1 | /////////////////////////////////////////////////////////// |
---|
2 | version = "$Id$"; |
---|
3 | category="Noncommutative"; |
---|
4 | info=" |
---|
5 | LIBRARY: ncfactor.lib Tools for factorization in some noncommutative algebras |
---|
6 | AUTHORS: Albert Heinle, albert.heinle@rwth-aachen.de |
---|
7 | @* Viktor Levandovskyy, levandov@math.rwth-aachen.de |
---|
8 | |
---|
9 | OVERVIEW: In this library, new methods for factorization on polynomials |
---|
10 | are implemented for two algebras, both generated by two generators (Weyl and |
---|
11 | shift algebras) over a field K. Recall, that the first Weyl algebra over K |
---|
12 | is generated by x,d obeying the relation d*x=x*d+1. |
---|
13 | @* The first shift algebra over K is generated by x,s obeying the relation s*x=x*s+s. |
---|
14 | @* More detailled description of the algorithms can be found at |
---|
15 | @url{http://www.math.rwth-aachen.de/\~Albert.Heinle}. |
---|
16 | |
---|
17 | Guide: We are interested in computing a tree of factorizations, that is at the moment |
---|
18 | a list of all found factorizations is returned. It may contain factorizations, which |
---|
19 | are further reducible. |
---|
20 | |
---|
21 | PROCEDURES: |
---|
22 | facFirstWeyl(h); factorization in the first Weyl algebra |
---|
23 | testNCfac(l[,h]); tests factorizations from a given list for correctness |
---|
24 | facSubWeyl(h,X,D); factorization in the first Weyl algebra as a subalgebra |
---|
25 | facFirstShift(h); factorization in the first shift algebra |
---|
26 | "; |
---|
27 | |
---|
28 | LIB "general.lib"; |
---|
29 | LIB "nctools.lib"; |
---|
30 | LIB "involut.lib"; |
---|
31 | LIB "freegb.lib"; // for isVar |
---|
32 | |
---|
33 | proc tst_ncfactor() |
---|
34 | { |
---|
35 | example facFirstWeyl; |
---|
36 | example facFirstShift; |
---|
37 | example facSubWeyl; |
---|
38 | example testNCfac; |
---|
39 | } |
---|
40 | |
---|
41 | ///////////////////////////////////////////////////// |
---|
42 | //==================================================* |
---|
43 | //deletes double-entries in a list of factorization |
---|
44 | //without evaluating the product. |
---|
45 | static proc delete_dublicates_noteval(list l) |
---|
46 | {//proc delete_dublicates_noteval |
---|
47 | list result= l; |
---|
48 | int j; int k; int i; |
---|
49 | int deleted = 0; |
---|
50 | int is_equal; |
---|
51 | for (i = 1; i<= size(l); i++) |
---|
52 | {//Iterate over the different factorizations |
---|
53 | for (j = i+1; j<= size(l); j++) |
---|
54 | {//Compare the i'th factorization to the j'th |
---|
55 | if (size(l[i])!= size(l[j])) |
---|
56 | {//different sizes => not equal |
---|
57 | j++; |
---|
58 | continue; |
---|
59 | }//different sizes => not equal |
---|
60 | is_equal = 1; |
---|
61 | for (k = 1; k <= size(l[i]);k++) |
---|
62 | {//Compare every entry |
---|
63 | if (l[i][k]!=l[j][k]) |
---|
64 | { |
---|
65 | is_equal = 0; |
---|
66 | break; |
---|
67 | } |
---|
68 | }//Compare every entry |
---|
69 | if (is_equal == 1) |
---|
70 | {//Delete this entry, because there is another equal one int the list |
---|
71 | result = delete(result, i-deleted); |
---|
72 | deleted = deleted+1; |
---|
73 | break; |
---|
74 | }//Delete this entry, because there is another equal one int the list |
---|
75 | }//Compare the i'th factorization to the j'th |
---|
76 | }//Iterate over the different factorizations |
---|
77 | return(result); |
---|
78 | }//proc delete_dublicates_noteval |
---|
79 | |
---|
80 | //================================================== |
---|
81 | //deletes the double-entries in a list with |
---|
82 | //evaluating the products |
---|
83 | static proc delete_dublicates_eval(list l) |
---|
84 | {//proc delete_dublicates_eval |
---|
85 | list result=l; |
---|
86 | int j; int k; int i; |
---|
87 | int deleted = 0; |
---|
88 | int is_equal; |
---|
89 | for (i = 1; i<= size(result); i++) |
---|
90 | {//Iterating over all elements in result |
---|
91 | for (j = i+1; j<= size(result); j++) |
---|
92 | {//comparing with the other elements |
---|
93 | if (product(result[i]) == product(result[j])) |
---|
94 | {//There are two equal results; throw away that one with the smaller size |
---|
95 | if (size(result[i])>=size(result[j])) |
---|
96 | {//result[i] has more entries |
---|
97 | result = delete(result,j); |
---|
98 | continue; |
---|
99 | }//result[i] has more entries |
---|
100 | else |
---|
101 | {//result[j] has more entries |
---|
102 | result = delete(result,i); |
---|
103 | i--; |
---|
104 | break; |
---|
105 | }//result[j] has more entries |
---|
106 | }//There are two equal results; throw away that one with the smaller size |
---|
107 | }//comparing with the other elements |
---|
108 | }//Iterating over all elements in result |
---|
109 | return(result); |
---|
110 | }//proc delete_dublicates_eval |
---|
111 | |
---|
112 | |
---|
113 | //==================================================* |
---|
114 | //given a list of factors g and a desired size nof, the following |
---|
115 | //procedure combines the factors, such that we recieve a |
---|
116 | //list of the length nof. |
---|
117 | static proc combinekfinlf(list g, int nof, intvec limits) //nof stands for "number of factors" |
---|
118 | {//Procedure combinekfinlf |
---|
119 | list result; |
---|
120 | int i; int j; int k; //iteration variables |
---|
121 | list fc; //fc stands for "factors combined" |
---|
122 | list temp; //a temporary store for factors |
---|
123 | def nofgl = size(g); //nofgl stands for "number of factors of the given list" |
---|
124 | if (nofgl == 0) |
---|
125 | {//g was the empty list |
---|
126 | return(result); |
---|
127 | }//g was the empty list |
---|
128 | if (nof <= 0) |
---|
129 | {//The user wants to recieve a negative number or no element as a result |
---|
130 | return(result); |
---|
131 | }//The user wants to recieve a negative number or no element as a result |
---|
132 | if (nofgl == nof) |
---|
133 | {//There are no factors to combine |
---|
134 | if (limitcheck(g,limits)) |
---|
135 | { |
---|
136 | result = result + list(g); |
---|
137 | } |
---|
138 | return(result); |
---|
139 | }//There are no factors to combine |
---|
140 | if (nof == 1) |
---|
141 | {//User wants to get just one factor |
---|
142 | if (limitcheck(list(product(g)),limits)) |
---|
143 | { |
---|
144 | result = result + list(list(product(g))); |
---|
145 | } |
---|
146 | return(result); |
---|
147 | }//User wants to get just one factor |
---|
148 | for (i = nof; i > 1; i--) |
---|
149 | {//computing the possibilities that have at least one original factor from g |
---|
150 | for (j = i; j>=1; j--) |
---|
151 | {//shifting the window of combinable factors to the left |
---|
152 | //fc below stands for "factors combined" |
---|
153 | fc = combinekfinlf(list(g[(j)..(j+nofgl - i)]),nof - i + 1,limits); |
---|
154 | for (k = 1; k<=size(fc); k++) |
---|
155 | {//iterating over the different solutions of the smaller problem |
---|
156 | if (j>1) |
---|
157 | {//There are g_i before the combination |
---|
158 | if (j+nofgl -i < nofgl) |
---|
159 | {//There are g_i after the combination |
---|
160 | temp = list(g[1..(j-1)]) + fc[k] + list(g[(j+nofgl-i+1)..nofgl]); |
---|
161 | }//There are g_i after the combination |
---|
162 | else |
---|
163 | {//There are no g_i after the combination |
---|
164 | temp = list(g[1..(j-1)]) + fc[k]; |
---|
165 | }//There are no g_i after the combination |
---|
166 | }//There are g_i before the combination |
---|
167 | if (j==1) |
---|
168 | {//There are no g_i before the combination |
---|
169 | if (j+ nofgl -i <nofgl) |
---|
170 | {//There are g_i after the combination |
---|
171 | temp = fc[k]+ list(g[(j + nofgl - i +1)..nofgl]); |
---|
172 | }//There are g_i after the combination |
---|
173 | }//There are no g_i before the combination |
---|
174 | if (limitcheck(temp,limits)) |
---|
175 | { |
---|
176 | result = result + list(temp); |
---|
177 | } |
---|
178 | }//iterating over the different solutions of the smaller problem |
---|
179 | }//shifting the window of combinable factors to the left |
---|
180 | }//computing the possibilities that have at least one original factor from g |
---|
181 | for (i = 2; i<=nofgl/nof;i++) |
---|
182 | {//getting the other possible results |
---|
183 | result = result + combinekfinlf(list(product(list(g[1..i])))+list(g[(i+1)..nofgl]),nof,limits); |
---|
184 | }//getting the other possible results |
---|
185 | result = delete_dublicates_noteval(result); |
---|
186 | return(result); |
---|
187 | }//Procedure combinekfinlf |
---|
188 | |
---|
189 | |
---|
190 | //==================================================* |
---|
191 | //merges two sets of factors ignoring common |
---|
192 | //factors |
---|
193 | static proc merge_icf(list l1, list l2, intvec limits) |
---|
194 | {//proc merge_icf |
---|
195 | list g; |
---|
196 | list f; |
---|
197 | int i; int j; |
---|
198 | if (size(l1)==0) |
---|
199 | { |
---|
200 | return(list()); |
---|
201 | } |
---|
202 | if (size(l2)==0) |
---|
203 | { |
---|
204 | return(list()); |
---|
205 | } |
---|
206 | if (size(l2)<=size(l1)) |
---|
207 | {//l1 will be our g, l2 our f |
---|
208 | g = l1; |
---|
209 | f = l2; |
---|
210 | }//l1 will be our g, l2 our f |
---|
211 | else |
---|
212 | {//l1 will be our f, l2 our g |
---|
213 | g = l2; |
---|
214 | f = l1; |
---|
215 | }//l1 will be our f, l2 our g |
---|
216 | def result = combinekfinlf(g,size(f),limits); |
---|
217 | for (i = 1 ; i<= size(result); i++) |
---|
218 | {//Adding the factors of f to every possibility listed in temp |
---|
219 | for (j = 1; j<= size(f); j++) |
---|
220 | { |
---|
221 | result[i][j] = result[i][j]+f[j]; |
---|
222 | } |
---|
223 | if(!limitcheck(result[i],limits)) |
---|
224 | { |
---|
225 | result = delete(result,i); |
---|
226 | i--; |
---|
227 | } |
---|
228 | }//Adding the factors of f to every possibility listed in temp |
---|
229 | return(result); |
---|
230 | }//proc merge_icf |
---|
231 | |
---|
232 | //==================================================* |
---|
233 | //merges two sets of factors with respect to the occurrence |
---|
234 | //of common factors |
---|
235 | static proc merge_cf(list l1, list l2, intvec limits) |
---|
236 | {//proc merge_cf |
---|
237 | list g; |
---|
238 | list f; |
---|
239 | int i; int j; |
---|
240 | list pre; |
---|
241 | list post; |
---|
242 | list candidate; |
---|
243 | list temp; |
---|
244 | int temppos; |
---|
245 | if (size(l1)==0) |
---|
246 | {//the first list is empty |
---|
247 | return(list()); |
---|
248 | }//the first list is empty |
---|
249 | if(size(l2)==0) |
---|
250 | {//the second list is empty |
---|
251 | return(list()); |
---|
252 | }//the second list is empty |
---|
253 | if (size(l2)<=size(l1)) |
---|
254 | {//l1 will be our g, l2 our f |
---|
255 | g = l1; |
---|
256 | f = l2; |
---|
257 | }//l1 will be our g, l2 our f |
---|
258 | else |
---|
259 | {//l1 will be our f, l2 our g |
---|
260 | g = l2; |
---|
261 | f = l1; |
---|
262 | }//l1 will be our f, l2 our g |
---|
263 | list M; |
---|
264 | for (i = 2; i<size(f); i++) |
---|
265 | {//finding common factors of f and g... |
---|
266 | for (j=2; j<size(g);j++) |
---|
267 | {//... with g |
---|
268 | if (f[i] == g[j]) |
---|
269 | {//we have an equal pair |
---|
270 | M = M + list(list(i,j)); |
---|
271 | }//we have an equal pair |
---|
272 | }//... with g |
---|
273 | }//finding common factors of f and g... |
---|
274 | if (g[1]==f[1]) |
---|
275 | {//Checking for the first elements to be equal |
---|
276 | M = M + list(list(1,1)); |
---|
277 | }//Checking for the first elements to be equal |
---|
278 | if (g[size(g)]==f[size(f)]) |
---|
279 | {//Checking for the last elements to be equal |
---|
280 | M = M + list(list(size(f),size(g))); |
---|
281 | }//Checking for the last elements to be equal |
---|
282 | list result;//= list(list()); |
---|
283 | while(size(M)>0) |
---|
284 | {//set of equal pairs is not empty |
---|
285 | temp = M[1]; |
---|
286 | temppos = 1; |
---|
287 | for (i = 2; i<=size(M); i++) |
---|
288 | {//finding the minimal element of M |
---|
289 | if (M[i][1]<=temp[1]) |
---|
290 | {//a possible candidate that is smaller than temp could have been found |
---|
291 | if (M[i][1]==temp[1]) |
---|
292 | {//In this case we must look at the second number |
---|
293 | if (M[i][2]< temp[2]) |
---|
294 | {//the candidate is smaller |
---|
295 | temp = M[i]; |
---|
296 | temppos = i; |
---|
297 | }//the candidate is smaller |
---|
298 | }//In this case we must look at the second number |
---|
299 | else |
---|
300 | {//The candidate is definately smaller |
---|
301 | temp = M[i]; |
---|
302 | temppos = i; |
---|
303 | }//The candidate is definately smaller |
---|
304 | }//a possible candidate that is smaller than temp could have been found |
---|
305 | }//finding the minimal element of M |
---|
306 | M = delete(M, temppos); |
---|
307 | if(temp[1]>1) |
---|
308 | {//There are factors to combine before the equal factor |
---|
309 | if (temp[1]<size(f)) |
---|
310 | {//The most common case |
---|
311 | //first the combinations ignoring common factors |
---|
312 | pre = merge_icf(list(f[1..(temp[1]-1)]),list(g[1..(temp[2]-1)]),limits); |
---|
313 | post = merge_icf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
---|
314 | for (i = 1; i <= size(pre); i++) |
---|
315 | {//all possible pre's... |
---|
316 | for (j = 1; j<= size(post); j++) |
---|
317 | {//...combined with all possible post's |
---|
318 | candidate = pre[i]+list(f[temp[1]])+post[j]; |
---|
319 | if (limitcheck(candidate,limits)) |
---|
320 | { |
---|
321 | result = result + list(candidate); |
---|
322 | } |
---|
323 | }//...combined with all possible post's |
---|
324 | }//all possible pre's... |
---|
325 | //Now the combinations with respect to common factors |
---|
326 | post = merge_cf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
---|
327 | if (size(post)>0) |
---|
328 | {//There are factors to combine |
---|
329 | for (i = 1; i <= size(pre); i++) |
---|
330 | {//all possible pre's... |
---|
331 | for (j = 1; j<= size(post); j++) |
---|
332 | {//...combined with all possible post's |
---|
333 | candidate= pre[i]+list(f[temp[1]])+post[j]; |
---|
334 | if (limitcheck(candidate,limits)) |
---|
335 | { |
---|
336 | result = result + list(candidate); |
---|
337 | } |
---|
338 | }//...combined with all possible post's |
---|
339 | }//all possible pre's... |
---|
340 | }//There are factors to combine |
---|
341 | }//The most common case |
---|
342 | else |
---|
343 | {//the last factor is the common one |
---|
344 | pre = merge_icf(list(f[1..(temp[1]-1)]),list(g[1..(temp[2]-1)]),limits); |
---|
345 | for (i = 1; i<= size(pre); i++) |
---|
346 | {//iterating over the possible pre-factors |
---|
347 | candidate = pre[i]+list(f[temp[1]]); |
---|
348 | if (limitcheck(candidate,limits)) |
---|
349 | { |
---|
350 | result = result + list(candidate); |
---|
351 | } |
---|
352 | }//iterating over the possible pre-factors |
---|
353 | }//the last factor is the common one |
---|
354 | }//There are factors to combine before the equal factor |
---|
355 | else |
---|
356 | {//There are no factors to combine before the equal factor |
---|
357 | if (temp[1]<size(f)) |
---|
358 | {//Just a check for security |
---|
359 | //first without common factors |
---|
360 | post=merge_icf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
---|
361 | for (i = 1; i<=size(post); i++) |
---|
362 | { |
---|
363 | candidate = list(f[temp[1]])+post[i]; |
---|
364 | if (limitcheck(candidate,limits)) |
---|
365 | { |
---|
366 | result = result + list(candidate); |
---|
367 | } |
---|
368 | } |
---|
369 | //Now with common factors |
---|
370 | post = merge_cf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
---|
371 | if(size(post)>0) |
---|
372 | {//we could find other combinations |
---|
373 | for (i = 1; i<=size(post); i++) |
---|
374 | { |
---|
375 | candidate = list(f[temp[1]])+post[i]; |
---|
376 | if (limitcheck(candidate,limits)) |
---|
377 | { |
---|
378 | result = result + list(candidate); |
---|
379 | } |
---|
380 | } |
---|
381 | }//we could find other combinations |
---|
382 | }//Just a check for security |
---|
383 | }//There are no factors to combine before the equal factor |
---|
384 | }//set of equal pairs is not empty |
---|
385 | return(result); |
---|
386 | }//proc merge_cf |
---|
387 | |
---|
388 | |
---|
389 | //==================================================* |
---|
390 | //merges two sets of factors |
---|
391 | |
---|
392 | static proc mergence(list l1, list l2, intvec limits) |
---|
393 | {//Procedure mergence |
---|
394 | list g; |
---|
395 | list f; |
---|
396 | int l; int k; |
---|
397 | list F; |
---|
398 | if (size(l2)<=size(l1)) |
---|
399 | {//l1 will be our g, l2 our f |
---|
400 | g = l1; |
---|
401 | f = l2; |
---|
402 | }//l1 will be our g, l2 our f |
---|
403 | else |
---|
404 | {//l1 will be our f, l2 our g |
---|
405 | g = l2; |
---|
406 | f = l1; |
---|
407 | }//l1 will be our f, l2 our g |
---|
408 | list result; |
---|
409 | for (l = size(f); l>=1; l--) |
---|
410 | {//all possibilities to combine the factors of f |
---|
411 | F = combinekfinlf(f,l,limits); |
---|
412 | for (k = 1; k<= size(F);k++) |
---|
413 | {//for all possibilities of combinations of the factors of f |
---|
414 | result = result + merge_cf(F[k],g,limits); |
---|
415 | result = result + merge_icf(F[k],g,limits); |
---|
416 | }//for all possibilities of combinations of the factors of f |
---|
417 | }//all possibilities to combine the factors of f |
---|
418 | return(result); |
---|
419 | }//Procedure mergence |
---|
420 | |
---|
421 | |
---|
422 | //================================================== |
---|
423 | //Checks, whether a list of factors doesn't exceed the given limits |
---|
424 | static proc limitcheck(list g, intvec limits) |
---|
425 | {//proc limitcheck |
---|
426 | int i; |
---|
427 | if (size(limits)!=3) |
---|
428 | {//check the input |
---|
429 | return(0); |
---|
430 | }//check the input |
---|
431 | if(size(g)==0) |
---|
432 | { |
---|
433 | return(0); |
---|
434 | } |
---|
435 | def prod = product(g); |
---|
436 | def limg = intvec(deg(prod,intvec(1,1)) ,deg(prod,intvec(1,0)),deg(prod,intvec(0,1))); |
---|
437 | for (i = 1; i<=size(limg);i++) |
---|
438 | {//the final check |
---|
439 | if(limg[i]>limits[i]) |
---|
440 | { |
---|
441 | return(0); |
---|
442 | } |
---|
443 | }//the final check |
---|
444 | return(1); |
---|
445 | }//proc limitcheck |
---|
446 | |
---|
447 | |
---|
448 | //==================================================* |
---|
449 | //one factorization of a homogeneous polynomial |
---|
450 | //in the first Weyl Algebra |
---|
451 | static proc homogfacFirstWeyl(poly h) |
---|
452 | "USAGE: homogfacFirstWeyl(h); h is a homogeneous polynomial in the |
---|
453 | first Weyl algebra with respect to the weight vector [-1,1] |
---|
454 | RETURN: list |
---|
455 | PURPOSE: Computes a factorization of a homogeneous polynomial h with |
---|
456 | respect to the weight vector [-1,1] in the first Weyl algebra |
---|
457 | THEORY: @code{homogfacFirstWeyl} returns a list with a factorization of the given, |
---|
458 | [-1,1]-homogeneous polynomial. If the degree of the polynomial is k with |
---|
459 | k positive, the last k entries in the output list are the second |
---|
460 | variable. If k is positive, the last k entries will be x. The other |
---|
461 | entries will be irreducible polynomials of degree zero or 1 resp. -1. |
---|
462 | SEE ALSO: homogfacFirstWeyl_all |
---|
463 | "{//proc homogfacFirstWeyl |
---|
464 | def r = basering; |
---|
465 | poly hath; |
---|
466 | int i; int j; |
---|
467 | if (!homogwithorder(h,intvec(-1,1))) |
---|
468 | {//The given polynomial is not homogeneous |
---|
469 | return(list()); |
---|
470 | }//The given polynomial is not homogeneous |
---|
471 | if (h==0) |
---|
472 | { |
---|
473 | return(list(0)); |
---|
474 | } |
---|
475 | list result; |
---|
476 | int m = deg(h,intvec(-1,1)); |
---|
477 | if (m!=0) |
---|
478 | {//The degree is not zero |
---|
479 | if (m <0) |
---|
480 | {//There are more x than y |
---|
481 | hath = lift(var(1)^(-m),h)[1,1]; |
---|
482 | for (i = 1; i<=-m; i++) |
---|
483 | { |
---|
484 | result = result + list(var(1)); |
---|
485 | } |
---|
486 | }//There are more x than y |
---|
487 | else |
---|
488 | {//There are more y than x |
---|
489 | hath = lift(var(2)^m,h)[1,1]; |
---|
490 | for (i = 1; i<=m;i++) |
---|
491 | { |
---|
492 | result = result + list(var(2)); |
---|
493 | } |
---|
494 | }//There are more y than x |
---|
495 | }//The degree is not zero |
---|
496 | else |
---|
497 | {//The degree is zero |
---|
498 | hath = h; |
---|
499 | }//The degree is zero |
---|
500 | //beginning to transform x^i*y^i in theta(theta-1)...(theta-i+1) |
---|
501 | list mons; |
---|
502 | for(i = 1; i<=size(hath);i++) |
---|
503 | {//Putting the monomials in a list |
---|
504 | mons = mons+list(hath[i]); |
---|
505 | }//Putting the monomials in a list |
---|
506 | ring tempRing = 0,(x,y,theta),dp; |
---|
507 | setring tempRing; |
---|
508 | map thetamap = r,x,y; |
---|
509 | list mons = thetamap(mons); |
---|
510 | poly entry; |
---|
511 | for (i = 1; i<=size(mons);i++) |
---|
512 | {//transforming the monomials as monomials in theta |
---|
513 | entry = leadcoef(mons[i]); |
---|
514 | for (j = 0; j<leadexp(mons[i])[2];j++) |
---|
515 | { |
---|
516 | entry = entry * (theta-j); |
---|
517 | } |
---|
518 | mons[i] = entry; |
---|
519 | }//transforming the monomials as monomials in theta |
---|
520 | list azeroresult = factorize(sum(mons)); |
---|
521 | list azeroresult_return_form; |
---|
522 | for (i = 1; i<=size(azeroresult[1]);i++) |
---|
523 | {//rewrite the result of the commutative factorization |
---|
524 | for (j = 1; j <= azeroresult[2][i];j++) |
---|
525 | { |
---|
526 | azeroresult_return_form = azeroresult_return_form + list(azeroresult[1][i]); |
---|
527 | } |
---|
528 | }//rewrite the result of the commutative factorization |
---|
529 | setring(r); |
---|
530 | map finalmap = tempRing,var(1),var(2),var(1)*var(2); |
---|
531 | list tempresult = finalmap(azeroresult_return_form); |
---|
532 | for (i = 1; i<=size(tempresult);i++) |
---|
533 | {//factorizations of theta resp. theta +1 |
---|
534 | if(tempresult[i]==var(1)*var(2)) |
---|
535 | { |
---|
536 | tempresult = insert(tempresult,var(1),i-1); |
---|
537 | i++; |
---|
538 | tempresult[i]=var(2); |
---|
539 | } |
---|
540 | if(tempresult[i]==var(2)*var(1)) |
---|
541 | { |
---|
542 | tempresult = insert(tempresult,var(2),i-1); |
---|
543 | i++; |
---|
544 | tempresult[i]=var(1); |
---|
545 | } |
---|
546 | }//factorizations of theta resp. theta +1 |
---|
547 | result = tempresult+result; |
---|
548 | return(result); |
---|
549 | }//proc homogfacFirstWeyl |
---|
550 | /* example */ |
---|
551 | /* { */ |
---|
552 | /* "EXAMPLE:";echo=2; */ |
---|
553 | /* ring R = 0,(x,y),Ws(-1,1); */ |
---|
554 | /* def r = nc_algebra(1,1); */ |
---|
555 | /* setring(r); */ |
---|
556 | /* poly h = (x^2*y^2+1)*(x^4); */ |
---|
557 | /* homogfacFirstWeyl(h); */ |
---|
558 | /* } */ |
---|
559 | |
---|
560 | //================================================== |
---|
561 | //Computes all possible homogeneous factorizations |
---|
562 | static proc homogfacFirstWeyl_all(poly h) |
---|
563 | "USAGE: homogfacFirstWeyl_all(h); h is a homogeneous polynomial in the first Weyl algebra |
---|
564 | with respect to the weight vector [-1,1] |
---|
565 | RETURN: list |
---|
566 | PURPOSE: Computes all factorizations of a homogeneous polynomial h with respect |
---|
567 | to the weight vector [-1,1] in the first Weyl algebra |
---|
568 | THEORY: @code{homogfacFirstWeyl} returns a list with all factorization of the given, |
---|
569 | homogeneous polynomial. It uses the output of homogfacFirstWeyl and permutes |
---|
570 | its entries with respect to the commutation rule. Furthermore, if a |
---|
571 | factor of degree zero is irreducible in K[\theta], but reducible in |
---|
572 | the first Weyl algebra, the permutations of this element with the other |
---|
573 | entries will also be computed. |
---|
574 | SEE ALSO: homogfacFirstWeyl |
---|
575 | "{//proc HomogfacFirstWeylAll |
---|
576 | if (deg(h,intvec(1,1)) <= 0 ) |
---|
577 | {//h is a constant |
---|
578 | return(list(list(h))); |
---|
579 | }//h is a constant |
---|
580 | def r = basering; |
---|
581 | list one_hom_fac; //stands for one homogeneous factorization |
---|
582 | int i; int j; int k; |
---|
583 | //Compute again a homogeneous factorization |
---|
584 | one_hom_fac = homogfacFirstWeyl(h); |
---|
585 | if (size(one_hom_fac) == 0) |
---|
586 | {//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
587 | return(list()); |
---|
588 | }//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
589 | //divide list in A0-Part and a list of x's resp. y's |
---|
590 | list list_not_azero = list(); |
---|
591 | list list_azero; |
---|
592 | list k_factor; |
---|
593 | int is_list_not_azero_empty = 1; |
---|
594 | int is_list_azero_empty = 1; |
---|
595 | k_factor = list(one_hom_fac[1]); |
---|
596 | if (absValue(deg(h,intvec(-1,1)))<size(one_hom_fac)-1) |
---|
597 | {//There is a nontrivial A_0-part |
---|
598 | list_azero = one_hom_fac[2..(size(one_hom_fac)-absValue(deg(h,intvec(-1,1))))]; |
---|
599 | is_list_azero_empty = 0; |
---|
600 | }//There is a nontrivial A_0 part |
---|
601 | for (i = 1; i<=size(list_azero)-1;i++) |
---|
602 | {//in homogfacFirstWeyl, we factorized theta, and this will be made undone |
---|
603 | if (list_azero[i] == var(1)) |
---|
604 | { |
---|
605 | if (list_azero[i+1]==var(2)) |
---|
606 | { |
---|
607 | list_azero[i] = var(1)*var(2); |
---|
608 | list_azero = delete(list_azero,i+1); |
---|
609 | } |
---|
610 | } |
---|
611 | if (list_azero[i] == var(2)) |
---|
612 | { |
---|
613 | if (list_azero[i+1]==var(1)) |
---|
614 | { |
---|
615 | list_azero[i] = var(2)*var(1); |
---|
616 | list_azero = delete(list_azero,i+1); |
---|
617 | } |
---|
618 | } |
---|
619 | }//in homogfacFirstWeyl, we factorized theta, and this will be made undone |
---|
620 | if(deg(h,intvec(-1,1))!=0) |
---|
621 | {//list_not_azero is not empty |
---|
622 | list_not_azero = |
---|
623 | one_hom_fac[(size(one_hom_fac)-absValue(deg(h,intvec(-1,1)))+1)..size(one_hom_fac)]; |
---|
624 | is_list_not_azero_empty = 0; |
---|
625 | }//list_not_azero is not empty |
---|
626 | //Map list_azero in K[theta] |
---|
627 | ring tempRing = 0,(x,y,theta), dp; |
---|
628 | setring(tempRing); |
---|
629 | poly entry; |
---|
630 | map thetamap = r,x,y; |
---|
631 | if(!is_list_not_azero_empty) |
---|
632 | {//Mapping in Singular is only possible, if the list before |
---|
633 | //contained at least one element of the other ring |
---|
634 | list list_not_azero = thetamap(list_not_azero); |
---|
635 | }//Mapping in Singular is only possible, if the list before |
---|
636 | //contained at least one element of the other ring |
---|
637 | if(!is_list_azero_empty) |
---|
638 | {//Mapping in Singular is only possible, if the list before |
---|
639 | //contained at least one element of the other ring |
---|
640 | list list_azero= thetamap(list_azero); |
---|
641 | }//Mapping in Singular is only possible, if the list before |
---|
642 | //contained at least one element of the other ring |
---|
643 | list k_factor = thetamap(k_factor); |
---|
644 | list tempmons; |
---|
645 | for(i = 1; i<=size(list_azero);i++) |
---|
646 | {//rewrite the polynomials in A1 as polynomials in K[theta] |
---|
647 | tempmons = list(); |
---|
648 | for (j = 1; j<=size(list_azero[i]);j++) |
---|
649 | { |
---|
650 | tempmons = tempmons + list(list_azero[i][j]); |
---|
651 | } |
---|
652 | for (j = 1 ; j<=size(tempmons);j++) |
---|
653 | { |
---|
654 | entry = leadcoef(tempmons[j]); |
---|
655 | for (k = 0; k < leadexp(tempmons[j])[2];k++) |
---|
656 | { |
---|
657 | entry = entry*(theta-k); |
---|
658 | } |
---|
659 | tempmons[j] = entry; |
---|
660 | } |
---|
661 | list_azero[i] = sum(tempmons); |
---|
662 | }//rewrite the polynomials in A1 as polynomials in K[theta] |
---|
663 | //Compute all permutations of the A0-part |
---|
664 | list result; |
---|
665 | int shift_sign; |
---|
666 | int shift; |
---|
667 | poly shiftvar; |
---|
668 | if (size(list_not_azero)!=0) |
---|
669 | {//Compute all possibilities to permute the x's resp. the y's in the list |
---|
670 | if (list_not_azero[1] == x) |
---|
671 | {//h had a negative weighted degree |
---|
672 | shift_sign = 1; |
---|
673 | shiftvar = x; |
---|
674 | }//h had a negative weighted degree |
---|
675 | else |
---|
676 | {//h had a positive weighted degree |
---|
677 | shift_sign = -1; |
---|
678 | shiftvar = y; |
---|
679 | }//h had a positive weighted degree |
---|
680 | result = permpp(list_azero + list_not_azero); |
---|
681 | for (i = 1; i<= size(result); i++) |
---|
682 | {//adjust the a_0-parts |
---|
683 | shift = 0; |
---|
684 | for (j=1; j<=size(result[i]);j++) |
---|
685 | { |
---|
686 | if (result[i][j]==shiftvar) |
---|
687 | { |
---|
688 | shift = shift + shift_sign; |
---|
689 | } |
---|
690 | else |
---|
691 | { |
---|
692 | result[i][j] = subst(result[i][j],theta,theta + shift); |
---|
693 | } |
---|
694 | } |
---|
695 | }//adjust the a_0-parts |
---|
696 | }//Compute all possibilities to permute the x's resp. the y's in the list |
---|
697 | else |
---|
698 | {//The result is just all the permutations of the a_0-part |
---|
699 | result = permpp(list_azero); |
---|
700 | }//The result is just all the permutations of the a_0 part |
---|
701 | if (size(result)==0) |
---|
702 | { |
---|
703 | return(result); |
---|
704 | } |
---|
705 | //Now we are going deeper and search for theta resp. theta + 1, substitute |
---|
706 | //them by xy resp. yx and go on permuting |
---|
707 | int found_theta; |
---|
708 | int thetapos; |
---|
709 | list leftpart; |
---|
710 | list rightpart; |
---|
711 | list lparts; |
---|
712 | list rparts; |
---|
713 | list tempadd; |
---|
714 | for (i = 1; i<=size(result) ; i++) |
---|
715 | {//checking every entry of result for theta or theta +1 |
---|
716 | found_theta = 0; |
---|
717 | for(j=1;j<=size(result[i]);j++) |
---|
718 | { |
---|
719 | if (result[i][j]==theta) |
---|
720 | {//the jth entry is theta and can be written as x*y |
---|
721 | thetapos = j; |
---|
722 | result[i]= insert(result[i],x,j-1); |
---|
723 | j++; |
---|
724 | result[i][j] = y; |
---|
725 | found_theta = 1; |
---|
726 | break; |
---|
727 | }//the jth entry is theta and can be written as x*y |
---|
728 | if(result[i][j] == theta +1) |
---|
729 | { |
---|
730 | thetapos = j; |
---|
731 | result[i] = insert(result[i],y,j-1); |
---|
732 | j++; |
---|
733 | result[i][j] = x; |
---|
734 | found_theta = 1; |
---|
735 | break; |
---|
736 | } |
---|
737 | } |
---|
738 | if (found_theta) |
---|
739 | {//One entry was theta resp. theta +1 |
---|
740 | leftpart = result[i]; |
---|
741 | leftpart = leftpart[1..thetapos]; |
---|
742 | rightpart = result[i]; |
---|
743 | rightpart = rightpart[(thetapos+1)..size(rightpart)]; |
---|
744 | lparts = list(leftpart); |
---|
745 | rparts = list(rightpart); |
---|
746 | //first deal with the left part |
---|
747 | if (leftpart[thetapos] == x) |
---|
748 | { |
---|
749 | shift_sign = 1; |
---|
750 | shiftvar = x; |
---|
751 | } |
---|
752 | else |
---|
753 | { |
---|
754 | shift_sign = -1; |
---|
755 | shiftvar = y; |
---|
756 | } |
---|
757 | for (j = size(leftpart); j>1;j--) |
---|
758 | {//drip x resp. y |
---|
759 | if (leftpart[j-1]==shiftvar) |
---|
760 | {//commutative |
---|
761 | j--; |
---|
762 | continue; |
---|
763 | }//commutative |
---|
764 | if (deg(leftpart[j-1],intvec(-1,1,0))!=0) |
---|
765 | {//stop here |
---|
766 | break; |
---|
767 | }//stop here |
---|
768 | //Here, we can only have a a0- part |
---|
769 | leftpart[j] = subst(leftpart[j-1],theta, theta + shift_sign); |
---|
770 | leftpart[j-1] = shiftvar; |
---|
771 | lparts = lparts + list(leftpart); |
---|
772 | }//drip x resp. y |
---|
773 | //and now deal with the right part |
---|
774 | if (rightpart[1] == x) |
---|
775 | { |
---|
776 | shift_sign = 1; |
---|
777 | shiftvar = x; |
---|
778 | } |
---|
779 | else |
---|
780 | { |
---|
781 | shift_sign = -1; |
---|
782 | shiftvar = y; |
---|
783 | } |
---|
784 | for (j = 1 ; j < size(rightpart); j++) |
---|
785 | { |
---|
786 | if (rightpart[j+1] == shiftvar) |
---|
787 | { |
---|
788 | j++; |
---|
789 | continue; |
---|
790 | } |
---|
791 | if (deg(rightpart[j+1],intvec(-1,1,0))!=0) |
---|
792 | { |
---|
793 | break; |
---|
794 | } |
---|
795 | rightpart[j] = subst(rightpart[j+1], theta, theta - shift_sign); |
---|
796 | rightpart[j+1] = shiftvar; |
---|
797 | rparts = rparts + list(rightpart); |
---|
798 | } |
---|
799 | //And now, we put all possibilities together |
---|
800 | tempadd = list(); |
---|
801 | for (j = 1; j<=size(lparts); j++) |
---|
802 | { |
---|
803 | for (k = 1; k<=size(rparts);k++) |
---|
804 | { |
---|
805 | tempadd = tempadd + list(lparts[j]+rparts[k]); |
---|
806 | } |
---|
807 | } |
---|
808 | tempadd = delete(tempadd,1); // The first entry is already in the list |
---|
809 | result = result + tempadd; |
---|
810 | continue; //We can may be not be done already with the ith entry |
---|
811 | }//One entry was theta resp. theta +1 |
---|
812 | }//checking every entry of result for theta or theta +1 |
---|
813 | //map back to the basering |
---|
814 | setring(r); |
---|
815 | map finalmap = tempRing, var(1), var(2),var(1)*var(2); |
---|
816 | list result = finalmap(result); |
---|
817 | for (i=1; i<=size(result);i++) |
---|
818 | {//adding the K factor |
---|
819 | result[i] = k_factor + result[i]; |
---|
820 | }//adding the k-factor |
---|
821 | result = delete_dublicates_noteval(result); |
---|
822 | return(result); |
---|
823 | }//proc HomogfacFirstWeylAll |
---|
824 | /* example */ |
---|
825 | /* { */ |
---|
826 | /* "EXAMPLE:";echo=2; */ |
---|
827 | /* ring R = 0,(x,y),Ws(-1,1); */ |
---|
828 | /* def r = nc_algebra(1,1); */ |
---|
829 | /* setring(r); */ |
---|
830 | /* poly h = (x^2*y^2+1)*(x^4); */ |
---|
831 | /* homogfacFirstWeyl_all(h); */ |
---|
832 | /* } */ |
---|
833 | |
---|
834 | //==================================================* |
---|
835 | //Computes all permutations of a given list |
---|
836 | static proc perm(list l) |
---|
837 | {//proc perm |
---|
838 | int i; int j; |
---|
839 | list tempresult; |
---|
840 | list result; |
---|
841 | if (size(l)==0) |
---|
842 | { |
---|
843 | return(list()); |
---|
844 | } |
---|
845 | if (size(l)==1) |
---|
846 | { |
---|
847 | return(list(l)); |
---|
848 | } |
---|
849 | for (i = 1; i<=size(l); i++ ) |
---|
850 | { |
---|
851 | tempresult = perm(delete(l,i)); |
---|
852 | for (j = 1; j<=size(tempresult);j++) |
---|
853 | { |
---|
854 | tempresult[j] = list(l[i])+tempresult[j]; |
---|
855 | } |
---|
856 | result = result+tempresult; |
---|
857 | } |
---|
858 | return(result); |
---|
859 | }//proc perm |
---|
860 | |
---|
861 | //================================================== |
---|
862 | //computes all permutations of a given list by |
---|
863 | //ignoring equal entries (faster than perm) |
---|
864 | static proc permpp(list l) |
---|
865 | {//proc permpp |
---|
866 | int i; int j; |
---|
867 | list tempresult; |
---|
868 | list l_without_double; |
---|
869 | list l_without_double_pos; |
---|
870 | int double_entry; |
---|
871 | list result; |
---|
872 | if (size(l)==0) |
---|
873 | { |
---|
874 | return(list()); |
---|
875 | } |
---|
876 | if (size(l)==1) |
---|
877 | { |
---|
878 | return(list(l)); |
---|
879 | } |
---|
880 | for (i = 1; i<=size(l);i++) |
---|
881 | {//Filling the list with unique entries |
---|
882 | double_entry = 0; |
---|
883 | for (j = 1; j<=size(l_without_double);j++) |
---|
884 | { |
---|
885 | if (l_without_double[j] == l[i]) |
---|
886 | { |
---|
887 | double_entry = 1; |
---|
888 | break; |
---|
889 | } |
---|
890 | } |
---|
891 | if (!double_entry) |
---|
892 | { |
---|
893 | l_without_double = l_without_double + list(l[i]); |
---|
894 | l_without_double_pos = l_without_double_pos + list(i); |
---|
895 | } |
---|
896 | }//Filling the list with unique entries |
---|
897 | for (i = 1; i<=size(l_without_double); i++ ) |
---|
898 | { |
---|
899 | tempresult = permpp(delete(l,l_without_double_pos[i])); |
---|
900 | for (j = 1; j<=size(tempresult);j++) |
---|
901 | { |
---|
902 | tempresult[j] = list(l_without_double[i])+tempresult[j]; |
---|
903 | } |
---|
904 | result = result+tempresult; |
---|
905 | } |
---|
906 | return(result); |
---|
907 | }//proc permpp |
---|
908 | |
---|
909 | //================================================== |
---|
910 | //factorization of the first Weyl Algebra |
---|
911 | |
---|
912 | //The following procedure just serves the purpose to |
---|
913 | //transform the input into an appropriate input for |
---|
914 | //the procedure sfacwa, where the ring must contain the |
---|
915 | //variables in a certain order. |
---|
916 | proc facFirstWeyl(poly h) |
---|
917 | "USAGE: facFirstWeyl(h); h a polynomial in the first Weyl algebra |
---|
918 | RETURN: list |
---|
919 | PURPOSE: compute all factorizations of a polynomial in the first Weyl algebra |
---|
920 | THEORY: Implements the new algorithm by A. Heinle and V. Levandovskyy, see the thesis of A. Heinle |
---|
921 | ASSUME: basering in the first Weyl algebra |
---|
922 | NOTE: Every entry of the output list is a list with factors for one possible factorization. |
---|
923 | The first factor is always a constant (1, if no nontrivial constant could be excluded). |
---|
924 | EXAMPLE: example facFirstWeyl; shows examples |
---|
925 | SEE ALSO: facSubWeyl, testNCfac, facFirstShift |
---|
926 | "{//proc facFirstWeyl |
---|
927 | //Redefine the ring in my standard form |
---|
928 | if (!isWeyl()) |
---|
929 | {//Our basering is not the Weyl algebra |
---|
930 | return(list()); |
---|
931 | }//Our basering is not the Weyl algebra |
---|
932 | if(nvars(basering)!=2) |
---|
933 | {//Our basering is the Weyl algebra, but not the first |
---|
934 | return(list()); |
---|
935 | }//Our basering is the Weyl algebra, but not the first |
---|
936 | list result = list(); |
---|
937 | int i;int j; int k; int l; //counter |
---|
938 | if (ringlist(basering)[6][1,2] == -1) //manual of ringlist will tell you why |
---|
939 | { |
---|
940 | def r = basering; |
---|
941 | ring tempRing = ringlist(r)[1][1],(x,y),Ws(-1,1); // very strange: |
---|
942 | // setting Wp(-1,1) leads to SegFault; to clarify why!!! |
---|
943 | def NTR = nc_algebra(1,1); |
---|
944 | setring NTR ; |
---|
945 | map transf = r, var(2), var(1); |
---|
946 | list resulttemp = sfacwa(transf(h)); |
---|
947 | setring(r); |
---|
948 | map transfback = NTR, var(2),var(1); |
---|
949 | result = transfback(resulttemp); |
---|
950 | } |
---|
951 | else { result = sfacwa(h);} |
---|
952 | list recursivetemp; |
---|
953 | for(i = 1; i<=size(result);i++) |
---|
954 | {//recursively factorize factors |
---|
955 | if(size(result[i])>2) |
---|
956 | {//Nontrivial factorization |
---|
957 | for (j=2;j<=size(result[i]);j++) |
---|
958 | {//Factorize every factor |
---|
959 | recursivetemp = facFirstWeyl(result[i][j]); |
---|
960 | if(size(recursivetemp)>1) |
---|
961 | {//we have a nontrivial factorization |
---|
962 | for(k=1; k<=size(recursivetemp);k++) |
---|
963 | {//insert factorized factors |
---|
964 | if(size(recursivetemp[k])>2) |
---|
965 | {//nontrivial |
---|
966 | result = insert(result,result[i],i); |
---|
967 | for(l = size(recursivetemp[k]);l>=2;l--) |
---|
968 | { |
---|
969 | result[i+1] = insert(result[i+1],recursivetemp[k][l],j); |
---|
970 | } |
---|
971 | result[i+1] = delete(result[i+1],j); |
---|
972 | }//nontrivial |
---|
973 | }//insert factorized factors |
---|
974 | }//we have a nontrivial factorization |
---|
975 | }//Factorize every factor |
---|
976 | }//Nontrivial factorization |
---|
977 | }//recursively factorize factors |
---|
978 | if (size(result)==0) |
---|
979 | {//only the trivial factorization could be found |
---|
980 | result = list(list(1,h)); |
---|
981 | }//only the trivial factorization could be found |
---|
982 | //now, refine the possible redundant list |
---|
983 | return( delete_dublicates_noteval(result) ); |
---|
984 | }//proc facFirstWeyl |
---|
985 | example |
---|
986 | { |
---|
987 | "EXAMPLE:";echo=2; |
---|
988 | ring R = 0,(x,y),dp; |
---|
989 | def r = nc_algebra(1,1); |
---|
990 | setring(r); |
---|
991 | poly h = (x^2*y^2+x)*(x+1); |
---|
992 | facFirstWeyl(h); |
---|
993 | } |
---|
994 | |
---|
995 | //This is the main program |
---|
996 | static proc sfacwa(poly h) |
---|
997 | "USAGE: sfacwa(h); h is a polynomial in the first Weyl algebra |
---|
998 | RETURN: list |
---|
999 | PURPOSE: Computes a factorization of a polynomial h in the first Weyl algebra |
---|
1000 | THEORY: @code{sfacwa} returns a list with some factorizations of the given |
---|
1001 | polynomial. The possibilities of the factorization of the highest |
---|
1002 | homogeneous part and those of the lowest will be merged. If during this |
---|
1003 | procedure a factorization of the polynomial occurs, it will be added to |
---|
1004 | the output list. For a more detailed description visit |
---|
1005 | @url{http://www.math.rwth-aachen.de/\~Albert.Heinle} |
---|
1006 | SEE ALSO: homogfacFirstWeyl_all, homogfacFirstWeyl |
---|
1007 | "{//proc sfacwa |
---|
1008 | if(homogwithorder(h,intvec(-1,1))) |
---|
1009 | { |
---|
1010 | return(homogfacFirstWeyl_all(h)); |
---|
1011 | } |
---|
1012 | def r = basering; |
---|
1013 | map invo = basering,-var(1),var(2); |
---|
1014 | int i; int j; int k; |
---|
1015 | intvec limits = deg(h,intvec(1,1)) ,deg(h,intvec(1,0)),deg(h,intvec(0,1)); |
---|
1016 | def prod; |
---|
1017 | //end finding the limits |
---|
1018 | poly maxh = jet(h,deg(h,intvec(-1,1)),intvec(-1,1))-jet(h,deg(h,intvec(-1,1))-1,intvec(-1,1)); |
---|
1019 | poly minh = jet(h,deg(h,intvec(1,-1)),intvec(1,-1))-jet(h,deg(h,intvec(1,-1))-1,intvec(1,-1)); |
---|
1020 | list result; |
---|
1021 | list temp; |
---|
1022 | list homogtemp; |
---|
1023 | list M; list hatM; |
---|
1024 | list f1 = homogfacFirstWeyl_all(maxh); |
---|
1025 | list f2 = homogfacFirstWeyl_all(minh); |
---|
1026 | int is_equal; |
---|
1027 | poly hath; |
---|
1028 | for (i = 1; i<=size(f1);i++) |
---|
1029 | {//Merge all combinations |
---|
1030 | for (j = 1; j<=size(f2); j++) |
---|
1031 | { |
---|
1032 | M = M + mergence(f1[i],f2[j],limits); |
---|
1033 | } |
---|
1034 | }//Merge all combinations |
---|
1035 | for (i = 1 ; i<= size(M); i++) |
---|
1036 | {//filter valid combinations |
---|
1037 | if (product(M[i]) == h) |
---|
1038 | {//We have one factorization |
---|
1039 | result = result + list(M[i]); |
---|
1040 | M = delete(M,i); |
---|
1041 | continue; |
---|
1042 | }//We have one factorization |
---|
1043 | else |
---|
1044 | { |
---|
1045 | if (deg(h,intvec(-1,1))<=deg(h-product(M[i]),intvec(-1,1))) |
---|
1046 | { |
---|
1047 | M = delete(M,i); |
---|
1048 | continue; |
---|
1049 | } |
---|
1050 | if (deg(h,intvec(1,-1))<=deg(h-product(M[i]),intvec(1,-1))) |
---|
1051 | { |
---|
1052 | M = delete(M,i); |
---|
1053 | continue; |
---|
1054 | } |
---|
1055 | } |
---|
1056 | }//filter valid combinations |
---|
1057 | M = delete_dublicates_eval(M); |
---|
1058 | while(size(M)>0) |
---|
1059 | {//work on the elements of M |
---|
1060 | hatM = list(); |
---|
1061 | for(i = 1; i<=size(M); i++) |
---|
1062 | {//iterate over all elements of M |
---|
1063 | hath = h-product(M[i]); |
---|
1064 | temp = list(); |
---|
1065 | //First check for common inhomogeneous factors between hath and h |
---|
1066 | if (involution(NF(involution(hath,invo), std(involution(ideal(M[i][1]),invo))),invo)==0) |
---|
1067 | {//hath and h have a common factor on the left |
---|
1068 | j = 1; |
---|
1069 | f1 = M[i]; |
---|
1070 | if (j+1<=size(f1)) |
---|
1071 | {//Checking for more than one common factor |
---|
1072 | while(involution(NF(involution(hath,invo),std(involution(ideal(product(f1[1..(j+1)])),invo))),invo)==0) |
---|
1073 | { |
---|
1074 | if (j+1<size(f1)) |
---|
1075 | { |
---|
1076 | j++; |
---|
1077 | } |
---|
1078 | else |
---|
1079 | { |
---|
1080 | break; |
---|
1081 | } |
---|
1082 | } |
---|
1083 | }//Checking for more than one common factor |
---|
1084 | f2 = list(f1[1..j])+list(involution(lift(involution(product(f1[1..j]),invo),involution(hath,invo))[1,1],invo)); |
---|
1085 | temp = temp + merge_cf(f2,f1,limits); |
---|
1086 | }//hath and h have a common factor on the left |
---|
1087 | if (reduce(hath, std(ideal(M[i][size(M[i])])))==0) |
---|
1088 | {//hath and h have a common factor on the right |
---|
1089 | j = size(M[i]); |
---|
1090 | f1 = M[i]; |
---|
1091 | if (j-1>0) |
---|
1092 | {//Checking for more than one factor |
---|
1093 | while(reduce(hath,std(ideal(product(f1[(j-1)..size(f1)]))))==0) |
---|
1094 | { |
---|
1095 | if (j-1>1) |
---|
1096 | { |
---|
1097 | j--; |
---|
1098 | } |
---|
1099 | else |
---|
1100 | { |
---|
1101 | break; |
---|
1102 | } |
---|
1103 | } |
---|
1104 | }//Checking for more than one factor |
---|
1105 | f2 = list(lift(product(f1[j..size(f1)]),hath)[1,1])+list(f1[j..size(f1)]); |
---|
1106 | temp = temp + merge_cf(f2,M[i],limits); |
---|
1107 | }//hath and h have a common factor on the right |
---|
1108 | //and now the homogeneous |
---|
1109 | maxh = jet(hath,deg(hath,intvec(-1,1)),intvec(-1,1))-jet(hath,deg(hath,intvec(-1,1))-1,intvec(-1,1)); |
---|
1110 | minh = jet(hath,deg(hath,intvec(1,-1)),intvec(1,-1))-jet(hath,deg(hath,intvec(1,-1))-1,intvec(1,-1)); |
---|
1111 | f1 = homogfacFirstWeyl_all(maxh); |
---|
1112 | f2 = homogfacFirstWeyl_all(minh); |
---|
1113 | for (j = 1; j<=size(f1);j++) |
---|
1114 | { |
---|
1115 | for (k=1; k<=size(f2);k++) |
---|
1116 | { |
---|
1117 | homogtemp = mergence(f1[j],f2[k],limits); |
---|
1118 | } |
---|
1119 | } |
---|
1120 | for (j = 1; j<= size(homogtemp); j++) |
---|
1121 | { |
---|
1122 | temp = temp + mergence(homogtemp[j],M[i],limits); |
---|
1123 | } |
---|
1124 | for (j = 1; j<=size(temp); j++) |
---|
1125 | {//filtering invalid entries in temp |
---|
1126 | if(product(temp[j])==h) |
---|
1127 | {//This is already a result |
---|
1128 | result = result + list(temp[j]); |
---|
1129 | temp = delete(temp,j); |
---|
1130 | continue; |
---|
1131 | }//This is already a result |
---|
1132 | if (deg(hath,intvec(-1,1))<=deg(hath-product(temp[j]),intvec(-1,1))) |
---|
1133 | { |
---|
1134 | temp = delete(temp,j); |
---|
1135 | continue; |
---|
1136 | } |
---|
1137 | }//filtering invalid entries in temp |
---|
1138 | hatM = hatM + temp; |
---|
1139 | }//iterate over all elements of M |
---|
1140 | M = hatM; |
---|
1141 | for (i = 1; i<=size(M);i++) |
---|
1142 | {//checking for complete factorizations |
---|
1143 | if (h == product(M[i])) |
---|
1144 | { |
---|
1145 | result = result + list(M[i]); |
---|
1146 | M = delete(M,i); |
---|
1147 | continue; |
---|
1148 | } |
---|
1149 | }//checking for complete factorizations |
---|
1150 | M = delete_dublicates_eval(M); |
---|
1151 | }//work on the elements of M |
---|
1152 | //In the case, that there is none, write a constant factor before the factor of interest. |
---|
1153 | for (i = 1 ; i<=size(result);i++) |
---|
1154 | {//add a constant factor |
---|
1155 | if (deg(result[i][1],intvec(1,1))!=0) |
---|
1156 | { |
---|
1157 | result[i] = insert(result[i],1); |
---|
1158 | } |
---|
1159 | }//add a constant factor |
---|
1160 | result = delete_dublicates_noteval(result); |
---|
1161 | return(result); |
---|
1162 | }//proc sfacwa |
---|
1163 | |
---|
1164 | |
---|
1165 | //================================================== |
---|
1166 | /*Singular has no way implemented to test polynomials |
---|
1167 | for homogenity with respect to a weight vector. |
---|
1168 | The following procedure does exactly this*/ |
---|
1169 | static proc homogwithorder(poly h, intvec weights) |
---|
1170 | {//proc homogwithorder |
---|
1171 | if(size(weights) != nvars(basering)) |
---|
1172 | {//The user does not know how many variables the current ring has |
---|
1173 | return(0); |
---|
1174 | }//The user does not know how many variables the current ring has |
---|
1175 | int i; |
---|
1176 | int dofp = deg(h,weights); //degree of polynomial |
---|
1177 | for (i = 1; i<=size(h);i++) |
---|
1178 | { |
---|
1179 | if (deg(h[i],weights)!=dofp) |
---|
1180 | { |
---|
1181 | return(0); |
---|
1182 | } |
---|
1183 | } |
---|
1184 | return(1); |
---|
1185 | }//proc homogwithorder |
---|
1186 | |
---|
1187 | //================================================== |
---|
1188 | //Testfac: Given a list with different factorizations of |
---|
1189 | // one polynomial, the following procedure checks |
---|
1190 | // whether they all refer to the same polynomial. |
---|
1191 | // If they do, the output will be a list, that contains |
---|
1192 | // the product of each factorization. If not, the empty |
---|
1193 | // list will be returned. |
---|
1194 | // If the optional argument # is given (i.e. the polynomial |
---|
1195 | // which is factorized by the elements of the given list), |
---|
1196 | // then we look, if the entries are factorizations of p |
---|
1197 | // and if not, a list with the products subtracted by p |
---|
1198 | // will be returned |
---|
1199 | proc testNCfac(list l, list #) |
---|
1200 | "USAGE: testNCfac(l[,p,b]); l is a list, p is an optional poly, b is 1 or 0 |
---|
1201 | RETURN: Case 1: No optional argument. In this case the output is 1, if the |
---|
1202 | entries in the given list represent the same polynomial or 0 |
---|
1203 | otherwise. |
---|
1204 | Case 2: One optional argument p is given. In this case it returns 1, |
---|
1205 | if all the entries in l are factorizations of p, otherwise 0. |
---|
1206 | Case 3: Second optional b is given. In this case a list is returned |
---|
1207 | containing the difference between the product of each entry in |
---|
1208 | l and p. |
---|
1209 | ASSUME: basering is the first Weyl algebra, the entries of l are polynomials |
---|
1210 | PURPOSE: Checks whether a list of factorizations contains factorizations of |
---|
1211 | the same element in the first Weyl algebra |
---|
1212 | THEORY: @code{testNCfac} multiplies out each factorization and checks whether |
---|
1213 | each factorization was a factorization of the same element. |
---|
1214 | @* - if there is only a list given, the output will be 0, if it |
---|
1215 | does not contain factorizations of the same element. Otherwise the output |
---|
1216 | will be 1. |
---|
1217 | @* - if there is a polynomial in the second argument, then the procedure checks |
---|
1218 | whether the given list contains factorizations of this polynomial. If it |
---|
1219 | does, then the output depends on the third argument. If it is not given, |
---|
1220 | the procedure will check whether the factorizations in the list |
---|
1221 | l are associated to this polynomial and return either 1 or 0, respectively. |
---|
1222 | If the third argument is given, the output will be a list with |
---|
1223 | the length of the given one and in each entry is the product of one |
---|
1224 | entry in l subtracted by the polynomial. |
---|
1225 | EXAMPLE: example testNCfac; shows examples |
---|
1226 | SEE ALSO: facFirstWeyl, facSubWeyl, facFirstShift |
---|
1227 | "{//proc testfac |
---|
1228 | if (size(l)==0) |
---|
1229 | {//The empty list is given |
---|
1230 | return(list()); |
---|
1231 | }//The empty list is given |
---|
1232 | if (size(#)>2) |
---|
1233 | {//We want max. one optional argument |
---|
1234 | return(list()); |
---|
1235 | }//We want max. one optional argument |
---|
1236 | list result; |
---|
1237 | int i; int j; |
---|
1238 | if (size(#)==0) |
---|
1239 | {//No optional argument is given |
---|
1240 | int valid = 1; |
---|
1241 | for (i = size(l);i>=1;i--) |
---|
1242 | {//iterate over the elements of the given list |
---|
1243 | if (size(result)>0) |
---|
1244 | { |
---|
1245 | if (product(l[i])!=result[size(l)-i]) |
---|
1246 | { |
---|
1247 | valid = 0; |
---|
1248 | break; |
---|
1249 | } |
---|
1250 | } |
---|
1251 | result = insert(result, product(l[i])); |
---|
1252 | }//iterate over the elements of the given list |
---|
1253 | return(valid); |
---|
1254 | }//No optional argument is given |
---|
1255 | else |
---|
1256 | { |
---|
1257 | int valid = 1; |
---|
1258 | for (i = size(l);i>=1;i--) |
---|
1259 | {//iterate over the elements of the given list |
---|
1260 | if (product(l[i])!=#[1]) |
---|
1261 | { |
---|
1262 | valid = 0; |
---|
1263 | } |
---|
1264 | result = insert(result, product(l[i])-#[1]); |
---|
1265 | }//iterate over the elements of the given list |
---|
1266 | if(size(#)==2){return(result);} |
---|
1267 | return(valid); |
---|
1268 | } |
---|
1269 | }//proc testfac |
---|
1270 | example |
---|
1271 | { |
---|
1272 | "EXAMPLE:";echo=2; |
---|
1273 | ring r = 0,(x,y),dp; |
---|
1274 | def R = nc_algebra(1,1); |
---|
1275 | setring R; |
---|
1276 | poly h = (x^2*y^2+1)*(x^2); |
---|
1277 | def t1 = facFirstWeyl(h); |
---|
1278 | //fist a correct list |
---|
1279 | testNCfac(t1); |
---|
1280 | //now a correct list with the factorized polynomial |
---|
1281 | testNCfac(t1,h); |
---|
1282 | //now we put in an incorrect list without a polynomial |
---|
1283 | t1[3][3] = y; |
---|
1284 | testNCfac(t1); |
---|
1285 | // take h as additional input |
---|
1286 | testNCfac(t1,h); |
---|
1287 | // take h as additional input and output list of differences |
---|
1288 | testNCfac(t1,h,1); |
---|
1289 | } |
---|
1290 | //================================================== |
---|
1291 | //Procedure facSubWeyl: |
---|
1292 | //This procedure serves the purpose to compute a |
---|
1293 | //factorization of a given polynomial in a ring, whose subring |
---|
1294 | //is the first Weyl algebra. The polynomial must only contain |
---|
1295 | //the two arguments, which are also given by the user. |
---|
1296 | |
---|
1297 | proc facSubWeyl(poly h, X, D) |
---|
1298 | "USAGE: facSubWeyl(h,x,y); h, X, D polynomials |
---|
1299 | RETURN: list |
---|
1300 | ASSUME: X and D are variables of a basering, which satisfy DX = XD +1. |
---|
1301 | @* That is, they generate the copy of the first Weyl algebra in a basering. |
---|
1302 | @* Moreover, h is a polynomial in X and D only. |
---|
1303 | PURPOSE: compute factorizations of the polynomial, which depends on X and D. |
---|
1304 | EXAMPLE: example facSubWeyl; shows examples |
---|
1305 | SEE ALSO: facFirstWeyl, testNCfac, facFirstShift |
---|
1306 | "{ |
---|
1307 | // basering can be anything having a Weyl algebra as subalgebra |
---|
1308 | def @r = basering; |
---|
1309 | //We begin to check the input for assumptions |
---|
1310 | // which are: X,D are vars of the basering, |
---|
1311 | if ( (isVar(X)!=1) || (isVar(D)!=1) || (size(X)>1) || (size(D)>1) || |
---|
1312 | (leadcoef(X) != number(1)) || (leadcoef(D) != number(1)) ) |
---|
1313 | { |
---|
1314 | ERROR("expected pure variables as generators of a subalgebra"); |
---|
1315 | } |
---|
1316 | // Weyl algebra: |
---|
1317 | poly w = D*X-X*D-1; // [D,X]=1 |
---|
1318 | poly u = D*X-X*D+1; // [X,D]=1 |
---|
1319 | if (u*w!=0) |
---|
1320 | { |
---|
1321 | // that is no combination gives Weyl |
---|
1322 | ERROR("2nd and 3rd argument do not generate a Weyl algebra"); |
---|
1323 | } |
---|
1324 | // one of two is correct |
---|
1325 | int isReverted = 0; // Reverted Weyl if dx=xd-1 holds |
---|
1326 | if (u==0) |
---|
1327 | { |
---|
1328 | isReverted = 1; |
---|
1329 | } |
---|
1330 | // else: do nothing |
---|
1331 | // DONE with assumptions, Input successfully checked |
---|
1332 | intvec lexpofX = leadexp(X); |
---|
1333 | intvec lexpofD = leadexp(D); |
---|
1334 | int varnumX=1; |
---|
1335 | int varnumD=1; |
---|
1336 | while(lexpofX[varnumX] != 1) |
---|
1337 | { |
---|
1338 | varnumX++; |
---|
1339 | } |
---|
1340 | while(lexpofD[varnumD] != 1) |
---|
1341 | { |
---|
1342 | varnumD++; |
---|
1343 | } |
---|
1344 | /* VL : to add printlevel stuff */ |
---|
1345 | |
---|
1346 | if (isReverted) |
---|
1347 | { |
---|
1348 | ring firstweyl = 0,(var(varnumD),var(varnumX)),dp; |
---|
1349 | def Firstweyl = nc_algebra(1,1); |
---|
1350 | setring Firstweyl; |
---|
1351 | ideal M = 0:nvars(@r); |
---|
1352 | M[varnumX]=var(2); |
---|
1353 | M[varnumD]=var(1); |
---|
1354 | map Q = @r,M; |
---|
1355 | poly h= Q(h); |
---|
1356 | } |
---|
1357 | else |
---|
1358 | { // that is unReverted |
---|
1359 | ring firstweyl = 0,(var(varnumX),var(varnumD)),dp; |
---|
1360 | def Firstweyl = nc_algebra(1,1); |
---|
1361 | setring Firstweyl; |
---|
1362 | poly h= imap(@r,h); |
---|
1363 | } |
---|
1364 | list result = facFirstWeyl(h); |
---|
1365 | setring @r; |
---|
1366 | list result; |
---|
1367 | if (isReverted) |
---|
1368 | { |
---|
1369 | // map swap back |
---|
1370 | ideal M; M[1] = var(varnumD); M[2] = var(varnumX); |
---|
1371 | map S = Firstweyl, M; |
---|
1372 | result = S(result); |
---|
1373 | } |
---|
1374 | else |
---|
1375 | { |
---|
1376 | // that is unReverted |
---|
1377 | result = imap(Firstweyl,result); |
---|
1378 | } |
---|
1379 | return(result); |
---|
1380 | }//proc facSubWeyl |
---|
1381 | example |
---|
1382 | { |
---|
1383 | "EXAMPLE:";echo=2; |
---|
1384 | ring r = 0,(x,y,z),dp; |
---|
1385 | matrix D[3][3]; D[1,3]=-1; |
---|
1386 | def R = nc_algebra(1,D); // x,z generate Weyl subalgebra |
---|
1387 | setring R; |
---|
1388 | poly h = (x^2*z^2+x)*x; |
---|
1389 | list fact1 = facSubWeyl(h,x,z); |
---|
1390 | // compare with facFirstWeyl: |
---|
1391 | ring s = 0,(z,x),dp; |
---|
1392 | def S = nc_algebra(1,1); setring S; |
---|
1393 | poly h = (x^2*z^2+x)*x; |
---|
1394 | list fact2 = facFirstWeyl(h); |
---|
1395 | map F = R,x,0,z; |
---|
1396 | list fact1 = F(fact1); // it is identical to list fact2 |
---|
1397 | testNCfac(fact1); // check the correctness again |
---|
1398 | } |
---|
1399 | //================================================== |
---|
1400 | |
---|
1401 | //================================================== |
---|
1402 | //************From here: Shift-Algebra************** |
---|
1403 | //================================================== |
---|
1404 | //==================================================* |
---|
1405 | //one factorization of a homogeneous polynomial |
---|
1406 | //in the first Shift Algebra |
---|
1407 | static proc homogfacFirstShift(poly h) |
---|
1408 | {//proc homogfacFirstShift |
---|
1409 | def r = basering; |
---|
1410 | poly hath; |
---|
1411 | int i; int j; |
---|
1412 | if (!homogwithorder(h,intvec(0,1))) |
---|
1413 | {//The given polynomial is not homogeneous |
---|
1414 | return(list()); |
---|
1415 | }//The given polynomial is not homogeneous |
---|
1416 | if (h==0) |
---|
1417 | { |
---|
1418 | return(list(0)); |
---|
1419 | } |
---|
1420 | list result; |
---|
1421 | int m = deg(h,intvec(0,1)); |
---|
1422 | if (m>0) |
---|
1423 | {//The degree is not zero |
---|
1424 | hath = lift(var(2)^m,h)[1,1]; |
---|
1425 | for (i = 1; i<=m;i++) |
---|
1426 | { |
---|
1427 | result = result + list(var(2)); |
---|
1428 | } |
---|
1429 | }//The degree is not zero |
---|
1430 | else |
---|
1431 | {//The degree is zero |
---|
1432 | hath = h; |
---|
1433 | }//The degree is zero |
---|
1434 | ring tempRing = 0,(x),dp; |
---|
1435 | setring tempRing; |
---|
1436 | map thetamap = r,x,1; |
---|
1437 | poly hath = thetamap(hath); |
---|
1438 | list azeroresult = factorize(hath); |
---|
1439 | list azeroresult_return_form; |
---|
1440 | for (i = 1; i<=size(azeroresult[1]);i++) |
---|
1441 | {//rewrite the result of the commutative factorization |
---|
1442 | for (j = 1; j <= azeroresult[2][i];j++) |
---|
1443 | { |
---|
1444 | azeroresult_return_form = azeroresult_return_form + list(azeroresult[1][i]); |
---|
1445 | } |
---|
1446 | }//rewrite the result of the commutative factorization |
---|
1447 | setring(r); |
---|
1448 | map finalmap = tempRing,var(1); |
---|
1449 | list tempresult = finalmap(azeroresult_return_form); |
---|
1450 | result = tempresult+result; |
---|
1451 | return(result); |
---|
1452 | }//proc homogfacFirstShift |
---|
1453 | |
---|
1454 | //================================================== |
---|
1455 | //Computes all possible homogeneous factorizations |
---|
1456 | static proc homogfacFirstShift_all(poly h) |
---|
1457 | {//proc HomogfacFirstShiftAll |
---|
1458 | if (deg(h,intvec(1,1)) <= 0 ) |
---|
1459 | {//h is a constant |
---|
1460 | return(list(list(h))); |
---|
1461 | }//h is a constant |
---|
1462 | def r = basering; |
---|
1463 | list one_hom_fac; //stands for one homogeneous factorization |
---|
1464 | int i; int j; int k; |
---|
1465 | int shiftcounter; |
---|
1466 | //Compute again a homogeneous factorization |
---|
1467 | one_hom_fac = homogfacFirstShift(h); |
---|
1468 | one_hom_fac = delete(one_hom_fac,1); |
---|
1469 | if (size(one_hom_fac) == 0) |
---|
1470 | {//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
1471 | return(list()); |
---|
1472 | }//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
1473 | list result = permpp(one_hom_fac); |
---|
1474 | for (i = 1; i<=size(result);i++) |
---|
1475 | { |
---|
1476 | shiftcounter = 0; |
---|
1477 | for (j = 1; j<=size(result[i]); j++) |
---|
1478 | { |
---|
1479 | if (result[i][j]==var(2)) |
---|
1480 | { |
---|
1481 | shiftcounter++; |
---|
1482 | } |
---|
1483 | else |
---|
1484 | { |
---|
1485 | result[i][j] = subst(result[i][j], var(1), var(1)-shiftcounter); |
---|
1486 | } |
---|
1487 | } |
---|
1488 | result[i] = insert(result[i],1); |
---|
1489 | } |
---|
1490 | result = delete_dublicates_noteval(result); |
---|
1491 | return(result); |
---|
1492 | }//proc HomogfacFirstShiftAll |
---|
1493 | |
---|
1494 | //================================================== |
---|
1495 | //factorization of the first Shift Algebra |
---|
1496 | proc facFirstShift(poly h) |
---|
1497 | "USAGE: facFirstShift(h); h a polynomial in the first shift algebra |
---|
1498 | RETURN: list |
---|
1499 | PURPOSE: compute all factorizations of a polynomial in the first shift algebra |
---|
1500 | THEORY: Implements the new algorithm by A. Heinle and V. Levandovskyy, see the thesis of A. Heinle |
---|
1501 | ASSUME: basering in the first shift algebra |
---|
1502 | NOTE: Every entry of the output list is a list with factors for one possible factorization. |
---|
1503 | EXAMPLE: example facFirstShift; shows examples |
---|
1504 | SEE ALSO: testNCfac, facFirstWeyl, facSubWeyl |
---|
1505 | "{//facFirstShift |
---|
1506 | if(nvars(basering)!=2) |
---|
1507 | {//Our basering is the Shift algebra, but not the first |
---|
1508 | ERROR("Basering is not the first shift algebra"); |
---|
1509 | return(list()); |
---|
1510 | }//Our basering is the Shift algebra, but not the first |
---|
1511 | def r = basering; |
---|
1512 | setring r; |
---|
1513 | list LR = ringlist(r); |
---|
1514 | number @n = leadcoef(LR[5][1,2]); |
---|
1515 | poly @p = LR[6][1,2]; |
---|
1516 | if ( @n!=number(1) ) |
---|
1517 | { |
---|
1518 | ERROR("Basering is not the first shift algebra"); |
---|
1519 | return(list()); |
---|
1520 | } |
---|
1521 | list result = list(); |
---|
1522 | int i;int j; int k; int l; //counter |
---|
1523 | // create a ring with the ordering which makes shift algebra |
---|
1524 | // graded |
---|
1525 | // def r = basering; // done before |
---|
1526 | ring tempRing = LR[1][1],(x,s),(a(0,1),Dp); |
---|
1527 | def tempRingnc = nc_algebra(1,s); |
---|
1528 | setring r; |
---|
1529 | // information on relations |
---|
1530 | if (@p == -var(1)) // reverted shift algebra |
---|
1531 | { |
---|
1532 | setring(tempRingnc); |
---|
1533 | map transf = r, var(2), var(1); |
---|
1534 | setring(r); |
---|
1535 | map transfback = tempRingnc, var(2),var(1); |
---|
1536 | // result = transfback(resulttemp); |
---|
1537 | } |
---|
1538 | else |
---|
1539 | { |
---|
1540 | if ( @p == var(2)) // usual shift algebra |
---|
1541 | { |
---|
1542 | setring(tempRingnc); |
---|
1543 | map transf = r, var(1), var(2); |
---|
1544 | // result = facshift(h); |
---|
1545 | setring(r); |
---|
1546 | map transfback = tempRingnc, var(1),var(2); |
---|
1547 | } |
---|
1548 | else |
---|
1549 | { |
---|
1550 | ERROR("Basering is not the first shift algebra"); |
---|
1551 | return(list()); |
---|
1552 | } |
---|
1553 | } |
---|
1554 | // main calls |
---|
1555 | setring(tempRingnc); |
---|
1556 | list resulttemp = facshift(transf(h)); |
---|
1557 | setring(r); |
---|
1558 | result = transfback(resulttemp); |
---|
1559 | |
---|
1560 | list recursivetemp; |
---|
1561 | for(i = 1; i<=size(result);i++) |
---|
1562 | {//recursively factorize factors |
---|
1563 | if(size(result[i])>2) |
---|
1564 | {//Nontrivial factorization |
---|
1565 | for (j=2;j<=size(result[i]);j++) |
---|
1566 | {//Factorize every factor |
---|
1567 | recursivetemp = facFirstShift(result[i][j]); |
---|
1568 | if(size(recursivetemp)>1) |
---|
1569 | {//we have a nontrivial factorization |
---|
1570 | for(k=1; k<=size(recursivetemp);k++) |
---|
1571 | {//insert factorized factors |
---|
1572 | if(size(recursivetemp[k])>2) |
---|
1573 | {//nontrivial |
---|
1574 | result = insert(result,result[i],i); |
---|
1575 | for(l = size(recursivetemp[k]);l>=2;l--) |
---|
1576 | { |
---|
1577 | result[i+1] = insert(result[i+1],recursivetemp[k][l],j); |
---|
1578 | } |
---|
1579 | result[i+1] = delete(result[i+1],j); |
---|
1580 | }//nontrivial |
---|
1581 | }//insert factorized factors |
---|
1582 | }//we have a nontrivial factorization |
---|
1583 | }//Factorize every factor |
---|
1584 | }//Nontrivial factorization |
---|
1585 | }//recursively factorize factors |
---|
1586 | //now, refine the possible redundant list |
---|
1587 | return( delete_dublicates_noteval(result) ); |
---|
1588 | }//facFirstShift |
---|
1589 | example |
---|
1590 | { |
---|
1591 | "EXAMPLE:";echo=2; |
---|
1592 | ring R = 0,(x,s),dp; |
---|
1593 | def r = nc_algebra(1,s); |
---|
1594 | setring(r); |
---|
1595 | poly h = (s^2*x+x)*s; |
---|
1596 | facFirstShift(h); |
---|
1597 | } |
---|
1598 | |
---|
1599 | static proc facshift(poly h) |
---|
1600 | "USAGE: facshift(h); h is a polynomial in the first Shift algebra |
---|
1601 | RETURN: list |
---|
1602 | PURPOSE: Computes a factorization of a polynomial h in the first Shift algebra |
---|
1603 | THEORY: @code{facshift} returns a list with some factorizations of the given |
---|
1604 | polynomial. The possibilities of the factorization of the highest |
---|
1605 | homogeneous part and those of the lowest will be merged. If during this |
---|
1606 | procedure a factorization of the polynomial occurs, it will be added to |
---|
1607 | the output list. For a more detailled description visit |
---|
1608 | @url{http://www.math.rwth-aachen.de/\~Albert.Heinle} |
---|
1609 | SEE ALSO: homogfacFirstShift_all, homogfacFirstShift |
---|
1610 | "{//proc facshift |
---|
1611 | if(homogwithorder(h,intvec(0,1))) |
---|
1612 | { |
---|
1613 | return(homogfacFirstShift_all(h)); |
---|
1614 | } |
---|
1615 | def r = basering; |
---|
1616 | map invo = basering,-var(1),-var(2); |
---|
1617 | int i; int j; int k; |
---|
1618 | intvec limits = deg(h,intvec(1,1)) ,deg(h,intvec(1,0)),deg(h,intvec(0,1)); |
---|
1619 | def prod; |
---|
1620 | //end finding the limits |
---|
1621 | poly maxh = jet(h,deg(h,intvec(0,1)),intvec(0,1))-jet(h,deg(h,intvec(0,1))-1,intvec(0,1)); |
---|
1622 | poly minh = jet(h,deg(h,intvec(0,-1)),intvec(0,-1))-jet(h,deg(h,intvec(0,-1))-1,intvec(0,-1)); |
---|
1623 | list result; |
---|
1624 | list temp; |
---|
1625 | list homogtemp; |
---|
1626 | list M; list hatM; |
---|
1627 | list f1 = homogfacFirstShift_all(maxh); |
---|
1628 | list f2 = homogfacFirstShift_all(minh); |
---|
1629 | int is_equal; |
---|
1630 | poly hath; |
---|
1631 | for (i = 1; i<=size(f1);i++) |
---|
1632 | {//Merge all combinations |
---|
1633 | for (j = 1; j<=size(f2); j++) |
---|
1634 | { |
---|
1635 | M = M + mergence(f1[i],f2[j],limits); |
---|
1636 | } |
---|
1637 | }//Merge all combinations |
---|
1638 | for (i = 1 ; i<= size(M); i++) |
---|
1639 | {//filter valid combinations |
---|
1640 | if (product(M[i]) == h) |
---|
1641 | {//We have one factorization |
---|
1642 | result = result + list(M[i]); |
---|
1643 | M = delete(M,i); |
---|
1644 | continue; |
---|
1645 | }//We have one factorization |
---|
1646 | else |
---|
1647 | { |
---|
1648 | if (deg(h,intvec(0,1))<=deg(h-product(M[i]),intvec(0,1))) |
---|
1649 | { |
---|
1650 | M = delete(M,i); |
---|
1651 | continue; |
---|
1652 | } |
---|
1653 | if (deg(h,intvec(0,-1))<=deg(h-product(M[i]),intvec(0,-1))) |
---|
1654 | { |
---|
1655 | M = delete(M,i); |
---|
1656 | continue; |
---|
1657 | } |
---|
1658 | } |
---|
1659 | }//filter valid combinations |
---|
1660 | M = delete_dublicates_eval(M); |
---|
1661 | while(size(M)>0) |
---|
1662 | {//work on the elements of M |
---|
1663 | hatM = list(); |
---|
1664 | for(i = 1; i<=size(M); i++) |
---|
1665 | {//iterate over all elements of M |
---|
1666 | hath = h-product(M[i]); |
---|
1667 | temp = list(); |
---|
1668 | //First check for common inhomogeneous factors between hath and h |
---|
1669 | if (involution(NF(involution(hath,invo), std(involution(ideal(M[i][1]),invo))),invo)==0) |
---|
1670 | {//hath and h have a common factor on the left |
---|
1671 | j = 1; |
---|
1672 | f1 = M[i]; |
---|
1673 | if (j+1<=size(f1)) |
---|
1674 | {//Checking for more than one common factor |
---|
1675 | while(involution(NF(involution(hath,invo),std(involution(ideal(product(f1[1..(j+1)])),invo))),invo)==0) |
---|
1676 | { |
---|
1677 | if (j+1<size(f1)) |
---|
1678 | { |
---|
1679 | j++; |
---|
1680 | } |
---|
1681 | else |
---|
1682 | { |
---|
1683 | break; |
---|
1684 | } |
---|
1685 | } |
---|
1686 | }//Checking for more than one common factor |
---|
1687 | if (deg(product(f1[1..j]),intvec(1,1))!=0) |
---|
1688 | { |
---|
1689 | f2 = list(f1[1..j])+list(involution(lift(involution(product(f1[1..j]),invo),involution(hath,invo))[1,1],invo)); |
---|
1690 | } |
---|
1691 | else |
---|
1692 | { |
---|
1693 | f2 = list(f1[1..j])+list(involution(lift(product(f1[1..j]),involution(hath,invo))[1,1],invo)); |
---|
1694 | } |
---|
1695 | temp = temp + merge_cf(f2,f1,limits); |
---|
1696 | }//hath and h have a common factor on the left |
---|
1697 | if (reduce(hath, std(ideal(M[i][size(M[i])])))==0) |
---|
1698 | {//hath and h have a common factor on the right |
---|
1699 | j = size(M[i]); |
---|
1700 | f1 = M[i]; |
---|
1701 | if (j-1>0) |
---|
1702 | {//Checking for more than one factor |
---|
1703 | while(reduce(hath,std(ideal(product(f1[(j-1)..size(f1)]))))==0) |
---|
1704 | { |
---|
1705 | if (j-1>1) |
---|
1706 | { |
---|
1707 | j--; |
---|
1708 | } |
---|
1709 | else |
---|
1710 | { |
---|
1711 | break; |
---|
1712 | } |
---|
1713 | } |
---|
1714 | }//Checking for more than one factor |
---|
1715 | f2 = list(lift(product(f1[j..size(f1)]),hath)[1,1])+list(f1[j..size(f1)]); |
---|
1716 | temp = temp + merge_cf(f2,M[i],limits); |
---|
1717 | }//hath and h have a common factor on the right |
---|
1718 | //and now the homogeneous |
---|
1719 | maxh = jet(hath,deg(hath,intvec(0,1)),intvec(0,1))-jet(hath,deg(hath,intvec(0,1))-1,intvec(0,1)); |
---|
1720 | minh = jet(hath,deg(hath,intvec(0,-1)),intvec(0,-1))-jet(hath,deg(hath,intvec(0,-1))-1,intvec(0,-1)); |
---|
1721 | f1 = homogfacFirstShift_all(maxh); |
---|
1722 | f2 = homogfacFirstShift_all(minh); |
---|
1723 | for (j = 1; j<=size(f1);j++) |
---|
1724 | { |
---|
1725 | for (k=1; k<=size(f2);k++) |
---|
1726 | { |
---|
1727 | homogtemp = mergence(f1[j],f2[k],limits); |
---|
1728 | } |
---|
1729 | } |
---|
1730 | for (j = 1; j<= size(homogtemp); j++) |
---|
1731 | { |
---|
1732 | temp = temp + mergence(homogtemp[j],M[i],limits); |
---|
1733 | } |
---|
1734 | for (j = 1; j<=size(temp); j++) |
---|
1735 | {//filtering invalid entries in temp |
---|
1736 | if(product(temp[j])==h) |
---|
1737 | {//This is already a result |
---|
1738 | result = result + list(temp[j]); |
---|
1739 | temp = delete(temp,j); |
---|
1740 | continue; |
---|
1741 | }//This is already a result |
---|
1742 | if (deg(hath,intvec(0,1))<=deg(hath-product(temp[j]),intvec(0,1))) |
---|
1743 | { |
---|
1744 | temp = delete(temp,j); |
---|
1745 | continue; |
---|
1746 | } |
---|
1747 | }//filtering invalid entries in temp |
---|
1748 | hatM = hatM + temp; |
---|
1749 | }//iterate over all elements of M |
---|
1750 | M = hatM; |
---|
1751 | for (i = 1; i<=size(M);i++) |
---|
1752 | {//checking for complete factorizations |
---|
1753 | if (h == product(M[i])) |
---|
1754 | { |
---|
1755 | result = result + list(M[i]); |
---|
1756 | M = delete(M,i); |
---|
1757 | continue; |
---|
1758 | } |
---|
1759 | }//checking for complete factorizations |
---|
1760 | M = delete_dublicates_eval(M); |
---|
1761 | }//work on the elements of M |
---|
1762 | //In the case, that there is none, write a constant factor before the factor of interest. |
---|
1763 | for (i = 1 ; i<=size(result);i++) |
---|
1764 | {//add a constant factor |
---|
1765 | if (deg(result[i][1],intvec(1,1))!=0) |
---|
1766 | { |
---|
1767 | result[i] = insert(result[i],1); |
---|
1768 | } |
---|
1769 | }//add a constant factor |
---|
1770 | result = delete_dublicates_noteval(result); |
---|
1771 | return(result); |
---|
1772 | }//proc facshift |
---|
1773 | |
---|
1774 | static proc refineFactList(list L) |
---|
1775 | { |
---|
1776 | // assume: list L is an output of factorization proc |
---|
1777 | // doing: remove doubled entries |
---|
1778 | int s = size(L); int sm; |
---|
1779 | int i,j,k,cnt; |
---|
1780 | list M, U, A, B; |
---|
1781 | A = L; |
---|
1782 | k = 0; |
---|
1783 | cnt = 1; |
---|
1784 | for (i=1; i<=s; i++) |
---|
1785 | { |
---|
1786 | if (size(A[i]) != 0) |
---|
1787 | { |
---|
1788 | M = A[i]; |
---|
1789 | // "probing with"; M; i; |
---|
1790 | B[cnt] = M; cnt++; |
---|
1791 | for (j=i+1; j<=s; j++) |
---|
1792 | { |
---|
1793 | if ( isEqualList(M,A[j]) ) |
---|
1794 | { |
---|
1795 | k++; |
---|
1796 | // U consists of intvecs with equal pairs |
---|
1797 | U[k] = intvec(i,j); |
---|
1798 | A[j] = 0; |
---|
1799 | } |
---|
1800 | } |
---|
1801 | } |
---|
1802 | } |
---|
1803 | kill A,U,M; |
---|
1804 | return(B); |
---|
1805 | } |
---|
1806 | example |
---|
1807 | { |
---|
1808 | "EXAMPLE:";echo=2; |
---|
1809 | ring R = 0,(x,s),dp; |
---|
1810 | def r = nc_algebra(1,1); |
---|
1811 | setring(r); |
---|
1812 | list l,m; |
---|
1813 | l = list(1,s2+1,x,s,x+s); |
---|
1814 | m = l,list(1,s,x,s,x),l; |
---|
1815 | refineFactList(m); |
---|
1816 | } |
---|
1817 | |
---|
1818 | static proc isEqualList(list L, list M) |
---|
1819 | { |
---|
1820 | // int boolean: 1=yes, 0 =no : test whether two lists are identical |
---|
1821 | int s = size(L); |
---|
1822 | if (size(M)!=s) { return(0); } |
---|
1823 | int j=1; |
---|
1824 | while ( (L[j]==M[j]) && (j<s) ) |
---|
1825 | { |
---|
1826 | j++; |
---|
1827 | } |
---|
1828 | if (L[j]==M[j]) |
---|
1829 | { |
---|
1830 | return(1); |
---|
1831 | } |
---|
1832 | return(0); |
---|
1833 | } |
---|
1834 | example |
---|
1835 | { |
---|
1836 | "EXAMPLE:";echo=2; |
---|
1837 | ring R = 0,(x,s),dp; |
---|
1838 | def r = nc_algebra(1,1); |
---|
1839 | setring(r); |
---|
1840 | list l,m; |
---|
1841 | l = list(1,s2+1,x,s,x+s); |
---|
1842 | m = l; |
---|
1843 | isEqualList(m,l); |
---|
1844 | } |
---|
1845 | |
---|
1846 | /* |
---|
1847 | Example polynomials where one can find factorizations: K<x,y |yx=xy+1> |
---|
1848 | (x^2+y)*(x^2+y); |
---|
1849 | (x^2+x)*(x^2+y); |
---|
1850 | (x^3+x+1)*(x^4+y*x+2); |
---|
1851 | (x^2*y+y)*(y+x*y); |
---|
1852 | y^3+x*y^3+2*y^2+2*(x+1)*y^2+y+(x+2)*y; //Example 5 Grigoriev-Schwarz. |
---|
1853 | (y+1)*(y+1)*(y+x*y); //Landau Example projected to the first dimension. |
---|
1854 | */ |
---|
1855 | |
---|
1856 | |
---|
1857 | /* very hard things from Martin Lee: |
---|
1858 | // ex1, ex2 |
---|
1859 | ring s = 0,(z,x),Ws(-1,1); |
---|
1860 | def S = nc_algebra(1,1); setring S; |
---|
1861 | poly a = 10z5x4+26z4x5+47z5x2-97z4x3; //Abgebrochen nach einer Stunde; yes, it takes long |
---|
1862 | def l= facFirstWeyl (a); l; |
---|
1863 | kill l; |
---|
1864 | poly b = -5328z8x5-5328z7x6+720z9x2+720z8x3-16976z7x4-38880z6x5-5184z7x3-5184z6x4-3774z5x5+2080z8x+5760z7x2-6144z6x3-59616z5x4+3108z3x6-4098z6x2-25704z5x3-21186z4x4+8640z6x-17916z4x3+22680z2x5+2040z5x-4848z4x2-9792z3x3+3024z2x4-10704z3x2-3519z2x3+34776zx4+12096zx3+2898x4-5040z2x+8064x3+6048x2; //Abgebrochen nach 1.5 Stunden; seems to be very complicated |
---|
1865 | def l= facFirstWeyl (b); l; |
---|
1866 | |
---|
1867 | // ex3: there was difference in answers => fixed |
---|
1868 | LIB "ncfactor.lib"; |
---|
1869 | ring r = 0,(x,y,z),dp; |
---|
1870 | matrix D[3][3]; D[1,3]=-1; |
---|
1871 | def R = nc_algebra(1,D); |
---|
1872 | setring R; |
---|
1873 | poly g= 7*z4*x+62*z3+26*z; |
---|
1874 | def l1= facSubWeyl (g, x, z); |
---|
1875 | l1; |
---|
1876 | //---- other ring |
---|
1877 | ring s = 0,(x,z),dp; |
---|
1878 | def S = nc_algebra(1,-1); setring S; |
---|
1879 | poly g= 7*z4*x+62*z3+26*z; |
---|
1880 | def l2= facFirstWeyl (g); |
---|
1881 | l2; |
---|
1882 | map F = R,x,0,z; |
---|
1883 | list l1 = F(l1); |
---|
1884 | l1; |
---|
1885 | //---- so the answers look different, check them! |
---|
1886 | testNCfac(l2); // ok |
---|
1887 | testNCfac(l1); // was not ok, but now it's been fixed!!! |
---|
1888 | |
---|
1889 | // selbst D und X so vertauschen dass sie erfuellt ist : ist gemacht |
---|
1890 | |
---|
1891 | */ |
---|
1892 | |
---|
1893 | /* |
---|
1894 | // bug from M Lee |
---|
1895 | LIB "ncfactor.lib"; |
---|
1896 | ring s = 0,(z,x),dp; |
---|
1897 | def S = nc_algebra(1,1); setring S; |
---|
1898 | poly f= -60z4x2-54z4-56zx3-59z2x-64; |
---|
1899 | def l= facFirstWeyl (f); |
---|
1900 | l; // before: empty list; after fix: 1 entry, f is irreducible |
---|
1901 | poly g = 75z3x2+92z3+24; |
---|
1902 | def l= facFirstWeyl (g); |
---|
1903 | l; //before: empty list, now: correct |
---|
1904 | */ |
---|
1905 | |
---|
1906 | /* more things from Martin Lee; fixed |
---|
1907 | ring R = 0,(x,s),dp; |
---|
1908 | def r = nc_algebra(1,s); |
---|
1909 | setring(r); |
---|
1910 | poly h = (s2*x+x)*s; |
---|
1911 | h= h* (x+s); |
---|
1912 | def l= facFirstShift(h); |
---|
1913 | l; // contained doubled entries: not anymore, fixed! |
---|
1914 | |
---|
1915 | ring R = 0,(x,s),dp; |
---|
1916 | def r = nc_algebra(1,-1); |
---|
1917 | setring(r); |
---|
1918 | poly h = (s2*x+x)*s; |
---|
1919 | h= h* (x+s); |
---|
1920 | def l= facFirstWeyl(h); |
---|
1921 | l; // contained doubled entries: not anymore, fixed! |
---|
1922 | |
---|
1923 | */ |
---|