[0df59c8] | 1 | /////////////////////////////////////////////////////////////////////// |
---|
[4e654a2] | 2 | version = "$Id$"; |
---|
[0df59c8] | 3 | |
---|
| 4 | // last changed 21.5.12 C.G. reversal wieder eingefuegt (standalone) |
---|
| 5 | category = "general"; |
---|
| 6 | info = |
---|
| 7 | " |
---|
| 8 | LIBRARY: decomp.lib Functional Decomposition of Polynomials |
---|
| 9 | AUTHOR: Christian Gorzel, University of Muenster |
---|
| 10 | email: gorzelc@math.uni-muenster.de |
---|
| 11 | |
---|
| 12 | |
---|
| 13 | OVERVIEW: |
---|
| 14 | @texinfo |
---|
| 15 | This library implements functional uni-multivariate decomposition |
---|
| 16 | of multivariate polynomials. |
---|
| 17 | |
---|
| 18 | A (multivariate) polynomial f is a composite if it can be written as |
---|
| 19 | @math{g \\circ h} where g is univariate and h is multivariate, |
---|
| 20 | where @math{\\deg(g), \\deg(h)>1}. |
---|
| 21 | |
---|
| 22 | Uniqueness for monic polynomials is up to linear coordinate change |
---|
| 23 | @tex |
---|
| 24 | $g\\circ h = g(x/c -d) \\circ c(h(x)+d)$. |
---|
| 25 | @end tex |
---|
| 26 | |
---|
| 27 | If f is a composite, then @code{decompose(f);} returns an ideal (g,h); |
---|
| 28 | such that @math{\\deg(g) < \\deg(f)} is maximal, (@math{\\deg(h)\\geq 2}). |
---|
| 29 | The polynomial h is, by the maximality of @math{\\deg(g)}, not a composite. |
---|
| 30 | |
---|
| 31 | The polynomial g is univariate in the (first) variable vvar of f, |
---|
| 32 | such that deg_vvar(f) is maximal. |
---|
| 33 | |
---|
| 34 | @code{decompose(f,1);} computes a full decomposition, i.e. if f is a |
---|
| 35 | composite, then an ideal @math{(g_1,\\dots ,g_m,h)} is returned, where |
---|
| 36 | @math{g_i} are univariate and each entry is primitive such that |
---|
| 37 | @math{f=g_1\\circ \\dots \\circ g_m\\circ h}. |
---|
| 38 | |
---|
| 39 | If f is not a composite, for instance if @math{\\deg(f)} is prime, |
---|
| 40 | then @code{decompose(f);} returns f. |
---|
| 41 | |
---|
| 42 | The command @code{decompose} is the inverse: @code{compose(decompose(f,1))==f}. |
---|
| 43 | |
---|
| 44 | Recall, that Chebyshev polynomials of the first kind commute by composition. @* |
---|
| 45 | |
---|
| 46 | The decomposition algorithms work in the tame case, that is if |
---|
| 47 | char(basering)=0 or p:=char(basering) > 0 but deg(g) is not divisible by |
---|
| 48 | p. |
---|
| 49 | Additionally, it works for monic polynomials over @math{Z} and in some |
---|
| 50 | cases for monic polyomials over coefficient rings. @* See |
---|
| 51 | @code{is_composite} for examples. (It also works over the reals but |
---|
| 52 | there it seems not be numerical stable.) @* |
---|
| 53 | |
---|
| 54 | More information on the univariate resp. multivariate case. @* |
---|
| 55 | |
---|
| 56 | Univariate decomposition is created, with the additional assumption |
---|
| 57 | @math{\\deg(g), \\deg(h)>1}. @* |
---|
| 58 | |
---|
| 59 | A multivariate polynomial f is a composite, if f can be written as |
---|
| 60 | @math{g \\circ h}, where @math{g} is a univariate polynomial and @math{h} |
---|
| 61 | is multivariate. Note, that unlike in the univariate case, the polynomial |
---|
| 62 | @math{h} may be of degree @math{1}. @* |
---|
| 63 | E.g. @math{f = (x+y)^2+ 2(x+y) +1} is the composite of |
---|
| 64 | @math{g = x^2+2x+1} and @math{h = x+y}. @* |
---|
| 65 | |
---|
| 66 | If @code{nvars(basering)>1}, then, by default, a single-variable |
---|
| 67 | multivariate polynomial is not considered to be the same as in the |
---|
| 68 | one-variable polynomial ring; it will always be decomposed. That is: @* |
---|
| 69 | @code{> ring r1=0,x,dp;} @* |
---|
| 70 | @code{> decompose(x3+2x+1);} @* |
---|
| 71 | @code{x3+2x+1} @* |
---|
| 72 | but: @* |
---|
| 73 | @code{> ring r2=0,(x,y),dp;} @* |
---|
| 74 | @code{> decompose(x3+2x+1);} @* |
---|
| 75 | @code{_[1]=x3+2x+1} @* |
---|
| 76 | @code{_[2]=x} @* |
---|
| 77 | |
---|
| 78 | In particular: @* |
---|
| 79 | @code{is_composite(x3+2x+1)==1;} in @code{ring r1} but @* |
---|
| 80 | @code{is_composite(x3+2x+1)==0;} in @code{ring r2}. @* |
---|
| 81 | |
---|
| 82 | This is justified by interpreting the polynomial decomposition as an |
---|
| 83 | affine Stein factorization of the mapping @math{f:k^n \\to k, n\\geq 2}. |
---|
| 84 | |
---|
| 85 | The behaviour can changed by the some global variables. |
---|
| 86 | |
---|
| 87 | @code{int DECMETH;} choose von zur Gathen's or Kozen-Landau's method. |
---|
| 88 | @* @code{int MINS;} compute f = g o h, such that h(0) = 0. @* |
---|
| 89 | @code{int IMPROVE;} simplify the coefficients of g and h if f is |
---|
| 90 | not monic. @* |
---|
| 91 | @code{int DEGONE;} single-variable multivariate are |
---|
| 92 | considered uni-variate. @* |
---|
| 93 | |
---|
| 94 | See @code{decompopts;} for more information. |
---|
| 95 | |
---|
| 96 | Additional information is displayed if @code{printlevel > 0}. |
---|
| 97 | @end texinfo |
---|
| 98 | REFERENCES: |
---|
| 99 | @texinfo |
---|
| 100 | @tex |
---|
| 101 | D. Kozen, S. Landau: Polynomial Decomposition Algorithms, \\par |
---|
| 102 | \\quad \\qquad J. Symb. Comp. (1989), 7, 445-456. \\par |
---|
| 103 | J. von zu Gathen: Functional Decomposition of Polynomials: the Tame Case,\\par |
---|
| 104 | \\quad \\qquad J. Symb. Comp. (1990), 9, 281-299. \\par |
---|
| 105 | J. von zur Gathen, J. Gerhard: Modern computer algebra, \\par |
---|
| 106 | \\quad \\qquad Cambridge University Press, Cambridge, 2003. |
---|
| 107 | @end tex |
---|
| 108 | @end texinfo |
---|
| 109 | PROCEDURES: |
---|
| 110 | // decompunivmonic(f,r); |
---|
| 111 | // decompmultivmonic(f,var,s); |
---|
| 112 | decompopts([\"reset\"]); displays resp. resets global options |
---|
| 113 | decompose(f[,1]); [complete] functional decomposition of poly f |
---|
| 114 | is_composite(f); predicate, is f a composite polynomial? |
---|
| 115 | chebyshev(n[,1]); the nth Chebyshev polynomial of the first kind |
---|
| 116 | compose(f1,..,fn); compose f1 (f2 (...(fn))), f_i polys of ideal |
---|
| 117 | |
---|
| 118 | AUXILIARY PROCEDURES: |
---|
| 119 | makedistinguished(f,var); transforms f to a var-distinguished polynomial |
---|
| 120 | // divisors(n[,1]); intvec [increasing] of the divisors d of n |
---|
| 121 | // gcdv(v); the gcd of the entries in intvec v |
---|
| 122 | // maxdegs(f); maximal degree for each variable of the poly f |
---|
| 123 | // randomintvec(n,a,b[,1]); random intvec size n, [non-zero] entries in {a,b} |
---|
| 124 | KEYWORDS: Functional decomposition |
---|
| 125 | "; |
---|
| 126 | |
---|
| 127 | /* |
---|
| 128 | decompunivpoly(poly f,list #) // f = goh; r = deg g, s = deg h; |
---|
| 129 | |
---|
| 130 | Ablauf ist: |
---|
| 131 | |
---|
| 132 | decompose(f) |
---|
| 133 | | check whether f is the composite by a monomial |
---|
| 134 | | check whether f is univariate |
---|
| 135 | | transformation to a distinguished polynomial |
---|
| 136 | decompmultivmonic(f,vvar,r) |
---|
| 137 | decompunivmonic(f,r) // detect vvar by maxdegs |
---|
| 138 | |lift univariate decomposition |
---|
| 139 | | back-transformation |
---|
| 140 | | fulldecompose, iterate |
---|
| 141 | | decompuniv for g |
---|
| 142 | |
---|
| 143 | */ |
---|
| 144 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 145 | |
---|
| 146 | |
---|
| 147 | proc decompopts(list #) |
---|
| 148 | "USAGE: decompopts(); or decompopts(\"reset\"); |
---|
| 149 | RETURN: nothing |
---|
| 150 | NOTE: |
---|
| 151 | @texinfo |
---|
| 152 | in the first case, it shows the setting of the control parameters;@* |
---|
| 153 | in the second case, it kills the user-defined control parameters and@* |
---|
| 154 | resets to the default setting which will then |
---|
| 155 | be diplayed. @* @* |
---|
| 156 | int DECMETH; Method for computing the univariate decomposition@* |
---|
| 157 | 0 : (default) Kozen-Landau @* |
---|
| 158 | 1 : von zur Gathen @* |
---|
| 159 | |
---|
| 160 | int IMPROVE Choice of coefficients for the decomposition @* |
---|
| 161 | @math{(g_1,\ldots,g_l,h)} of a non-monic polynomials f. @* |
---|
| 162 | 0 : leadcoef(@math{g_1}) = leadcoef(@math{f}) |
---|
| 163 | and @math{g_2,\ldots,g_l,h} are monic @* |
---|
| 164 | 1 : (default), content(@math{g_i}) = 1 @* |
---|
| 165 | |
---|
| 166 | int MINS @* |
---|
| 167 | @math{f=g\circ h, (g_1,\ldots,g_m,h)} of a non-monic polynomials f.@* |
---|
| 168 | 0 : g(0) = f(0), h(0) = 0 [ueberlegen fuer complete] @* |
---|
| 169 | 1 : (default), g(0)=0, h(0) = f(0) @* |
---|
| 170 | 2 : Tschirnhaus @* |
---|
| 171 | |
---|
| 172 | int DECORD; The order in which the decomposition will be computed@* |
---|
| 173 | 0 : minfirst @* |
---|
| 174 | 1 : (default) maxfirst @* |
---|
| 175 | |
---|
| 176 | int DEGONE; decompose also polynomials built on linear ones @* |
---|
| 177 | 0 : (default) @* |
---|
| 178 | 1 : |
---|
| 179 | @end texinfo |
---|
| 180 | EXAMPLE: example decompopts; shows an example |
---|
| 181 | " |
---|
| 182 | { |
---|
| 183 | /* |
---|
| 184 | siehe Erlaeuterungen, globale Variablen wie im Header angegeben, |
---|
| 185 | suchen mit CTRL-S Top:: |
---|
| 186 | diese eintragen |
---|
| 187 | */ |
---|
| 188 | if (size(#)) |
---|
| 189 | { |
---|
| 190 | if (string(#[1]) == "reset") |
---|
| 191 | { |
---|
| 192 | if (defined(DECMETH)) {kill DECMETH;} |
---|
| 193 | // if (defined(DECORD)) {kill DECORD;} |
---|
| 194 | if (defined(MINS)) {kill MINS;} |
---|
| 195 | if (defined(IMPROVE)) {kill IMPROVE;} |
---|
| 196 | } |
---|
| 197 | } |
---|
| 198 | |
---|
| 199 | if (voice==2) |
---|
| 200 | { |
---|
| 201 | ""; |
---|
| 202 | " === Global variables for decomp.lib === "; |
---|
| 203 | ""; |
---|
| 204 | |
---|
| 205 | if (!defined(DECMETH)) {" -- DECMETH (int) not defined, implicitly 1";} |
---|
| 206 | else |
---|
| 207 | { |
---|
| 208 | if (DECMETH!=0 and DECMETH!=1) { DECMETH=1; } |
---|
| 209 | " -- DECMETH =", DECMETH; |
---|
| 210 | } |
---|
| 211 | /* |
---|
| 212 | if (!defined(DECORD)) {" -- DECORD (int) not defined, implicitly 1";} |
---|
| 213 | else |
---|
| 214 | { |
---|
| 215 | if (DECORD!=0 and DECORD!=1) { DECORD=1; } |
---|
| 216 | " -- (int) DECORD =", DECORD; |
---|
| 217 | } |
---|
| 218 | */ |
---|
| 219 | if (!defined(MINS)) {" -- MINS (int) not defined, implicitly 0";} |
---|
| 220 | else |
---|
| 221 | { |
---|
| 222 | if (MINS!=0 and MINS!=1) { MINS = 0; } |
---|
| 223 | " -- (int) MINS =", MINS; |
---|
| 224 | } |
---|
| 225 | |
---|
| 226 | if (!defined(IMPROVE)) {" -- IMPROVE (int) not defined, implicitly 1";} |
---|
| 227 | else |
---|
| 228 | { |
---|
| 229 | if (IMPROVE!=0 and IMPROVE!=1) { IMPROVE=1; } |
---|
| 230 | " -- (int) IMPROVE =", IMPROVE; |
---|
| 231 | } |
---|
| 232 | } |
---|
| 233 | } |
---|
| 234 | example; |
---|
| 235 | { "EXAMPLE:"; echo =2; |
---|
| 236 | decompopts(); |
---|
| 237 | } |
---|
| 238 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 239 | |
---|
| 240 | //static |
---|
| 241 | proc decompmonom(poly f, list #) |
---|
| 242 | "USAGE: decompmonom(f[,vvar]); f poly, vvar poly |
---|
| 243 | PURPOSE: compute a maximal decomposition in case that |
---|
| 244 | f = g o h, where g is univariate and h is a single monomial |
---|
| 245 | RETURN: ideal, (g,h); g univariate, h monomial if such a decomposition exist, |
---|
| 246 | poly, the input, otherwise |
---|
| 247 | ASSUME: f is non-constant |
---|
| 248 | EXAMPLE: example decompmonom; shows an example |
---|
| 249 | " |
---|
| 250 | { |
---|
| 251 | int i,k; |
---|
| 252 | poly g; |
---|
| 253 | |
---|
| 254 | poly vvar = var(1); |
---|
| 255 | if (size(#)) { vvar = var(rvar(#[1])); } |
---|
| 256 | |
---|
| 257 | //poly vvar = maxdeg(f); |
---|
| 258 | poly zeropart = jet(f,0); |
---|
| 259 | poly ff = f - zeropart; |
---|
| 260 | int mindeg = -deg(ff,-1:nvars(basering)); |
---|
| 261 | poly minff = jet(ff,mindeg); |
---|
| 262 | if (size(minff)>1) { return(f); } |
---|
| 263 | intvec minv = leadexp(minff); |
---|
| 264 | minv = minv/gcdv(minv); |
---|
| 265 | for (i=1;i<=size(ff);i++) |
---|
| 266 | { |
---|
| 267 | k = divintvecs(leadexp(ff[i]),minv); |
---|
| 268 | if (k==0) { return(f); } |
---|
| 269 | else { g = g + leadcoef(ff[i])*vvar^k; } |
---|
| 270 | } |
---|
| 271 | g = g + zeropart; |
---|
| 272 | dbprint("* Sucessfully multivariate decomposed by a monomial"+newline); |
---|
| 273 | return(ideal(g,monomial(minv))); |
---|
| 274 | } |
---|
| 275 | example |
---|
| 276 | { "EXAMPLE:"; echo =2; |
---|
| 277 | ring r = 0,(x,y),dp; |
---|
| 278 | poly f = subst((x2+x3)^150,x,x2y3); |
---|
| 279 | decompmonom(f); |
---|
| 280 | |
---|
| 281 | ring rxyz = 0,(x,y,z),dp; |
---|
| 282 | poly g = 1+x2+x3+x5; |
---|
| 283 | poly G = subst(g,x,x7y5z3); |
---|
| 284 | ideal I = decompmonom(G^50); |
---|
| 285 | I[2]; |
---|
| 286 | } |
---|
| 287 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 288 | |
---|
| 289 | static proc divintvecs(intvec v,intvec w) |
---|
| 290 | "USAGE: divintvecs(v,w); v,w intvec, w!=0 |
---|
| 291 | RETURN: int, k if v = k*w, |
---|
| 292 | 0 otherwise |
---|
| 293 | NOTE: if w==0, then an Error message occurs |
---|
| 294 | EXAMPLE: example divintevcs; shows an example |
---|
| 295 | " |
---|
| 296 | { |
---|
| 297 | if (w==0) { |
---|
| 298 | ERROR("// Error: proc divintvecs: the second argument has to be non-zero."); |
---|
| 299 | return(0); |
---|
| 300 | } |
---|
| 301 | int i=1; |
---|
| 302 | while (w[i]==0) { i++; } |
---|
| 303 | int k = v[i] div w[i]; |
---|
| 304 | if (v == k*w) { return(k); } |
---|
| 305 | else { return(0); } |
---|
| 306 | } |
---|
| 307 | example |
---|
| 308 | { "EXAMPLE:"; echo =2; |
---|
| 309 | intvec v = 1,2,3; |
---|
| 310 | intvec w = 2,4,6; |
---|
| 311 | divintvecs(w,v); |
---|
| 312 | divintvecs(intvec(3,2,9),v); |
---|
| 313 | } |
---|
| 314 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 315 | |
---|
| 316 | static proc gcdv(intvec v) |
---|
| 317 | "USAGE: gcdv(v); intvec v |
---|
| 318 | RETURN: int, the gcd of the entries in v |
---|
| 319 | NOTE: if v=0, then gcdv(v)=1 @* |
---|
| 320 | this is different from Singular's builtin gcd, where gcd(0,0)==0 |
---|
| 321 | EXAMPLE: example gcdv; shows an example |
---|
| 322 | " |
---|
| 323 | { |
---|
| 324 | int ggt; |
---|
| 325 | int i,n; |
---|
| 326 | |
---|
| 327 | ggt = v[1]; |
---|
| 328 | for (i=2;i<=size(v);i++) |
---|
| 329 | { |
---|
| 330 | ggt = gcd(ggt,v[i]); |
---|
| 331 | } |
---|
| 332 | if (ggt==0) |
---|
| 333 | { |
---|
| 334 | ggt = 1; |
---|
| 335 | } |
---|
| 336 | return(ggt); |
---|
| 337 | } |
---|
| 338 | example |
---|
| 339 | { "EXAMPLE:"; echo =2; |
---|
| 340 | intvec v = 6,15,21; |
---|
| 341 | gcdv(v); |
---|
| 342 | gcdv(0:3); |
---|
| 343 | } |
---|
| 344 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 345 | |
---|
| 346 | static proc divisors(int n,list #) |
---|
| 347 | "USAGE: divisors(n); n int |
---|
| 348 | divisors(n,1); n int |
---|
| 349 | RETURN: intvec, the positive divisors of n @* |
---|
| 350 | in decreasing order (default) @* |
---|
| 351 | in increasing order in the second case |
---|
| 352 | EXAMPLE: example divisors; shows an example |
---|
| 353 | " |
---|
| 354 | { |
---|
| 355 | int i,j; |
---|
| 356 | intvec v = 1; |
---|
| 357 | |
---|
| 358 | list l = primefactors(n); |
---|
| 359 | list primesl = l[1]; |
---|
| 360 | list multl = l[2]; |
---|
| 361 | |
---|
| 362 | for (i=1;i<=size(primesl);i++) |
---|
| 363 | { |
---|
| 364 | for (j=1;j<=multl[i];j++) |
---|
| 365 | { v = v,primesl[i]*v;} |
---|
| 366 | } |
---|
| 367 | |
---|
| 368 | ring rhelp =0,x,dp; // sort the intvec |
---|
| 369 | poly h; |
---|
| 370 | for(i=1;i<=size(v);i++) |
---|
| 371 | { |
---|
| 372 | h = h+x^v[i]; |
---|
| 373 | } |
---|
| 374 | v=0; |
---|
| 375 | for(i=1;i<=size(h);i++) |
---|
| 376 | { |
---|
| 377 | v[i]=leadexp(h[i])[1]; |
---|
| 378 | } |
---|
| 379 | if (size(#)) { |
---|
| 380 | return(intvec(v[size(v)..1])); |
---|
| 381 | } |
---|
| 382 | |
---|
| 383 | return(v); |
---|
| 384 | } |
---|
| 385 | example |
---|
| 386 | { "EXAMPLE:"; echo = 2; |
---|
| 387 | divisors(30); |
---|
| 388 | divisors(-24,1); |
---|
| 389 | } |
---|
| 390 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 391 | // |
---|
| 392 | // Dies wirkt sich nur aus wenn Brueche vorhanden sind?! |
---|
| 393 | // Laeuft dann so statt cleardenom usw. problemlos ueber Z,Z_m |
---|
| 394 | // ansehen. |
---|
| 395 | // |
---|
| 396 | static proc improvecoef(poly g0,poly h0,number lc) |
---|
| 397 | "USAGE: improvecoef(g0,h0,lc); g0, h0 poly; lc number |
---|
| 398 | RETURN: poly, poly, number |
---|
| 399 | ASSUME: global ordering |
---|
| 400 | EXAMPLE: example improvecoef; shows an example |
---|
| 401 | " |
---|
| 402 | { |
---|
| 403 | int Zcoefs = find(charstr(basering),"integer"); |
---|
| 404 | poly vvar = var(univariate(g0)); |
---|
| 405 | number lch0 = leadcoef(h0); |
---|
| 406 | number denom; |
---|
| 407 | |
---|
| 408 | if (Zcoefs and lch0<0) // da cleardenom fuer integer buggy ist. |
---|
| 409 | { |
---|
| 410 | h0 = h0/(-1); |
---|
| 411 | denom = -1; |
---|
| 412 | } |
---|
| 413 | else |
---|
| 414 | { |
---|
| 415 | h0 = cleardenom(h0); |
---|
| 416 | denom = leadcoef(h0)/lch0; |
---|
| 417 | } |
---|
| 418 | g0 = subst(g0,vvar,1/denom*vvar); |
---|
| 419 | g0 = lc*g0; |
---|
| 420 | lc = leadcoef(g0); |
---|
| 421 | g0= 1/lc*g0; |
---|
| 422 | return(g0,h0,lc); |
---|
| 423 | } |
---|
| 424 | example |
---|
| 425 | { "EXAMPLE:"; echo = 2; |
---|
| 426 | ring r = 0,x,dp; |
---|
| 427 | |
---|
| 428 | poly g = 3x2+5x; |
---|
| 429 | poly h = 4x3+2/3x; |
---|
| 430 | number lc = 7; |
---|
| 431 | |
---|
| 432 | improvecoef(g,h,lc); |
---|
| 433 | } |
---|
| 434 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 435 | |
---|
| 436 | proc compose(list #) |
---|
| 437 | "USAGE: compose(f1,...,fn); f1,...,fn poly |
---|
| 438 | compose(I); I ideal, @* |
---|
| 439 | ASSUME: the ideal consists of n=ncols(I) >= 1 entries, @* |
---|
| 440 | where I[1],...,I[n-1] are univariate in the same variable @* |
---|
| 441 | but I[n] may be multivariate. |
---|
| 442 | RETURN: poly, the composition I[1](I[2](...I[n])) |
---|
| 443 | NOTE: this procedure is the inverse of decompose |
---|
| 444 | EXAMPLE: example compose; shows some examples |
---|
| 445 | SEE: decompose |
---|
| 446 | " |
---|
| 447 | { |
---|
| 448 | def d = basering; // Ohne dies kommt es zu Fehler, wenn auf Toplevel |
---|
| 449 | // ring r definiert ist. |
---|
| 450 | |
---|
| 451 | ideal I = ideal(#[1..size(#)]); |
---|
| 452 | int n=ncols(I); |
---|
| 453 | poly f=I[1]; |
---|
| 454 | map phisubst; |
---|
| 455 | ideal phiid = maxideal(1); |
---|
| 456 | |
---|
| 457 | int varnum = univariate(f); |
---|
| 458 | |
---|
| 459 | if (varnum<0) { |
---|
| 460 | " // the first polynomial is a constant"; |
---|
| 461 | return(f); |
---|
| 462 | } |
---|
| 463 | if (varnum==0 and n>1) { |
---|
| 464 | " // the first polynomial is not univariate"; |
---|
| 465 | return(f); |
---|
| 466 | } |
---|
| 467 | // Hier noch einen Test ergaenzen |
---|
| 468 | |
---|
| 469 | poly vvar = var(varnum); |
---|
| 470 | |
---|
| 471 | for(int i=2;i<=n;i++) |
---|
| 472 | { |
---|
| 473 | phiid[varnum]=I[i]; |
---|
| 474 | // phisubst=d,phiid; |
---|
| 475 | phisubst=basering,phiid; |
---|
| 476 | f = phisubst(f); |
---|
| 477 | } |
---|
| 478 | return(f); |
---|
| 479 | } |
---|
| 480 | example |
---|
| 481 | { "EXAMPLE:"; echo =2; |
---|
| 482 | ring r = 0,(x,y),dp; |
---|
| 483 | compose(x3+1,x2,y3+x); |
---|
| 484 | // or the input as one ideal |
---|
| 485 | compose(ideal(x3+1,x2,x3+y)); |
---|
| 486 | } |
---|
| 487 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 488 | |
---|
| 489 | proc is_composite(poly f) |
---|
| 490 | "USAGE: is_composite(f); f poly |
---|
| 491 | RETURN: int @* |
---|
| 492 | 1, if f is decomposable @* |
---|
| 493 | 0, if f is not decomposable @* |
---|
| 494 | -1, if char(basering)>0 and deg(f) is divisible by char(basering) but no |
---|
| 495 | decomposition has been found. |
---|
| 496 | NOTE: The last case means that it could exist a decomposition f=g o h with |
---|
| 497 | char(basering)|deg(g), but this wild case cannot be decided by the algorithm.@* |
---|
| 498 | Some additional information will be displayed when called by the user. |
---|
| 499 | EXAMPLE: example is_composite; shows some examples |
---|
| 500 | " |
---|
| 501 | { |
---|
| 502 | int d = deg(f,nvars(basering)); |
---|
| 503 | int cb = char(basering); |
---|
| 504 | |
---|
| 505 | if (d<1) |
---|
| 506 | { |
---|
| 507 | " The polynomial is constant "; |
---|
| 508 | return(0); |
---|
| 509 | } |
---|
| 510 | if (d==1) |
---|
| 511 | { |
---|
| 512 | " The polynomial is linear "; |
---|
| 513 | return(0); |
---|
| 514 | } |
---|
| 515 | |
---|
| 516 | if (nvars(basering)==1 and d==prime(d)) |
---|
| 517 | { |
---|
| 518 | " The degree is prime."; |
---|
| 519 | return(0); |
---|
| 520 | } |
---|
| 521 | |
---|
| 522 | if (nvars(basering)>1 and univariate(f)) // and not(defined(DEGONE)) |
---|
| 523 | { |
---|
| 524 | return(1); |
---|
| 525 | } |
---|
| 526 | |
---|
| 527 | // else try to decompose |
---|
| 528 | int nc = ncols(ideal(decompose(f))); |
---|
| 529 | |
---|
| 530 | if (cb > 0) // check the not covered wild case |
---|
| 531 | { |
---|
| 532 | if ((d mod cb == 0) and (nc == 1)) |
---|
| 533 | { |
---|
| 534 | if (voice==2) |
---|
| 535 | { |
---|
| 536 | "// -- Warning: wild case, cannot decide whether the polynomial has a"; |
---|
| 537 | "// -- decomposition goh with deg(g) divisible by char(basering) = " |
---|
| 538 | + string(cb) + "."; |
---|
| 539 | } |
---|
| 540 | return(-1); |
---|
| 541 | } |
---|
| 542 | } |
---|
| 543 | // in the tame case, decompose gives the correct result |
---|
| 544 | return(nc>1); |
---|
| 545 | } |
---|
| 546 | example |
---|
| 547 | { "EXAMPLE:"; echo =2; |
---|
| 548 | |
---|
| 549 | ring r0 = 0,x,dp; |
---|
| 550 | is_composite(x4+5x2+6); // biquadratic polynomial |
---|
| 551 | |
---|
| 552 | is_composite(2x2+x+1); // prime degree |
---|
| 553 | // ----------------------------------------------------------------------- |
---|
| 554 | // polynomial ring with several variables |
---|
| 555 | ring R = 0,(x,y),dp; |
---|
| 556 | // ----------------------------------------------------------------------- |
---|
| 557 | // single-variable multivariate polynomials |
---|
| 558 | is_composite(2x+1); |
---|
| 559 | is_composite(2x2+x+1); |
---|
| 560 | // ----------------------------------------------------------------------- |
---|
| 561 | // prime characteristic |
---|
| 562 | ring r7 = 7,x,dp; |
---|
| 563 | is_composite(compose(ideal(x2+x,x14))); // is_composite(x14+x7); |
---|
| 564 | is_composite(compose(ideal(x14+x,x2))); // is_composite(x14+x2); |
---|
| 565 | |
---|
| 566 | } |
---|
| 567 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 568 | |
---|
| 569 | proc decompose(poly f,list #) |
---|
| 570 | "USAGE: decompose(f); f poly |
---|
| 571 | decompose(f,1); f poly |
---|
| 572 | RETURN: poly, the input, if f is not a composite |
---|
| 573 | ideal, if the input is a composite |
---|
| 574 | NOTE: computes a full decomposition if called by the second variant |
---|
| 575 | EXAMPLE: example decompose; shows some examples |
---|
| 576 | SEE: compose |
---|
| 577 | " |
---|
| 578 | { |
---|
| 579 | if (!defined(IMPROVE)){ int IMPROVE = 1; } |
---|
| 580 | if (!defined(MINFIRST)){ int MINFIRST = 0; } |
---|
| 581 | int fulldecompose; |
---|
| 582 | |
---|
| 583 | if (size(#)) { // cf. ERROR-msg in randomintvec |
---|
| 584 | if (typeof(#[1])=="int") { |
---|
| 585 | fulldecompose = (#[1]==1); |
---|
| 586 | } |
---|
| 587 | } |
---|
| 588 | |
---|
| 589 | int m,iscomposed; |
---|
| 590 | int globalord = 1; |
---|
| 591 | ideal I; |
---|
| 592 | |
---|
| 593 | // --- preparatory stuff ---------------------------------------------------- |
---|
| 594 | // The degree is not independent of the term order |
---|
| 595 | int n = deg(f,1:nvars(basering)); |
---|
| 596 | int varnum = univariate(f); // to avoid transformation if f is univariate |
---|
| 597 | |
---|
| 598 | // if (deg(f)<=1) {return(f);} //steigt automatisch bei der for-schleife aus m = 2 |
---|
| 599 | if (n==prime(n) and nvars(basering)==1 |
---|
| 600 | // or (varnum>0 and nvars(basering)) |
---|
| 601 | ) {return(f);} |
---|
| 602 | |
---|
| 603 | if (varnum<0) |
---|
| 604 | { |
---|
| 605 | ERROR("// -- Error proc decompoly: the polynomial is constant."); |
---|
| 606 | } |
---|
| 607 | //-------------------------------------------------------------------------- |
---|
| 608 | |
---|
| 609 | int minfirst = MINFIRST!=0; |
---|
| 610 | list mdeg; |
---|
| 611 | intvec maxdegv,degcand; |
---|
| 612 | |
---|
| 613 | // -- switch to global order, necessary for division -- // Weiter nach oben |
---|
| 614 | if (typeof(attrib(basering,"global"))!="int") { |
---|
| 615 | globalord = 0; |
---|
| 616 | } |
---|
| 617 | else { |
---|
| 618 | globalord = attrib(basering,"global"); |
---|
| 619 | } |
---|
| 620 | |
---|
| 621 | if (!globalord) { |
---|
| 622 | def d = basering; |
---|
| 623 | list ll = ringlist(basering); |
---|
| 624 | ll[3] = list(list("dp",1:nvars(basering)),list("C",0)); |
---|
| 625 | def rneu = ring(ll); |
---|
| 626 | setring rneu; |
---|
| 627 | poly f = fetch(d,f); |
---|
| 628 | ideal I; |
---|
| 629 | } |
---|
| 630 | // ----------------------------------------------------------------------- |
---|
| 631 | |
---|
| 632 | map phiback; |
---|
| 633 | poly f0,g0,h0,vvar; |
---|
| 634 | number lc; |
---|
| 635 | ideal J; // wird erst in fulldecompose benoetigt |
---|
| 636 | |
---|
| 637 | // --- Determine the candidates for deg(g) a decreasing sequence of divisors |
---|
| 638 | poly lf = jet(f,n)-jet(f,n-1); |
---|
| 639 | //"lf = ",lf; |
---|
| 640 | if (size(lf)==1) // the leading homogeneous part is a monomial |
---|
| 641 | { |
---|
| 642 | degcand = divisors(gcdv(leadexp(lf))); |
---|
| 643 | } |
---|
| 644 | else |
---|
| 645 | { |
---|
| 646 | degcand = divisors(n); // Das ist absteigend |
---|
| 647 | } |
---|
| 648 | |
---|
| 649 | if(printlevel>0) {degcand;} |
---|
| 650 | |
---|
| 651 | // --- preparatory steps for the multivariate case ------------------------- |
---|
| 652 | |
---|
| 653 | if (varnum>0) // -- univariate polynomial |
---|
| 654 | { |
---|
| 655 | vvar = var(varnum); |
---|
| 656 | f0 = f; // save f |
---|
| 657 | } |
---|
| 658 | else // i.e. multivariate (varnum==0),the case varnum < 0 is excluded above |
---|
| 659 | { |
---|
| 660 | // -- find variable with maximal degree |
---|
| 661 | mdeg = maxdegs(f); |
---|
| 662 | maxdegv = mdeg[2]; |
---|
| 663 | varnum = maxdegv[2]; |
---|
| 664 | vvar = var(varnum); |
---|
| 665 | phiback = maxideal(1); |
---|
| 666 | |
---|
| 667 | // special case, the polynomial is a composite of a single monomial //20.6.10 |
---|
| 668 | if (qhweight(f)!=0) { I = decompmonom(f,vvar); } |
---|
| 669 | iscomposed = size(I)>1; |
---|
| 670 | if (iscomposed) // 3.6.11 - dies decompmonom |
---|
| 671 | { //I; |
---|
| 672 | ideal J = decompunivmonic(I[1],deg(I[1])); |
---|
| 673 | I[2]= subst(J[2],vvar,I[2]); |
---|
| 674 | I[1] = J[1]; |
---|
| 675 | //I; |
---|
| 676 | } |
---|
| 677 | |
---|
| 678 | if (!iscomposed) // -- transform into a distinguished polynomial |
---|
| 679 | { |
---|
| 680 | f0,phiback = makedistinguished(f,vvar); |
---|
| 681 | } |
---|
| 682 | } |
---|
| 683 | // ------ Start computation ------------------------------------------------ |
---|
| 684 | // -- normalize and save the leading coefficient |
---|
| 685 | lc = 1; |
---|
| 686 | //f0; |
---|
| 687 | //"vvar = ",vvar; |
---|
| 688 | |
---|
| 689 | // --- 11.4.11 hier auch noch gewichteten Grad beruecksichtigen ? -- |
---|
| 690 | |
---|
| 691 | if (!iscomposed) { lc = leadcoef(coeffs(f0,vvar)[deg(f0)+1,1]); } // 20.6.10 |
---|
| 692 | |
---|
| 693 | // if Z, Z_m, and f is not monic (and content !=1) // if (f0/lc*lc!=f0) |
---|
| 694 | if (find(charstr(basering),"integer") and not(lc==1 or lc==-1)) // 6.4.11 |
---|
| 695 | { |
---|
| 696 | ERROR("// -- Error proc decompose: Can not decompose non-monic polynomial over Z!"); |
---|
| 697 | } |
---|
| 698 | |
---|
| 699 | if (lc!=1){ f0 = 1/number(lc)*f0;} // --- normalize the polynomial |
---|
| 700 | |
---|
| 701 | // -- Now the input is prepared to be monic and vvar-distinguished |
---|
| 702 | //---------------------------------------------------------------- |
---|
| 703 | m = 1; |
---|
| 704 | |
---|
| 705 | // --- Special case: a multivariate can be composite of a linear polynom |
---|
| 706 | if (univariate(f) and nvars(basering)==1) // 11.8.09 d.h. |
---|
| 707 | { // --- if univariate ---------------------------------------- |
---|
| 708 | if(minfirst) {degcand = divisors(n,1);} // dies ist aufsteigend |
---|
| 709 | m = 2; // skip first entry |
---|
| 710 | } |
---|
| 711 | // if decomposed as the decomposition with a monomial |
---|
| 712 | // then skip the multivariate process // 20.6.10 detected as decompmonomial |
---|
| 713 | if (iscomposed) { degcand = 1; } |
---|
| 714 | |
---|
| 715 | if (printlevel>0 and !iscomposed) { "* Degree candidates are", degcand; } |
---|
| 716 | |
---|
| 717 | // -- check succesively for each candidate |
---|
| 718 | // whether f is decomposable with deg g = r |
---|
| 719 | |
---|
| 720 | for(;m<size(degcand);m++) // decreasing |
---|
| 721 | { //r = degcand[m]; |
---|
| 722 | I = decompmultivmonic(f0,vvar,degcand[m]); |
---|
| 723 | if (size(I)>1) |
---|
| 724 | { |
---|
| 725 | iscomposed = 1; |
---|
| 726 | break; |
---|
| 727 | } |
---|
| 728 | } |
---|
| 729 | // -- all candidates have be checked but f is primitive |
---|
| 730 | if(!iscomposed) { |
---|
| 731 | if (!globalord) { setring d; } // restore old ring |
---|
| 732 | dbprint("** not decomposable: linear / not tame / prime degree --"); |
---|
| 733 | return(f); |
---|
| 734 | } |
---|
| 735 | |
---|
| 736 | // -- the monic vvar-distinguished polynomial f0 is decomposed ------- |
---|
| 737 | // -- retransformation for the multivariate case --------------------- |
---|
| 738 | g0,h0 = I; |
---|
| 739 | |
---|
| 740 | if (!univariate(f)) { h0 = phiback(h0);} |
---|
| 741 | |
---|
| 742 | if (IMPROVE) { g0,h0,lc=improvecoef(g0,h0,lc);} // ueber switch |
---|
| 743 | I = h0; |
---|
| 744 | |
---|
| 745 | // -- Full decomposition: try to decompose g further ------------------ |
---|
| 746 | if (fulldecompose) { |
---|
| 747 | dbprint(newline+"** Compute a complete decomposition"); |
---|
| 748 | while (iscomposed) { |
---|
| 749 | iscomposed=0; |
---|
| 750 | degcand=divisors(deg(g0,1:nvars(basering))); // absteigend |
---|
| 751 | if (printlevel> 0) { "** Degree candidates are now: ", degcand; } |
---|
| 752 | for (m=2;m<size(degcand);m++) //OK, ergibt lexicographically .. |
---|
| 753 | { |
---|
| 754 | J =decompunivmonic(g0,degcand[m]); /* J =decompuniv(g0);*/ |
---|
| 755 | g0 = J[1]; |
---|
| 756 | h0=J[2]; |
---|
| 757 | iscomposed = deg(h0,1:nvars(basering))>1; |
---|
| 758 | if (iscomposed) { |
---|
| 759 | if (IMPROVE) { g0,h0,lc=improvecoef(g0,h0,lc); } // ueber switch |
---|
| 760 | I = h0,I; |
---|
| 761 | break; |
---|
| 762 | } |
---|
| 763 | } |
---|
| 764 | } |
---|
| 765 | dbprint("** completely decomposed"+newline); |
---|
| 766 | } |
---|
| 767 | I = lc*g0,I; |
---|
| 768 | if (!globalord) { |
---|
| 769 | setring d; |
---|
| 770 | I = fetch(rneu,I); |
---|
| 771 | } |
---|
| 772 | return(I); |
---|
| 773 | } |
---|
| 774 | example |
---|
| 775 | { "EXAMPLE:"; echo =2; |
---|
| 776 | ring r2 = 0,(x,y),dp; |
---|
| 777 | |
---|
| 778 | decompose(((x3+2y)^6+x3+2y)^4); |
---|
| 779 | |
---|
| 780 | // complete decomposition |
---|
| 781 | decompose(((x3+2y)^6+x3+2y)^4,1); |
---|
| 782 | // ----------------------------------------------------------------------- |
---|
| 783 | // decompose over the integers |
---|
| 784 | ring rZ = integer,x,dp; |
---|
| 785 | decompose(compose(ideal(x3,x2+2x,x3+2)),1); |
---|
| 786 | // ----------------------------------------------------------------------- |
---|
| 787 | // prime characteristic |
---|
| 788 | ring r7 = 7,x,dp; |
---|
| 789 | decompose(compose(ideal(x2+x,x7))); // tame case |
---|
| 790 | // ----------------------------------------------------------------------- |
---|
| 791 | decompose(compose(ideal(x7+x,x2))); // wild case |
---|
| 792 | // ----------------------------------------------------------------------- |
---|
| 793 | ring ry = (0,y),x,dp; // y is now a parameter |
---|
| 794 | compose(x2+yx+5,x5-2yx3+x); |
---|
| 795 | decompose(_); |
---|
| 796 | |
---|
| 797 | // Usage of variable IMPROVE |
---|
| 798 | ideal J = x2+10x, 64x7-112x5+56x3-7x, 4x3-3x; |
---|
| 799 | decompose(compose(J),1); |
---|
| 800 | int IMPROVE=0; |
---|
| 801 | exportto(Decomp,IMPROVE); |
---|
| 802 | decompose(compose(J),1); |
---|
| 803 | } |
---|
| 804 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 805 | /* ring rt =(0,t),x,dp; |
---|
| 806 | poly f = 36*x6+12*x4+15*x3+x2+5/2*x+(-t); |
---|
| 807 | decompose(f); |
---|
| 808 | */ |
---|
| 809 | |
---|
| 810 | |
---|
| 811 | // Dies gibt stets ein ideal zurueck, wenn f composite ist |
---|
| 812 | // gibt das polynom zurueck, wenn es primitiv ist |
---|
| 813 | // static |
---|
| 814 | proc decompmultivmonic(poly f,poly vvar,int r) |
---|
| 815 | "USAGE: decompmultivmonic(f,vvar,r); f,vvar poly; r int |
---|
| 816 | RETURN: ideal, I = ideal(g,h) if f = g o h with deg(g) = r@* |
---|
| 817 | poly f, if f is not a composite or char(basering) divides r |
---|
| 818 | ASSUME: f is monic and distinguished w.r.t. vvar, |
---|
| 819 | 1<=r<=deg(f) is a divisor of deg(f) |
---|
| 820 | and char(basering) does not divide r. |
---|
| 821 | EXAMPLE: example decompmultivmonic; shows an example |
---|
| 822 | " |
---|
| 823 | { |
---|
| 824 | def d = basering; |
---|
| 825 | int i,isprimitive; |
---|
| 826 | int m = nvars(basering); |
---|
| 827 | int n = deg(f); |
---|
| 828 | int varnum = rvar(vvar); |
---|
| 829 | intvec v = 1:m; // weight-vector for jet |
---|
| 830 | v[varnum]=0; |
---|
| 831 | int s = n div r; |
---|
| 832 | // r = deg g; s = deg h; |
---|
| 833 | |
---|
| 834 | poly f0 = f; |
---|
| 835 | poly h,h0,g,gp,fgp,k,t,u; |
---|
| 836 | ideal I,rem,phiid; |
---|
| 837 | list l; |
---|
| 838 | map phisubst; |
---|
| 839 | |
---|
| 840 | // -- entscheidet intern, abhaengig von der Anzahl der Ringvariablen, |
---|
| 841 | // -- ob f0 primitive ist. |
---|
| 842 | // " r = ",r; |
---|
| 843 | |
---|
| 844 | if (s*r!=n) |
---|
| 845 | { |
---|
| 846 | ERROR("// -- Error proc decompmultivmonic: r = "+string(r)+ |
---|
| 847 | " does not divide deg(f) = "+string(n)+"."); |
---|
| 848 | } |
---|
| 849 | |
---|
| 850 | int cb = char(basering); // oder dies in decompunivmonic |
---|
| 851 | if (cb>0) |
---|
| 852 | { |
---|
| 853 | if (r mod cb == 0) |
---|
| 854 | { |
---|
| 855 | if (voice == 2) |
---|
| 856 | { |
---|
| 857 | "// Warning: wild case in characteristic " + string(cb) + |
---|
| 858 | ". We cannot decide"; |
---|
| 859 | "// whether a decomposition goh with deg(g) = " + string(r)+ |
---|
| 860 | " exists.";""; |
---|
| 861 | } |
---|
| 862 | return(f); |
---|
| 863 | } |
---|
| 864 | } |
---|
| 865 | //--------------------------------------------------------------------------- |
---|
| 866 | |
---|
| 867 | for (i=1;i<=m;i++) |
---|
| 868 | { |
---|
| 869 | if (i!=varnum) {f0 = subst(f0,var(i),0);} |
---|
| 870 | } |
---|
| 871 | //" f0 = ",f0; |
---|
| 872 | // f0 ist nun das univariate |
---|
| 873 | |
---|
| 874 | // 24.3.09 // 11.8.09 nochmals ansehen |
---|
| 875 | if (r==deg(f0)) // the case of a linear multivarcomposite |
---|
| 876 | { |
---|
| 877 | dbprint("** try to decompose in linear h, deg g = "+string(r)); |
---|
| 878 | I = f0,vvar; // Das ist hier wichtig |
---|
| 879 | } |
---|
| 880 | else // find decomposition of the univariate f0 |
---|
| 881 | { |
---|
| 882 | I = decompunivmonic(f0,r); |
---|
| 883 | // dbprint(" ** monic decomposed");//" I = ";I; |
---|
| 884 | |
---|
| 885 | isprimitive=(deg(I[2])==1); |
---|
| 886 | if (isprimitive) {return(f);} |
---|
| 887 | } |
---|
| 888 | |
---|
| 889 | //---- proceed in the multivariate case |
---|
| 890 | //---- lift the univariate decomposition |
---|
| 891 | if (!univariate(f)) |
---|
| 892 | { |
---|
| 893 | dbprint("* Lift the univariate decomposition"); |
---|
| 894 | g,h0 = I; |
---|
| 895 | k = h0; |
---|
| 896 | gp = diff(g,vvar); |
---|
| 897 | |
---|
| 898 | // -- This is substitution ---- |
---|
| 899 | // t = substitute(gp,vvar,h0); |
---|
| 900 | phiid = maxideal(1); |
---|
| 901 | phiid[varnum]=h0; |
---|
| 902 | phisubst=basering,phiid; |
---|
| 903 | t = phisubst(gp); |
---|
| 904 | // -- substitution ende |
---|
| 905 | fgp = 1; |
---|
| 906 | i = 0; |
---|
| 907 | while(fgp!=0) |
---|
| 908 | { |
---|
| 909 | i++; |
---|
| 910 | // -- This is substitution ---- |
---|
| 911 | //gp = substitute(g,vvar,k); |
---|
| 912 | phiid[varnum]=k; |
---|
| 913 | phisubst=basering,phiid; |
---|
| 914 | gp = phisubst(g); |
---|
| 915 | // -- substitution ende |
---|
| 916 | |
---|
| 917 | fgp = f - gp; |
---|
| 918 | u = jet(fgp,i,v) - jet(fgp,i-1,v); // oder mit reduce(maxideal(x)) |
---|
| 919 | l = division(u,t); // die kleineren Terme abschneiden |
---|
| 920 | rem = l[2]; |
---|
| 921 | u = l[1][1,1]; // the factor |
---|
| 922 | if (rem!=0) |
---|
| 923 | { |
---|
| 924 | isprimitive = 1; |
---|
| 925 | break; |
---|
| 926 | } |
---|
| 927 | k = k + u; |
---|
| 928 | } |
---|
| 929 | h = k; |
---|
| 930 | I = g,h; |
---|
| 931 | //"decomposed as ="; |
---|
| 932 | //I; |
---|
| 933 | } |
---|
| 934 | if (isprimitive) { |
---|
| 935 | dbprint(">>> not multivariate decomposed"+newline); |
---|
| 936 | return(f); |
---|
| 937 | } |
---|
| 938 | else { |
---|
| 939 | dbprint("* Sucessfully multivariate decomposed"+newline); |
---|
| 940 | return(I); |
---|
| 941 | } |
---|
| 942 | } |
---|
| 943 | example |
---|
| 944 | { "EXAMPLE:"; echo = 2; |
---|
| 945 | ring r = 0,(x,y),lp; |
---|
| 946 | poly f = 3xy4 + 2xy2 + x5y3 + x + y6; |
---|
| 947 | decompmultivmonic(f,y,2); |
---|
| 948 | |
---|
| 949 | ring rx = 0,x,lp; |
---|
| 950 | decompmultivmonic(x8,x,4); |
---|
| 951 | } |
---|
| 952 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 953 | //static |
---|
| 954 | proc decompunivmonic(poly f,int r) |
---|
| 955 | "USAGE: decompunivmonic(f,r); f poly, r int |
---|
| 956 | RETURN: ideal, (g,h) such that f = goh and deg(g) = r |
---|
| 957 | poly f, if such a decomposition does not exist. |
---|
| 958 | ASSUME: f is univariate, r is a divisor of deg(f) @* |
---|
| 959 | and char(basering) does not divide r in case that char(basering) > 0. |
---|
| 960 | global order of the basering is assumed. |
---|
| 961 | EXAMPLE: example decompunivmonic; shows an example |
---|
| 962 | " |
---|
| 963 | { |
---|
| 964 | int d = deg(f); |
---|
| 965 | int s; // r = deg g; s = deg h; |
---|
| 966 | int minf,mins; |
---|
| 967 | int iscomposed = 1; |
---|
| 968 | |
---|
| 969 | if (!defined(MINS)) { int MINS = 0; } |
---|
| 970 | if (!defined(DECMETH)) { int DECMETH = 1; } |
---|
| 971 | int savedecmeth = DECMETH; |
---|
| 972 | int Zcoefs =charstr(basering)=="integer";//find(charstr(basering),"integer"); |
---|
| 973 | |
---|
| 974 | number cf; |
---|
| 975 | poly h,g; |
---|
| 976 | ideal I; |
---|
| 977 | matrix cc; |
---|
| 978 | |
---|
| 979 | // --- Check input and create the results for the simple cases |
---|
| 980 | |
---|
| 981 | if (deg(f)<1){return(ideal(f,var(1)));} // wird dies aufgerufen? |
---|
| 982 | //------------------------- |
---|
| 983 | |
---|
| 984 | int varnum = univariate(f); |
---|
| 985 | |
---|
| 986 | if (varnum==0) |
---|
| 987 | { |
---|
| 988 | "// -- The polynomial is not univariate"; |
---|
| 989 | return(f); |
---|
| 990 | } |
---|
| 991 | |
---|
| 992 | poly vvar = var(varnum); |
---|
| 993 | I = f,vvar; |
---|
| 994 | |
---|
| 995 | if (leadcoef(f)!=1) |
---|
| 996 | { |
---|
| 997 | "// -- Error proc decompunivmonic: the polynomial is not monic."; |
---|
| 998 | return(f); |
---|
| 999 | } |
---|
| 1000 | /* Dies einklammern, wenn (x+1)^2 zerlegt werden sollte |
---|
| 1001 | // aus decompose heraus, wird dies gar nicht aufgerufen! |
---|
| 1002 | if (deg(f)==1 or deg(f)==prime(deg(f))) |
---|
| 1003 | { |
---|
| 1004 | "// -- The polynomial is not a composite."; |
---|
| 1005 | return(I); |
---|
| 1006 | } |
---|
| 1007 | */ |
---|
| 1008 | /* ---------------------------------------------------- */ |
---|
| 1009 | s = d div r; |
---|
| 1010 | |
---|
| 1011 | if (d!=s*r) |
---|
| 1012 | { |
---|
| 1013 | ERROR("// -- Error proc decompunivmonic: the second argument does not divide deg f."); |
---|
| 1014 | } |
---|
| 1015 | int cb = char(basering); |
---|
| 1016 | if (cb>0) |
---|
| 1017 | { |
---|
| 1018 | if (r mod cb ==0) |
---|
| 1019 | { |
---|
| 1020 | "wild case: cannot determine a decomposition"; |
---|
| 1021 | return(I); |
---|
| 1022 | } |
---|
| 1023 | } |
---|
| 1024 | // ------------------------------------------------------------------------- |
---|
| 1025 | // The Newton iteration only works over coefficient *fields* |
---|
| 1026 | // Therefore use in this case the Kozen-Landau method i.e. set DECMETH = 1; |
---|
| 1027 | if (savedecmeth==0 and Zcoefs) { DECMETH=1; } |
---|
| 1028 | |
---|
| 1029 | // -- Start the computation ---------------------------------------------- |
---|
| 1030 | |
---|
| 1031 | dbprint("* STEP 1: Determine h"); |
---|
| 1032 | dbprint(" d = deg f = " +string(n) + " f = goh"," r = deg g = "+string(r), |
---|
| 1033 | " s = deg h = " +string(s)); |
---|
| 1034 | int tt = timer; |
---|
| 1035 | |
---|
| 1036 | if(DECMETH==1) { // Kozen-Landau |
---|
| 1037 | dbprint("* Kozen-Landau method"); |
---|
| 1038 | |
---|
| 1039 | // Determine ord(f); |
---|
| 1040 | //cc = coef(f,vvar); // extract coefficents of f |
---|
| 1041 | //print(cc); read(""); |
---|
| 1042 | |
---|
| 1043 | // dbprint("time: "+string(timer-tt)); tt = timer; |
---|
| 1044 | // minf = deg(cc[1,ncols(cc)]); // 11.8.09 Doch OK. |
---|
| 1045 | minf = -deg(f,-1:nvars(basering)); // this is local ord 15.3.10 |
---|
| 1046 | |
---|
| 1047 | // oder: mins = 1; if (minf) { .. dies .. } |
---|
| 1048 | mins = (minf div r) + (minf mod r) > 0; // i.e. ceil(minf/r) |
---|
| 1049 | |
---|
| 1050 | if (mins==0 and MINS) { mins=1; } // omit the constant term i.e. h(0) = 0 |
---|
| 1051 | |
---|
| 1052 | dbprint("** min f = "+string(minf) + " | min s = "+string(mins) + |
---|
| 1053 | " | s-mins = "+ string(s-mins)); |
---|
| 1054 | |
---|
| 1055 | // Dies wird wohl nicht benoetigt. |
---|
| 1056 | // int minr= (minf div s) + ((minf mod s)>0); // ceil |
---|
| 1057 | dbprint("** extract the coeffs "); |
---|
| 1058 | cc = coeffs(f,vvar); |
---|
| 1059 | |
---|
| 1060 | dbprint("time: "+ string(timer -tt)); |
---|
| 1061 | |
---|
| 1062 | h = vvar^s; |
---|
| 1063 | for (int j=1;j<=s-mins;j++) |
---|
| 1064 | { |
---|
| 1065 | /* |
---|
| 1066 | timer = 1;H = Power(h,r); "Power H"; timer; |
---|
| 1067 | timer = 1;G = h^r; "h^r"; timer; |
---|
| 1068 | */ |
---|
| 1069 | cf = (number(cc[d-j+1,1])-number(coeffs(h^r,vvar)[d-j+1,1])); |
---|
| 1070 | |
---|
| 1071 | // d-j+1,"cf =",cf, " r= ",r; |
---|
| 1072 | // dbprint("*** "+ string(d-j+1) + " cf = "+string(cf) + " r= "+string(r)); |
---|
| 1073 | |
---|
| 1074 | if (Zcoefs) { if (bigint(cf) mod r != 0) { iscomposed = 0; break; }} |
---|
| 1075 | cf = cf/r; |
---|
| 1076 | |
---|
| 1077 | //else { cf = cf/r; } |
---|
| 1078 | h = h + cf*vvar^(s-j); |
---|
| 1079 | // " h = ",h; |
---|
| 1080 | } |
---|
| 1081 | } else { |
---|
| 1082 | dbprint("* von zur Gathen-method"); |
---|
| 1083 | // "f=",f; |
---|
| 1084 | h = reversal(newtonrroot(reversal(f,d),r,s+!MINS),s,vvar); // verdreht OK |
---|
| 1085 | // " h = ",h; |
---|
| 1086 | dbprint("* END STEP 1: time: "+string(timer -tt)); |
---|
| 1087 | } |
---|
| 1088 | DECMETH=savedecmeth; // restore the original method |
---|
| 1089 | |
---|
| 1090 | if (iscomposed == 0) { |
---|
| 1091 | dbprint("** Failed in STEP 1: not decomposed with deg h = "+string(s)+newline); |
---|
| 1092 | return(I); |
---|
| 1093 | } |
---|
| 1094 | |
---|
| 1095 | // -- Step 2: try to rewrite f as a sum of powers of h --- |
---|
| 1096 | dbprint("* STEP 2: Determine g"); |
---|
| 1097 | poly H = h^r; |
---|
| 1098 | int dalt = r; |
---|
| 1099 | int ds; |
---|
| 1100 | number c; |
---|
| 1101 | while (d >= 0) // i.e. f!=0 |
---|
| 1102 | { |
---|
| 1103 | //dbprint("d = ",d); |
---|
| 1104 | ds = d div s; |
---|
| 1105 | if (ds * s !=d) // d mod s != 0, i.e. remaining f is a power of h |
---|
| 1106 | { |
---|
| 1107 | iscomposed = 0; |
---|
| 1108 | break; |
---|
| 1109 | } |
---|
| 1110 | c = leadcoef(f); |
---|
| 1111 | g = g + c*vvar^ds; |
---|
| 1112 | H = division(H,h^(dalt - ds))[1][1,1]; // 10.3.10 |
---|
| 1113 | // H = H / h^(dalt - ds); |
---|
| 1114 | f = f - c*H; |
---|
| 1115 | //"f = ",f; |
---|
| 1116 | |
---|
| 1117 | dalt = ds; |
---|
| 1118 | d = deg(f); |
---|
| 1119 | } |
---|
| 1120 | dbprint("* END STEP 2: time: "+string(timer -tt)); |
---|
| 1121 | if (iscomposed) |
---|
| 1122 | { |
---|
| 1123 | dbprint("** Sucessfully univariate decomposed with deg g = "+string(r)+newline); |
---|
| 1124 | I = g,h; |
---|
| 1125 | } else { |
---|
| 1126 | dbprint("** Failed in STEP 2: not decomposed with deg g = "+string(r)+newline); |
---|
| 1127 | } |
---|
| 1128 | |
---|
| 1129 | return(I); |
---|
| 1130 | } |
---|
| 1131 | example |
---|
| 1132 | { "EXAMPLE:"; echo = 2; |
---|
| 1133 | ring r=0,(x,y),dp; |
---|
| 1134 | decompunivmonic((x2+x+1)^3,3); |
---|
| 1135 | decompunivmonic((x2+x)^3,3); |
---|
| 1136 | |
---|
| 1137 | decompunivmonic((y2+y+1)^3,3); |
---|
| 1138 | } |
---|
| 1139 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1140 | // aus polyaux.lib |
---|
| 1141 | proc reversal(poly f,list #) |
---|
| 1142 | "USAGE: reversal(f); f poly |
---|
| 1143 | reversal(f,k); f poly, k int |
---|
| 1144 | reversal(f,k,vvar); f poly, k int, vvar poly (a ring variable) |
---|
| 1145 | RETURN: poly, the reversal x^k*f(1/x) of the input f |
---|
| 1146 | ASSUME: f is univariate and that k>=deg(f) |
---|
| 1147 | @* since no negative exponents are possible in Singular |
---|
| 1148 | @* if k<deg(f) then k = deg(f) is used |
---|
| 1149 | NOTE: reversal(f); is by default reversal(f,deg(f)); |
---|
| 1150 | the third variant is needed if f is a non-zero constant and k>0 @* |
---|
| 1151 | @* reversal is only idempotent, |
---|
| 1152 | @* if called twice with the deg(f) as second argument |
---|
| 1153 | EXAMPLE: example reversal; shows an example |
---|
| 1154 | " |
---|
| 1155 | { |
---|
| 1156 | int k = 0; |
---|
| 1157 | poly vvar = var(1); |
---|
| 1158 | |
---|
| 1159 | if (size(#)) { |
---|
| 1160 | k = #[1] - deg(f) ; |
---|
| 1161 | if (k<0) { k=0; } |
---|
| 1162 | if (size(#)==2){ // check whether second optional argument |
---|
| 1163 | vvar = var(univariate(#[2])); // is a ring variable |
---|
| 1164 | } |
---|
| 1165 | } |
---|
| 1166 | |
---|
| 1167 | int varnum = univariate(f); |
---|
| 1168 | |
---|
| 1169 | if (varnum==0) { |
---|
| 1170 | ERROR("// -- the input is not univariate."); |
---|
| 1171 | } |
---|
| 1172 | if (varnum<0) { // the polynomial is constant |
---|
| 1173 | return(f*vvar^k); |
---|
| 1174 | } |
---|
| 1175 | |
---|
| 1176 | def d = basering; |
---|
| 1177 | list l = ringlist(d); |
---|
| 1178 | list varl = l[2]; |
---|
| 1179 | varl = insert(varl,"@z",size(varl)); |
---|
| 1180 | l[2] = varl; |
---|
| 1181 | def rnew = ring(l); |
---|
| 1182 | setring rnew; |
---|
| 1183 | poly f = fetch(d,f); |
---|
| 1184 | f = subst(homog(f,@z),var(varnum),1,@z,var(varnum))*var(varnum)^k; |
---|
| 1185 | |
---|
| 1186 | setring d; |
---|
| 1187 | f = fetch(rnew,f); |
---|
| 1188 | return(f); |
---|
| 1189 | } |
---|
| 1190 | example |
---|
| 1191 | { "EXAMPLE:"; echo = 2; |
---|
| 1192 | ring r = 0,x,dp; |
---|
| 1193 | poly f = x3+2x+5; |
---|
| 1194 | reversal(f); |
---|
| 1195 | // the same as |
---|
| 1196 | reversal(f,3); |
---|
| 1197 | reversal(f,5); |
---|
| 1198 | |
---|
| 1199 | poly g = x3+2x; |
---|
| 1200 | reversal(g); |
---|
| 1201 | |
---|
| 1202 | // Not idempotent |
---|
| 1203 | reversal(reversal(g)); |
---|
| 1204 | |
---|
| 1205 | // idempotent |
---|
| 1206 | reversal(reversal(g,deg(g)),deg(g)); |
---|
| 1207 | // or for short |
---|
| 1208 | // reversal(reversal(g),deg(g)); |
---|
| 1209 | } |
---|
| 1210 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1211 | // aus polyaux.lib |
---|
| 1212 | proc newtonrroot(poly f,int r,int l) |
---|
| 1213 | "USAGE: newtonrroot(f,r,l); f poly; r int; l int |
---|
| 1214 | RETURN: poly h, the solution of h^r = f modulo vvar^l |
---|
| 1215 | ASSUME: f(0) = 1 |
---|
| 1216 | NOTE: this uses p-adic Newton iteration. It is the adaption of Algorithm 9.22@* |
---|
| 1217 | of von zur Gathen & Gerhard p. 264 for the special case: phi = Y^r - f |
---|
| 1218 | EXAMPLE: example newtonrroot; shows some examples |
---|
| 1219 | " |
---|
| 1220 | { |
---|
| 1221 | // phi = Y^r - f |
---|
| 1222 | |
---|
| 1223 | poly g = 1; // start polynomial |
---|
| 1224 | |
---|
| 1225 | poly s = 1/number(r); // initial solution |
---|
| 1226 | int i = 2; |
---|
| 1227 | //"s initial",s; |
---|
| 1228 | |
---|
| 1229 | while(i<l) { |
---|
| 1230 | // "iteration i",i; |
---|
| 1231 | |
---|
| 1232 | // g = (g -(g^r-f)*s) mod x^i; |
---|
| 1233 | g = jet((g -(g^r-f)*s), i-1); |
---|
| 1234 | // s = 2*s - (r*g^(r-1)*s^2) mod x^i; |
---|
| 1235 | s = jet(2*s - (r*g^(r-1)*s^2),i-1); |
---|
| 1236 | // "s is now ",s; |
---|
| 1237 | |
---|
| 1238 | i = 2*i; |
---|
| 1239 | } |
---|
| 1240 | //"return newtonrroot"; |
---|
| 1241 | //jet((g -(g^r-f)*s),l-1); |
---|
| 1242 | |
---|
| 1243 | return(jet((g -(g^r-f)*s),l-1)); |
---|
| 1244 | } |
---|
| 1245 | example |
---|
| 1246 | { "EXAMPLE:"; echo = 2; |
---|
| 1247 | ring r = 0,x,dp; |
---|
| 1248 | |
---|
| 1249 | ring r3 = 3,x,dp; |
---|
| 1250 | poly f = x+1; |
---|
| 1251 | // determine square root of f modulo x^4 |
---|
| 1252 | poly g = newtonrroot(f,2,4); |
---|
| 1253 | g; |
---|
| 1254 | g^2; |
---|
| 1255 | ring R = (0,b,c,d),x,ds; |
---|
| 1256 | // poly f = 1 + bx +cx2+dx3; |
---|
| 1257 | poly f = 1 + 5bx +5cx2+5dx3; |
---|
| 1258 | poly g2 = newtonrroot(f,2,4); |
---|
| 1259 | g2; |
---|
| 1260 | f-g2^2; |
---|
| 1261 | poly f5 = 1 +5*(bx+cx2+dx3); |
---|
| 1262 | poly g5 = newtonrroot(f5,5,4); |
---|
| 1263 | g5; |
---|
| 1264 | f5-g5^5; |
---|
| 1265 | // Multivariate polynomials |
---|
| 1266 | ring r = 0,(x,y,z),ds; |
---|
| 1267 | ring r2 =(0,a,b,c,d,e),(x,y),ds; |
---|
| 1268 | // poly f = 1 +ax+by+cx2+dxy+ey2; |
---|
| 1269 | poly f3 = 1 +9*(ax+by+cx2+dxy+ey2); |
---|
| 1270 | poly g3 = newtonrroot(f3,3,4); |
---|
| 1271 | jet(g3^3-f3,5); |
---|
| 1272 | } |
---|
| 1273 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1274 | |
---|
| 1275 | static proc randomintvec(int n,int a,int b,list #) |
---|
| 1276 | "USAGE: randomintvec(n,a,b); n,a,b int; |
---|
| 1277 | randomintvec(n,a,b,1); n,a,b int; |
---|
| 1278 | RETURN: intvec, say v, of length n |
---|
| 1279 | with entries a<=v[i]<=b, in the first case, resp. |
---|
| 1280 | with entries a<=v[i]<=b, where v[i]!=0, in the second case |
---|
| 1281 | NOTE: a<=b should be satisfied, otherwise always v[i]=b (due to random). |
---|
| 1282 | EXAMPLE: example randomintvec; shows some examples |
---|
| 1283 | " |
---|
| 1284 | { |
---|
| 1285 | int i,randint,nozeroes; |
---|
| 1286 | intvec v; |
---|
| 1287 | |
---|
| 1288 | if (size(#)) { |
---|
| 1289 | if (typeof(#[1])!="int") { |
---|
| 1290 | ERROR("4th argument can only be an integer, assumed 1."); |
---|
| 1291 | } |
---|
| 1292 | nozeroes = #[1]==1; |
---|
| 1293 | } |
---|
| 1294 | |
---|
| 1295 | for (i=1;i<=n;i++) |
---|
| 1296 | { |
---|
| 1297 | randint = random(a,b); |
---|
| 1298 | while (nozeroes and randint==0) { randint = random(a,b); } |
---|
| 1299 | v[i] = randint; |
---|
| 1300 | } |
---|
| 1301 | return(v); |
---|
| 1302 | } |
---|
| 1303 | example |
---|
| 1304 | { "EXAMPLE:"; echo = 1; |
---|
| 1305 | int randval = system("--random"); // store initial value |
---|
| 1306 | system("--random",0815); |
---|
| 1307 | echo = 2; |
---|
| 1308 | randomintvec(7,-1,1); // 7 entries in {-1,0,1} |
---|
| 1309 | randomintvec(7,-1,1,1); // 7 entries either -1 or 1 |
---|
| 1310 | randomintvec(3,-10,10); |
---|
| 1311 | echo = 1; |
---|
| 1312 | system("--random",randval); // reset random generator |
---|
| 1313 | } |
---|
| 1314 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1315 | |
---|
| 1316 | proc makedistinguished(poly f,poly vvar) |
---|
| 1317 | "USAGE: makedistinguished(f,vvar); f, vvar poly; where vvar is a ring variable |
---|
| 1318 | RETURN: (poly, ideal): the transformed polynomial and an ideal defining |
---|
| 1319 | the map which reverses the transformation. |
---|
| 1320 | PURPOSE: let vvar = var(1). Then f is transformed by a random linear |
---|
| 1321 | coordinate change |
---|
| 1322 | phi = (var(1), var(2)+c_2*vvar,...,var(n)+c_n*vvar) @* |
---|
| 1323 | such that phi(f) = f o phi becomes distinguished with respect |
---|
| 1324 | to vvar. That is, the new polynomial contains the monomial vvar^d, |
---|
| 1325 | where d is the degree of f. @* |
---|
| 1326 | If already f is distinguished w.r.t. vvar, then f is left unchanged |
---|
| 1327 | and the re-transformation is the identity. |
---|
| 1328 | NOTE 1: (this proc correctly works independent of the term ordering.) |
---|
| 1329 | to apply the reverse transformation, either define a map |
---|
| 1330 | or use substitute (to be loaded from poly.lib). |
---|
| 1331 | NOTE 2: If p=char(basering) > 0, then there exist polynomials of degree d>=p, |
---|
| 1332 | e.g. @math{(p-1)x^p y + xy^p}, that cannot be transformed to a |
---|
| 1333 | vvar-distinguished polynomial. @* |
---|
| 1334 | In this case, *p random trials will be made and the proc |
---|
| 1335 | may leave with an ERROR message. |
---|
| 1336 | EXAMPLE: example makedistinguished; shows some examples |
---|
| 1337 | " |
---|
| 1338 | { |
---|
| 1339 | def d = basering; // eigentlich ueberfluessig // wg Bug mit example part |
---|
| 1340 | map phi; // erforderlich |
---|
| 1341 | ideal Db= maxideal(1); |
---|
| 1342 | int n,b = nvars(basering),1; |
---|
| 1343 | intvec v= 0:n; |
---|
| 1344 | intvec w =v; |
---|
| 1345 | int varnum = rvar(vvar); |
---|
| 1346 | w[varnum]=1; // weight vector for deg |
---|
| 1347 | |
---|
| 1348 | poly g = f; |
---|
| 1349 | int degg = deg(g); |
---|
| 1350 | |
---|
| 1351 | int count = 1; // limit the number of trials in char(p) > 0 |
---|
| 1352 | //int count =2*char(basering); |
---|
| 1353 | |
---|
| 1354 | while(deg(g,w)!=degg and (count-2*char(basering))) // do a transformation |
---|
| 1355 | { |
---|
| 1356 | v = randomintvec(n,-b,b,1); // n non-zero entries |
---|
| 1357 | v[varnum] = 0; |
---|
| 1358 | phi = d,ideal(matrix(maxideal(1),n,1) + var(varnum)*v); // transformation; |
---|
| 1359 | g = phi(f); |
---|
| 1360 | b++; // increase the range for the random values |
---|
| 1361 | // count--; |
---|
| 1362 | count++; |
---|
| 1363 | } |
---|
| 1364 | if (deg(g,w)!=degg) { |
---|
| 1365 | ERROR("it could not be transform to a "+string(vvar)+"-distinguished polynomial."); |
---|
| 1366 | } |
---|
| 1367 | Db = ideal(matrix(maxideal(1),n,1) - var(varnum)*v); // back transformation |
---|
| 1368 | return(g,Db); |
---|
| 1369 | } |
---|
| 1370 | example |
---|
| 1371 | { "EXAMPLE:"; |
---|
| 1372 | int randval = system("--random"); // store initial value |
---|
| 1373 | system("--random",0815); |
---|
| 1374 | echo = 2; |
---|
| 1375 | |
---|
| 1376 | ring r = 0,(x,y),dp; |
---|
| 1377 | poly g; |
---|
| 1378 | map phi; |
---|
| 1379 | // ----------------------------------------------------------------------- |
---|
| 1380 | // Example 1: |
---|
| 1381 | poly f = 3xy4 + 2xy2 + x5y3 + x + y6; // degree 8 |
---|
| 1382 | // make the polynomial y-distinguished |
---|
| 1383 | g, phi = makedistinguished(f,y); |
---|
| 1384 | g; |
---|
| 1385 | phi; |
---|
| 1386 | |
---|
| 1387 | // to reverse the transformation apply the map |
---|
| 1388 | f == phi(g); |
---|
| 1389 | |
---|
| 1390 | // ----------------------------------------------------------------------- |
---|
| 1391 | // Example 2: |
---|
| 1392 | // The following polynomial is already x-distinguished |
---|
| 1393 | f = x6+y4+xy; |
---|
| 1394 | g,phi = makedistinguished(f,x); |
---|
| 1395 | g; // f is left unchanged |
---|
| 1396 | phi; // the transformation is the identity. |
---|
| 1397 | echo = 1; |
---|
| 1398 | |
---|
| 1399 | system("--random",randval); // reset random generator |
---|
| 1400 | // ----------------------------------------------------------------------- |
---|
| 1401 | echo = 2; |
---|
| 1402 | // Example 3: // polynomials which cannot be transformed |
---|
| 1403 | // If p=char(basering)>0, then (p-1)*x^p*y + x*y^p factorizes completely |
---|
| 1404 | // in linear factors, since (p-1)*x^p+x equiv 0 on F_p. Hence, |
---|
| 1405 | // such polynomials cannot be transformed to a distinguished polynomial. |
---|
| 1406 | |
---|
| 1407 | ring r3 = 3,(x,y),dp; |
---|
| 1408 | makedistinguished(2x3y+xy3,y); |
---|
| 1409 | } |
---|
| 1410 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1411 | |
---|
| 1412 | static proc maxdegs(poly f) |
---|
| 1413 | "USAGE: maxdegs(f); f poly |
---|
| 1414 | RETURN: list of two intvecs |
---|
| 1415 | _[1] intvec: degree for variable i, 1<=i<=nvars(basering) @* |
---|
| 1416 | _[2] intvec: max of _[1], index of first variable with this max degree |
---|
| 1417 | EXAMPLE: example maxdegs; shows an example |
---|
| 1418 | " |
---|
| 1419 | { |
---|
| 1420 | int i,n; |
---|
| 1421 | intvec degs,maxdeg; |
---|
| 1422 | list l; |
---|
| 1423 | |
---|
| 1424 | n = nvars(basering); |
---|
| 1425 | |
---|
| 1426 | for (i=1;i<=n;i++) |
---|
| 1427 | { |
---|
| 1428 | degs[i] = nrows(coeffs(f,var(i)))-1; |
---|
| 1429 | if (degs[i] > maxdeg) |
---|
| 1430 | { |
---|
| 1431 | maxdeg[1] = degs[i]; |
---|
| 1432 | maxdeg[2] = i; |
---|
| 1433 | } |
---|
| 1434 | } |
---|
| 1435 | return(list(degs,maxdeg)); |
---|
| 1436 | } |
---|
| 1437 | example |
---|
| 1438 | { "EXAMPLE:"; echo =2; |
---|
| 1439 | ring r = 0,(x,y,z),lp; |
---|
| 1440 | poly f = 3xy4 + 2xy2 + x5y3 + xz6 + y6; |
---|
| 1441 | maxdegs(f); |
---|
| 1442 | } |
---|
| 1443 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1444 | |
---|
| 1445 | proc chebyshev(int n,list #) |
---|
| 1446 | "USAGE: chebyshev(n); n int, n >= 0 |
---|
| 1447 | chebyshev(n,c); n int, n >= 0, c number, c!=0 |
---|
| 1448 | RETURN: poly, the [monic] nth Chebyshev polynomial of the first kind. @* |
---|
| 1449 | The polynomials are defined in the first variable, say x, of the |
---|
| 1450 | basering. |
---|
| 1451 | NOTE: @texinfo |
---|
| 1452 | The (generalized) Chebyshev polynomials of the first kind can be |
---|
| 1453 | defined by the recursion: |
---|
| 1454 | @tex |
---|
| 1455 | $C_0 = c,\ C_1 = x,\ C_n = 2/c\cdot x\cdot C_{n-1}-C_{n-2},\ n \geq 2,c\neq 0$. |
---|
| 1456 | @end tex |
---|
| 1457 | @end texinfo |
---|
| 1458 | These polynomials commute by composition: |
---|
| 1459 | @math{C_m \circ C_n = C_n\circ C_m}. @* |
---|
| 1460 | For c=1, we obtain the standard (non monic) Chebyshev polynomials |
---|
| 1461 | @math{T_n} which satisfy @math{T_n(x) = \cos(n \cdot \arccos(x))}. @* |
---|
| 1462 | For c=2 (default), we obtain the monic Chebyshev polynomials @math{P_n} |
---|
| 1463 | which satisfy the relation @math{P_n(x+ 1/x) = x^n+ 1/x^n}. @* |
---|
| 1464 | By default the monic Chebyshev polynomials are returned: |
---|
| 1465 | @math{P_n =}@code{chebyshev(n)} and @math{T_n=}@code{chebyshev(n,1)}.@* |
---|
| 1466 | It holds @math{P_n(x) = 2\cdot T_n(x/2)} and more generally |
---|
| 1467 | @math{C_n(c\cdot x) = c\cdot T_n(x)} @* |
---|
| 1468 | That is @code{subst(chebyshev(n,c),var(1),c*var(1))= c*chebyshev(n,1)}. |
---|
| 1469 | |
---|
| 1470 | If @code{char(basering) = 2}, then |
---|
| 1471 | @math{C_0 = 1, C_1 = x, C_2 = 1, C_3 = x}, and so on. |
---|
| 1472 | EXAMPLE: example chebyshev; shows some examples |
---|
| 1473 | " |
---|
| 1474 | { |
---|
| 1475 | number startv = 2; |
---|
| 1476 | |
---|
| 1477 | if (size(#)){ startv = #[1]; } |
---|
| 1478 | if (startv == 0) { startv = 1; } |
---|
| 1479 | |
---|
| 1480 | poly f0,f1 = startv,var(1); |
---|
| 1481 | poly fneu,falt = f1,f0; |
---|
| 1482 | poly fh; |
---|
| 1483 | |
---|
| 1484 | if (n<=0) {return(f0);} |
---|
| 1485 | if (n==1) {return(f1);} |
---|
| 1486 | |
---|
| 1487 | for(int i=2;i<=n;i++) |
---|
| 1488 | { |
---|
| 1489 | fh = 2/startv*var(1)*fneu - falt; |
---|
| 1490 | // fh = 2*var(1)*fneu - falt; |
---|
| 1491 | falt = fneu; |
---|
| 1492 | fneu = fh; |
---|
| 1493 | } |
---|
| 1494 | return(fh); |
---|
| 1495 | } |
---|
| 1496 | example |
---|
| 1497 | { "EXAMPLE:"; echo = 2; |
---|
| 1498 | ring r = 0,x,lp; |
---|
| 1499 | |
---|
| 1500 | // The monic Chebyshev polynomials |
---|
| 1501 | chebyshev(0); |
---|
| 1502 | chebyshev(1); |
---|
| 1503 | chebyshev(2); |
---|
| 1504 | chebyshev(3); |
---|
| 1505 | |
---|
| 1506 | // These polynomials commute |
---|
| 1507 | compose(chebyshev(2),chebyshev(6)) == |
---|
| 1508 | compose(chebyshev(6),chebyshev(2)); |
---|
| 1509 | |
---|
| 1510 | // The standard Chebyshev polynomials |
---|
| 1511 | chebyshev(0,1); |
---|
| 1512 | chebyshev(1,1); |
---|
| 1513 | chebyshev(2,1); |
---|
| 1514 | chebyshev(3,1); |
---|
| 1515 | // ----------------------------------------------------------------------- |
---|
| 1516 | // The relation for the various Chebyshev polynomials |
---|
| 1517 | 5*chebyshev(3,1)==subst(chebyshev(3,5),x,5x); |
---|
| 1518 | // ----------------------------------------------------------------------- |
---|
| 1519 | // char 2 case |
---|
| 1520 | ring r2 = 2,x,dp; |
---|
| 1521 | chebyshev(2); |
---|
| 1522 | chebyshev(3); |
---|
| 1523 | } |
---|
| 1524 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1525 | |
---|
| 1526 | /* |
---|
| 1527 | |
---|
| 1528 | // Examples for decomp.lib |
---|
| 1529 | |
---|
| 1530 | ring r02 = 0,(x,y),dp; |
---|
| 1531 | |
---|
| 1532 | decompose(compose(x6,chebyshev(4),x2+y3+x5y7),1); |
---|
| 1533 | |
---|
| 1534 | int MINS = 0; |
---|
| 1535 | decompose((xy+1)^7); |
---|
| 1536 | //_[1]=x7 |
---|
| 1537 | //_[2]=xy+1 |
---|
| 1538 | |
---|
| 1539 | decompose((x2y3+1)^7); |
---|
| 1540 | //_[1]=y7 |
---|
| 1541 | //_[2]=x2y3+1 |
---|
| 1542 | |
---|
| 1543 | MINS = 1; |
---|
| 1544 | ring r01 = 0,x,dp; |
---|
| 1545 | decompose((x+1)^7); |
---|
| 1546 | //x7+7x6+21x5+35x4+35x3+21x2+7x+1 |
---|
| 1547 | |
---|
| 1548 | decompunivmonic((x+1)^7,7); |
---|
| 1549 | //_[1]=x7 |
---|
| 1550 | //_[2]=x+1 |
---|
| 1551 | |
---|
| 1552 | int MINS =1; |
---|
| 1553 | decompunivmonic((x+1)^7,7); |
---|
| 1554 | //_[1]=x7+7x6+21x5+35x4+35x3+21x2+7x+1 |
---|
| 1555 | //_[2]=x |
---|
| 1556 | |
---|
| 1557 | // -- Example ------------- |
---|
| 1558 | |
---|
| 1559 | // Comparision Kozen-Landau vs. von zur Gathen |
---|
| 1560 | |
---|
| 1561 | ring r02 = 0,(x,y),dp; |
---|
| 1562 | |
---|
| 1563 | // printlevel = 5; |
---|
| 1564 | |
---|
| 1565 | decompopts("reset"); |
---|
| 1566 | |
---|
| 1567 | poly F = compose(x6,chebyshev(4)+3,8x2+y3+7x5y7+2); |
---|
| 1568 | deg(F); |
---|
| 1569 | |
---|
| 1570 | timer = 1;decompose(F,1);timer; |
---|
| 1571 | |
---|
| 1572 | int MINS = 1; |
---|
| 1573 | timer = 1;decompose(F,1);timer; |
---|
| 1574 | int IMPROVE =0; |
---|
| 1575 | timer = 1;decompose(F,1);timer; |
---|
| 1576 | |
---|
| 1577 | decompopts("reset"); |
---|
| 1578 | int DECMETH = 0; // von zur Gathen |
---|
| 1579 | |
---|
| 1580 | timer = 1;decompose(F,1);timer; |
---|
| 1581 | |
---|
| 1582 | decompopts("reset"); |
---|
| 1583 | |
---|
| 1584 | // -- Example ------------- |
---|
| 1585 | |
---|
| 1586 | ring rZ10 = (integer,10),x,dp; |
---|
| 1587 | chebyshev(2); |
---|
| 1588 | //x2+8 |
---|
| 1589 | chebyshev(3); |
---|
| 1590 | //x3+7x |
---|
| 1591 | |
---|
| 1592 | compose(chebyshev(2),chebyshev(3)); |
---|
| 1593 | //x6+4x4+9x2+8 |
---|
| 1594 | decompose(_); |
---|
| 1595 | int MINS =1; |
---|
| 1596 | decompose(compose(chebyshev(2),chebyshev(3))); |
---|
| 1597 | compose(_); |
---|
| 1598 | |
---|
| 1599 | decompopts("reset"); |
---|
| 1600 | |
---|
| 1601 | // -- Example ------------- |
---|
| 1602 | |
---|
| 1603 | ring rT =(0,y),x,dp; |
---|
| 1604 | compose(x2,x3+y,(y+1)*x2); |
---|
| 1605 | //(y6+6y5+15y4+20y3+15y2+6y+1)*x12+(2y4+6y3+6y2+2y)*x6+(y2) |
---|
| 1606 | |
---|
| 1607 | decompose(_,1); |
---|
| 1608 | //_[1]=(y6+6y5+15y4+20y3+15y2+6y+1)*x2 |
---|
| 1609 | //_[2]=x3+(y)/(y3+3y2+3y+1) |
---|
| 1610 | //_[3]=x2 |
---|
| 1611 | |
---|
| 1612 | int MINS =1; |
---|
| 1613 | compose(x2,x3+y,(y+1)*x2); |
---|
| 1614 | //(y6+6y5+15y4+20y3+15y2+6y+1)*x12+(2y4+6y3+6y2+2y)*x6+(y2) |
---|
| 1615 | |
---|
| 1616 | decompose(_,1); |
---|
| 1617 | //_[1]=(y6+6y5+15y4+20y3+15y2+6y+1)*x2+(2y4+6y3+6y2+2y)*x+(y2) |
---|
| 1618 | //_[2]=x3 |
---|
| 1619 | //_[3]=x2 |
---|
| 1620 | |
---|
| 1621 | //ring rt =(0,t),x,dp; |
---|
| 1622 | //compose(x2+tx+5,x5-2tx3+x); |
---|
| 1623 | //x10+(-4t)*x8+(4t2+2)*x6+(t)*x5+(-4t)*x4+(-2t2)*x3+x2+(t)*x+5 |
---|
| 1624 | |
---|
| 1625 | decompose(_); |
---|
| 1626 | //_[1]=x2+(-1/4t2+5) |
---|
| 1627 | //_[2]=x5+(-2t)*x3+x+(1/2t) |
---|
| 1628 | |
---|
| 1629 | int IMPROVE = 1; |
---|
| 1630 | compose(x2+tx+5,x5-2tx3+x); |
---|
| 1631 | //x10+(-4t)*x8+(4t2+2)*x6+(t)*x5+(-4t)*x4+(-2t2)*x3+x2+(t)*x+5 |
---|
| 1632 | |
---|
| 1633 | decompose(_); |
---|
| 1634 | //_[1]=x2+(-1/4t2+5) |
---|
| 1635 | //_[2]=x5+(-2t)*x3+x+(1/2t) |
---|
| 1636 | |
---|
| 1637 | int IMPROVE = 0; |
---|
| 1638 | compose(x2+tx+5,x5-2tx3+x); |
---|
| 1639 | //x10+(-4t)*x8+(4t2+2)*x6+(t)*x5+(-4t)*x4+(-2t2)*x3+x2+(t)*x+5 |
---|
| 1640 | decompose(_); |
---|
| 1641 | //_[1]=x2+(-1/4t2+5) |
---|
| 1642 | //_[2]=x5+(-2t)*x3+x+(1/2t) |
---|
| 1643 | |
---|
| 1644 | int MINS = 1; |
---|
| 1645 | compose(x2+tx+5,x5-2tx3+x); |
---|
| 1646 | //x10+(-4t)*x8+(4t2+2)*x6+(t)*x5+(-4t)*x4+(-2t2)*x3+x2+(t)*x+5 |
---|
| 1647 | |
---|
| 1648 | decompose(_); |
---|
| 1649 | //_[1]=x2+(t)*x+5 |
---|
| 1650 | //_[2]=x5+(-2t)*x3+x |
---|
| 1651 | |
---|
| 1652 | */ |
---|
| 1653 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1654 | // --- End of decomp.lib --------------------------------------------------- // |
---|
| 1655 | /////////////////////////////////////////////////////////////////////////////// |
---|