |
---|
| 1112 | |
---|
| 1113 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1114 | |
---|
| 1115 | proc NonLinearZeroSetMain(ideal I, intvec wt) |
---|
| 1116 | "USAGE: ZeroSetMainWork(I, wt, sVars); |
---|
[9173792] | 1117 | PURPOSE: solves the (nonlinear) univariate polynomials in I |
---|
[34b0314] | 1118 | of the ground field (without multiplicities). |
---|
[4e461ff] | 1119 | RETURN: list, all entries are polynomials |
---|
| 1120 | _[1] = list of solutions |
---|
| 1121 | _[2] = newA |
---|
| 1122 | _[3] = minpoly |
---|
| 1123 | _[4] - index of solved variables |
---|
| 1124 | _[5] = new representation of I |
---|
| 1125 | ASSUME: basering = Q[x_1,x_2,...,x_n,a], ideal 'mpoly' must be defined, |
---|
| 1126 | it might be 0 ! |
---|
| 1127 | NOTE: might change 'mpoly' !! |
---|
| 1128 | " |
---|
| 1129 | { |
---|
| 1130 | int i, nrSols, ok, n; |
---|
| 1131 | ideal generators; |
---|
| 1132 | list result, sols, index, vars, partSol; |
---|
| 1133 | map F; |
---|
| 1134 | poly f, newA; |
---|
| 1135 | string ringSTR; |
---|
| 1136 | |
---|
| 1137 | def NLZR = basering; |
---|
[6fe3a0] | 1138 | export(NLZR); |
---|
[4e461ff] | 1139 | |
---|
| 1140 | n = nvars(basering) - 1; |
---|
| 1141 | |
---|
| 1142 | generators = I; |
---|
| 1143 | newA = var(n + 1); |
---|
| 1144 | result[2] = newA; // default |
---|
| 1145 | nrSols = 0; |
---|
| 1146 | ok = 1; |
---|
| 1147 | i = 1; |
---|
| 1148 | while(ok) { |
---|
| 1149 | |
---|
| 1150 | // test if the i-th generator of I is univariate |
---|
| 1151 | |
---|
| 1152 | f = generators[i]; |
---|
| 1153 | vars = Variables(f, n); |
---|
| 1154 | if(size(vars) == 1) { |
---|
| 1155 | generators[i] = 0; |
---|
| 1156 | generators = simplify(generators, 2); // remove 0 |
---|
| 1157 | nrSols++; |
---|
| 1158 | index[nrSols] = vars[1]; // store index of solved variable |
---|
| 1159 | |
---|
| 1160 | // create univariate ring |
---|
| 1161 | |
---|
| 1162 | ringSTR = "ring RIS1 = 0, (" + string(var(vars[1])) + ", " + string(var(n+1)) + "), lp;"; |
---|
| 1163 | execute(ringSTR); |
---|
| 1164 | ideal mpoly = std(imap(NLZR, mpoly)); |
---|
| 1165 | list roots; |
---|
| 1166 | poly f = imap(NLZR, f); |
---|
[6fe3a0] | 1167 | export(RIS1); |
---|
[4e461ff] | 1168 | export(mpoly); |
---|
| 1169 | roots = RootsMain(f); |
---|
[6fe3a0] | 1170 | |
---|
[4e461ff] | 1171 | // get "old" basering with new minpoly |
---|
| 1172 | |
---|
| 1173 | setring(NLZR); |
---|
| 1174 | partSol = imap(RIS1, roots); |
---|
| 1175 | kill(RIS1); |
---|
| 1176 | if(mpoly[1] != partSol[3]) { // change minpoly |
---|
| 1177 | mpoly = std(partSol[3]); |
---|
| 1178 | F = NLZR, maxideal(1); |
---|
| 1179 | F[n + 1] = partSol[2]; |
---|
| 1180 | if(size(sols) > 0) {sols = SimplifyZeroset(F(sols)); } |
---|
| 1181 | newA = reduce(F(newA), mpoly); // normal form |
---|
| 1182 | result[2] = newA; |
---|
| 1183 | generators = SimplifyData(F(generators)); // does not remove 0's |
---|
| 1184 | } |
---|
| 1185 | sols = ExtendSolutions(sols, partSol[1]); |
---|
| 1186 | } // end univariate |
---|
| 1187 | else { |
---|
| 1188 | i = i + 1; |
---|
| 1189 | } |
---|
| 1190 | if(i > size(generators)) { ok = 0;} |
---|
| 1191 | } |
---|
| 1192 | result[1] = sols; |
---|
| 1193 | result[3] = mpoly; |
---|
| 1194 | result[4] = index; |
---|
| 1195 | result[5] = std(generators); |
---|
| 1196 | |
---|
| 1197 | kill(NLZR); |
---|
| 1198 | return(result); |
---|
| 1199 | } |
---|
| 1200 | |
---|
| 1201 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1202 | |
---|
| 1203 | static proc ExtendSolutions(list solutions, list newSolutions) |
---|
| 1204 | "USAGE: ExtendSolutions(sols, newSols); list sols, newSols; |
---|
[9173792] | 1205 | PURPOSE: extend the entries of 'sols' by the entries of 'newSols', |
---|
[4e461ff] | 1206 | each entry of 'newSols' is a number. |
---|
| 1207 | RETURN: list |
---|
| 1208 | ASSUME: basering = Q[x_1,...,x_n,a], ideal 'mpoly' must be defined, |
---|
| 1209 | it might be 0 ! |
---|
| 1210 | NOTE: used by 'NonLinearZeroSetMain' |
---|
| 1211 | " |
---|
| 1212 | { |
---|
| 1213 | int i, j, k, n, nrSols; |
---|
| 1214 | list newSols, temp; |
---|
| 1215 | |
---|
| 1216 | nrSols = size(solutions); |
---|
| 1217 | if(nrSols > 0) {n = size(solutions[1]);} |
---|
| 1218 | else { |
---|
| 1219 | n = 0; |
---|
| 1220 | nrSols = 1; |
---|
| 1221 | } |
---|
| 1222 | k = 0; |
---|
| 1223 | for(i = 1; i <= nrSols; i++) { |
---|
| 1224 | for(j = 1; j <= size(newSolutions); j++) { |
---|
| 1225 | k++; |
---|
| 1226 | if(n == 0) { temp[1] = newSolutions[j];} |
---|
| 1227 | else { |
---|
| 1228 | temp = solutions[i]; |
---|
| 1229 | temp[n + 1] = newSolutions[j]; |
---|
| 1230 | } |
---|
| 1231 | newSols[k] = temp; |
---|
| 1232 | } |
---|
| 1233 | } |
---|
| 1234 | return(newSols); |
---|
| 1235 | } |
---|
| 1236 | |
---|
| 1237 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1238 | |
---|
| 1239 | static proc MergeSolutions(list sol1, list index1, list sol2, list index2) |
---|
| 1240 | "USAGE: MergeSolutions(sol1, index1, sol2, index2); all parameters are lists |
---|
| 1241 | RETURN: list |
---|
| 1242 | PURPOSE: create a list of solutions of size n, each entry of 'sol2' must |
---|
[d12655] | 1243 | have size n. 'sol1' is one partial solution (from 'LinearZeroSetMain') |
---|
| 1244 | 'sol2' is a list of partial solutions (from 'NonLinearZeroSetMain') |
---|
[4e461ff] | 1245 | ASSUME: 'sol2' is not empty |
---|
[d12655] | 1246 | NOTE: used by 'ZeroSetMainWork' |
---|
[4e461ff] | 1247 | { |
---|
| 1248 | int i, j, k, m; |
---|
| 1249 | ideal sol; |
---|
| 1250 | list newSols; |
---|
| 1251 | |
---|
| 1252 | m = 0; |
---|
| 1253 | for(i = 1; i <= size(sol2); i++) { |
---|
| 1254 | m++; |
---|
| 1255 | newSols[m] = SubsMapIdeal(sol1 + sol2[i], index1 + index2, 0); |
---|
| 1256 | } |
---|
| 1257 | return(newSols); |
---|
| 1258 | } |
---|
| 1259 | |
---|
| 1260 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1261 | |
---|
| 1262 | static proc SubsMapIdeal(list sol, list index, int opt) |
---|
| 1263 | "USAGE: SubsMapIdeal(sol,index,opt); list sol, index; int opt; |
---|
[9173792] | 1264 | PURPOSE: built an ideal I as follows. |
---|
[4e461ff] | 1265 | if i is contained in 'index' then set I[i] = sol[i] |
---|
| 1266 | if i is not contained in 'index' then |
---|
| 1267 | - opt = 0: set I[i] = 0 |
---|
| 1268 | - opt = 1: set I[i] = var(i) |
---|
| 1269 | if opt = 1 and n = nvars(basering) then set I[n] = var(n). |
---|
| 1270 | RETURN: ideal |
---|
| 1271 | ASSUME: size(sol) = size(index) <= nvars(basering) |
---|
| 1272 | " |
---|
| 1273 | { |
---|
| 1274 | int k = 0; |
---|
| 1275 | ideal I; |
---|
| 1276 | for(int i = 1; i <= nvars(basering) - 1; i = i + 1) { // built subs. map |
---|
| 1277 | if(ContainedQ(index, i)) { |
---|
| 1278 | k++; |
---|
| 1279 | I[index[k]] = sol[k]; |
---|
| 1280 | } |
---|
| 1281 | else { |
---|
| 1282 | if(opt) { I[i] = var(i); } |
---|
| 1283 | else { I[i] = 0; } |
---|
| 1284 | } |
---|
| 1285 | } |
---|
| 1286 | if(opt) {I[nvars(basering)] = var(nvars(basering));} |
---|
| 1287 | return(I); |
---|
| 1288 | } |
---|
| 1289 | |
---|
| 1290 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1291 | |
---|
| 1292 | proc SimplifyZeroset(data) |
---|
| 1293 | "USAGE: SimplifyZeroset(data); list data |
---|
[9173792] | 1294 | PURPOSE: reduce the entries of the elements of 'data' w.r.t. the ideal 'mpoly' |
---|
[4e461ff] | 1295 | 'data' is a list of ideals/lists. |
---|
| 1296 | RETURN: list |
---|
| 1297 | ASSUME: basering = Q[x_1,...,x_n,a], order = lp |
---|
| 1298 | 'data' is a list of ideals |
---|
| 1299 | ideal 'mpoly' must be defined, it might be 0 ! |
---|
| 1300 | " |
---|
| 1301 | { |
---|
| 1302 | int i; |
---|
| 1303 | list result; |
---|
| 1304 | |
---|
| 1305 | for(i = 1; i <= size(data); i++) { |
---|
| 1306 | result[i] = SimplifyData(data[i]); |
---|
| 1307 | } |
---|
| 1308 | return(result); |
---|
| 1309 | } |
---|
| 1310 | |
---|
| 1311 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1312 | |
---|
| 1313 | proc Variables(poly f, int n) |
---|
| 1314 | "USAGE: Variables(f,n); poly f; int n; |
---|
[9173792] | 1315 | PURPOSE: list of variables among var(1),...,var(n) which occur in f. |
---|
[4e461ff] | 1316 | RETURN: list |
---|
| 1317 | ASSUME: n <= nvars(basering) |
---|
| 1318 | " |
---|
| 1319 | { |
---|
| 1320 | int i, nrV; |
---|
| 1321 | list index; |
---|
| 1322 | |
---|
| 1323 | nrV = 0; |
---|
| 1324 | for(i = 1; i <= n; i = i + 1) { |
---|
| 1325 | if(diff(f, var(i)) != 0) { nrV++; index[nrV] = i; } |
---|
| 1326 | } |
---|
| 1327 | return(index); |
---|
| 1328 | } |
---|
| 1329 | |
---|
| 1330 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1331 | |
---|
| 1332 | proc ContainedQ(data, f, list #) |
---|
[9173792] | 1333 | "USAGE: ContainedQ(data, f [, opt]); data=list; f=any type, opt=integer |
---|
| 1334 | PURPOSE: test if f is an element of data. |
---|
[4e461ff] | 1335 | RETURN: int |
---|
[9173792] | 1336 | 0 if f not contained in data |
---|
| 1337 | 1 if f contained in data |
---|
| 1338 | OPTIONS: opt = 0 : use '==' for comparing f with elements from data@* |
---|
| 1339 | opt = 1 : use @code{SameQ} for comparing f with elements from data |
---|
[4e461ff] | 1340 | " |
---|
| 1341 | { |
---|
| 1342 | int opt, i, found; |
---|
| 1343 | if(size(#) > 0) { opt = #[1];} |
---|
| 1344 | else { opt = 0; } |
---|
| 1345 | i = 1; |
---|
| 1346 | found = 0; |
---|
| 1347 | |
---|
| 1348 | while((!found) && (i <= size(data))) { |
---|
| 1349 | if(opt == 0) { |
---|
| 1350 | if(f == data[i]) { found = 1;} |
---|
| 1351 | else {i = i + 1;} |
---|
| 1352 | } |
---|
| 1353 | else { |
---|
| 1354 | if(SameQ(f, data[i])) { found = 1;} |
---|
| 1355 | else {i = i + 1;} |
---|
| 1356 | } |
---|
| 1357 | } |
---|
| 1358 | return(found); |
---|
| 1359 | } |
---|
| 1360 | |
---|
| 1361 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 1362 | |
---|
| 1363 | proc SameQ(a, b) |
---|
[9173792] | 1364 | "USAGE: SameQ(a, b); a,b=list/intvec |
---|
| 1365 | PURPOSE: test a == b elementwise, i.e., a[i] = b[i]. |
---|
[4e461ff] | 1366 | RETURN: int |
---|
| 1367 | 0 if a != b |
---|
| 1368 | 1 if a == b |
---|
| 1369 | " |
---|
| 1370 | { |
---|
| 1371 | if(typeof(a) == typeof(b)) { |
---|
| 1372 | if(typeof(a) == "list" || typeof(a) == "intvec") { |
---|
| 1373 | if(size(a) == size(b)) { |
---|
| 1374 | int i = 1; |
---|
| 1375 | int ok = 1; |
---|
| 1376 | while(ok && (i <= size(a))) { |
---|
| 1377 | if(a[i] == b[i]) { i = i + 1;} |
---|
| 1378 | else {ok = 0;} |
---|
| 1379 | } |
---|
| 1380 | return(ok); |
---|
| 1381 | } |
---|
| 1382 | else { return(0); } |
---|
| 1383 | } |
---|
| 1384 | else { return(a == b);} |
---|
| 1385 | } |
---|
| 1386 | else { return(0);} |
---|
| 1387 | } |
---|
| 1388 | |
---|
| 1389 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1390 | |
---|
| 1391 | static proc SimplifyPoly(poly f) |
---|
| 1392 | "USAGE: SimplifyPoly(f); poly f |
---|
[9173792] | 1393 | PURPOSE: reduces the coefficients of f w.r.t. the ideal 'moly' if they contain |
---|
[4e461ff] | 1394 | the algebraic number 'a'. |
---|
| 1395 | RETURN: poly |
---|
| 1396 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
| 1397 | ideal mpoly must be defined, it might be 0 ! |
---|
| 1398 | " |
---|
| 1399 | { |
---|
| 1400 | matrix coMx; |
---|
| 1401 | poly f1, vp; |
---|
| 1402 | |
---|
| 1403 | vp = 1; |
---|
| 1404 | for(int i = 1; i < nvars(basering); i++) { vp = vp * var(i);} |
---|
| 1405 | |
---|
| 1406 | coMx = coef(f, vp); |
---|
| 1407 | f1 = 0; |
---|
| 1408 | for(i = 1; i <= ncols(coMx); i++) { |
---|
| 1409 | f1 = f1 + coMx[1, i] * reduce(coMx[2, i], mpoly); |
---|
| 1410 | } |
---|
| 1411 | return(f1); |
---|
| 1412 | } |
---|
| 1413 | |
---|
| 1414 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1415 | |
---|
| 1416 | static proc SimplifyData(data) |
---|
| 1417 | "USAGE: SimplifyData(data); ideal/list data; |
---|
[9173792] | 1418 | PURPOSE: reduces the entries of 'data' w.r.t. the ideal 'mpoly' if they contain |
---|
[d12655] | 1419 | the algebraic number 'a' |
---|
[4e461ff] | 1420 | RETURN: ideal/list |
---|
| 1421 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
| 1422 | ideal 'mpoly' must be defined, it might be 0 ! |
---|
| 1423 | " |
---|
| 1424 | { |
---|
[6fe3a0] | 1425 | def altring=basering; |
---|
[4e461ff] | 1426 | int n; |
---|
| 1427 | poly f; |
---|
| 1428 | |
---|
| 1429 | if(typeof(data) == "ideal") { n = ncols(data); } |
---|
| 1430 | else { n = size(data);} |
---|
| 1431 | |
---|
| 1432 | for(int i = 1; i <= n; i++) { |
---|
| 1433 | f = data[i]; |
---|
| 1434 | data[i] = SimplifyPoly(f); |
---|
| 1435 | } |
---|
| 1436 | return(data); |
---|
| 1437 | } |
---|
| 1438 | |
---|
| 1439 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1440 | |
---|
| 1441 | static proc TransferRing(R) |
---|
| 1442 | "USAGE: TransferRing(R); |
---|
[9173792] | 1443 | PURPOSE: creates a new ring containing the same variables as R, but without |
---|
[4e461ff] | 1444 | parameters. If R contains a parameter then this parameter is added |
---|
[d12655] | 1445 | as the last variable and 'minpoly' is represented by the ideal 'mpoly' |
---|
| 1446 | If the basering does not contain a parameter then 'a' is added and |
---|
| 1447 | 'mpoly' = 0. |
---|
[4e461ff] | 1448 | RETURN: ring |
---|
| 1449 | ASSUME: R = K[x_1,...,x_n] where K = Q or K = Q(a). |
---|
| 1450 | NOTE: Creates the ring needed for all prodecures with name 'proc-name'Main |
---|
| 1451 | " |
---|
| 1452 | { |
---|
[6fe3a0] | 1453 | def altring=basering; |
---|
[4e461ff] | 1454 | string ringSTR, parName, minPoly; |
---|
| 1455 | |
---|
| 1456 | setring(R); |
---|
| 1457 | |
---|
| 1458 | if(npars(basering) == 0) { |
---|
| 1459 | parName = "a"; |
---|
| 1460 | minPoly = "0"; |
---|
| 1461 | } |
---|
| 1462 | else { |
---|
| 1463 | parName = parstr(basering); |
---|
| 1464 | minPoly = string(minpoly); |
---|
| 1465 | } |
---|
| 1466 | ringSTR = "ring TR = 0, (" + varstr(basering) + "," + parName + "), lp;"; |
---|
| 1467 | |
---|
| 1468 | execute(ringSTR); |
---|
| 1469 | execute("ideal mpoly = std(" + minPoly + ");"); |
---|
| 1470 | export(mpoly); |
---|
[6fe3a0] | 1471 | setring altring; |
---|
[4e461ff] | 1472 | return(TR); |
---|
| 1473 | } |
---|
| 1474 | |
---|
| 1475 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1476 | |
---|
| 1477 | static proc NewBaseRing() |
---|
| 1478 | "USAGE: NewBaseRing(); |
---|
[9173792] | 1479 | PURPOSE: creates a new ring, the last variable is added as a parameter. |
---|
[4e461ff] | 1480 | minpoly is set to mpoly[1]. |
---|
| 1481 | RETURN: ring |
---|
| 1482 | ASSUME: basering = Q[x_1,...,x_n, a], 'mpoly' must be defined |
---|
| 1483 | " |
---|
| 1484 | { |
---|
| 1485 | int n = nvars(basering); |
---|
| 1486 | int MP; |
---|
| 1487 | string ringSTR, parName, varString; |
---|
| 1488 | |
---|
| 1489 | def BR = basering; |
---|
| 1490 | if(mpoly[1] != 0) { |
---|
| 1491 | parName = "(0, " + string(var(n)) + ")"; |
---|
| 1492 | MP = 1; |
---|
| 1493 | } |
---|
| 1494 | else { |
---|
| 1495 | parName = "0"; |
---|
| 1496 | MP = 0; |
---|
| 1497 | } |
---|
| 1498 | |
---|
| 1499 | |
---|
| 1500 | for(int i = 1; i < n - 1; i++) { |
---|
| 1501 | varString = varString + string(var(i)) + ","; |
---|
| 1502 | } |
---|
| 1503 | varString = varString + string(var(n-1)); |
---|
| 1504 | |
---|
| 1505 | ringSTR = "ring TR = " + parName + ", (" + varString + "), lp;"; |
---|
| 1506 | execute(ringSTR); |
---|
| 1507 | if(MP) { minpoly = number(imap(BR, mpoly)[1]); } |
---|
[6fe3a0] | 1508 | setring BR; |
---|
[4e461ff] | 1509 | return(TR); |
---|
| 1510 | } |
---|
| 1511 | |
---|
| 1512 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1513 | |
---|
| 1514 | /* |
---|
| 1515 | Examples: |
---|
| 1516 | |
---|
| 1517 | |
---|
| 1518 | // order = 20; |
---|
| 1519 | ring S1 = 0, (s(1..3)), lp; |
---|
| 1520 | ideal I = s(2)*s(3), s(1)^2*s(2)+s(1)^2*s(3)-1, s(1)^2*s(3)^2-s(3), s(2)^4-s(3)^4+s(1)^2, s(1)^4+s(2)^3-s(3)^3, s(3)^5-s(1)^2*s(3); |
---|
[d12655] | 1521 | ideal mpoly = std(0); |
---|
[4e461ff] | 1522 | |
---|
| 1523 | // order = 10 |
---|
| 1524 | ring S2 = 0, (s(1..5)), lp; |
---|
| 1525 | ideal I = s(2)+s(3)-s(5), s(4)^2-s(5), s(1)*s(5)+s(3)*s(4)-s(4)*s(5), s(1)*s(4)+s(3)-s(5), s(3)^2-2*s(3)*s(5), s(1)*s(3)-s(1)*s(5)+s(4)*s(5), s(1)^2+s(4)^2-2*s(5), -s(1)+s(5)^3, s(3)*s(5)^2+s(4)-s(5)^3, s(1)*s(5)^2-1; |
---|
[d12655] | 1526 | ideal mpoly = std(0); |
---|
[4e461ff] | 1527 | |
---|
| 1528 | //order = 126 |
---|
| 1529 | ring S3 = 0, (s(1..5)), lp; |
---|
| 1530 | ideal I = s(3)*s(4), s(2)*s(4), s(1)*s(3), s(1)*s(2), s(3)^3+s(4)^3-1, s(2)^3+s(4)^3-1, s(1)^3-s(4)^3, s(4)^4-s(4), s(1)*s(4)^3-s(1), s(5)^7-1; |
---|
[d12655] | 1531 | ideal mpoly = std(0); |
---|
[4e461ff] | 1532 | |
---|
| 1533 | // order = 192 |
---|
| 1534 | ring S4 = 0, (s(1..4)), lp; |
---|
| 1535 | ideal I = s(2)*s(3)^2*s(4)+s(1)*s(3)*s(4)^2, s(2)^2*s(3)*s(4)+s(1)*s(2)*s(4)^2, s(1)*s(3)^3+s(2)*s(4)^3, s(1)*s(2)*s(3)^2+s(1)^2*s(3)*s(4), s(1)^2*s(3)^2-s(2)^2*s(4)^2, s(1)*s(2)^2*s(3)+s(1)^2*s(2)*s(4), s(1)^3*s(3)+s(2)^3*s(4), s(2)^4-s(3)^4, s(1)*s(2)^3+s(3)*s(4)^3, s(1)^2*s(2)^2-s(3)^2*s(4)^2, s(1)^3*s(2)+s(3)^3*s(4), s(1)^4-s(4)^4, s(3)^5*s(4)-s(3)*s(4)^5, s(3)^8+14*s(3)^4*s(4)^4+s(4)^8-1, 15*s(2)*s(3)*s(4)^7-s(1)*s(4)^8+s(1), 15*s(3)^4*s(4)^5+s(4)^9-s(4), 16*s(3)*s(4)^9-s(3)*s(4), 16*s(2)*s(4)^9-s(2)*s(4), 16*s(1)*s(3)*s(4)^8-s(1)*s(3), 16*s(1)*s(2)*s(4)^8-s(1)*s(2), 16*s(1)*s(4)^10-15*s(2)*s(3)*s(4)-16*s(1)*s(4)^2, 16*s(1)^2*s(4)^9-15*s(1)*s(2)*s(3)-16*s(1)^2*s(4), 16*s(4)^13+15*s(3)^4*s(4)-16*s(4)^5; |
---|
[d12655] | 1536 | ideal mpoly = std(0); |
---|
[4e461ff] | 1537 | |
---|
| 1538 | ring R = (0,a), (x,y,z), lp; |
---|
| 1539 | minpoly = a2 + 1; |
---|
| 1540 | ideal I1 = x2 - 1/2, a*z - 1, y - 2; |
---|
| 1541 | ideal I2 = x3 - 1/2, a*z2 - 3, y - 2*a; |
---|
[35f23d] | 1542 | |
---|
[d12655] | 1543 | */ |
---|