1 | // $Id: ring.lib,v 1.3 1997-09-12 07:40:36 Singular Exp $ |
---|
2 | //(GMG, last modified 03.11.95) |
---|
3 | /////////////////////////////////////////////////////////////////////////////// |
---|
4 | |
---|
5 | LIBRARY: ring.lib PROCEDURES FOR MANIPULATING RINGS AND MAPS |
---|
6 | |
---|
7 | changechar("R",c[,r]); make a copy R of basering [ring r] with new char c |
---|
8 | changeord("R",o[,r]); make a copy R of basering [ring r] with new ord o |
---|
9 | changevar("R",v[,r]); make a copy R of basering [ring r] with new vars v |
---|
10 | copyring("R"[,r]); make an exact copy R of basering [ring r] |
---|
11 | defring("R",c,n,v,o); define a ring R in specified char c, n vars v, ord o |
---|
12 | defrings(n[,p]); define ring Sn in n vars, char 32003 [p], ord ds |
---|
13 | defringp(n[,p]); define ring Pn in n vars, char 32003 [p], ord dp |
---|
14 | extendring("R",n,v,o); extend given ring by n vars v, ord o and name it R |
---|
15 | extendring1("R",n,v,o); similair to extendring but different ordering |
---|
16 | fetchall(R[,str]); fetch all objects of ring R to basering |
---|
17 | imapall(R[,str]); imap all objects of ring R to basering |
---|
18 | mapall(R,i[,str]); map all objects of ring R via ideal i to basering |
---|
19 | ringtensor("R",s,t,..);create ring R, tensor product of rings s,t,... |
---|
20 | substitute(id,p,list); substitute in id i-th factor of p by i-th poly of list |
---|
21 | swapvars(id,p1,p2); return id with variables p1 and p2 interchanged |
---|
22 | (parameters in square brackets [] are optional) |
---|
23 | |
---|
24 | LIB "inout.lib"; |
---|
25 | LIB "general.lib"; |
---|
26 | /////////////////////////////////////////////////////////////////////////////// |
---|
27 | |
---|
28 | proc changechar (string newr, string c, list #) |
---|
29 | USAGE: changechar(newr,c[,r]); newr,c=strings, r=ring |
---|
30 | CREATE: create a new ring with name `newr` and make it the basering if r is |
---|
31 | an existing ring [default: r=basering]. |
---|
32 | The new ring differs from the old ring only in the characteristic. |
---|
33 | If, say, (newr,c) = ("R","0,A") and the ring r exists, the new |
---|
34 | basering will have name R characteristic 0 and one parameter A. |
---|
35 | RETURN: No return value |
---|
36 | NOTE: Works for qrings if map from old_char to new_char is implemented |
---|
37 | This proc uses 'execute' or calls a procedure using 'execute'. |
---|
38 | If you use it in your own proc, let the local names of your proc |
---|
39 | start with @ (see the file HelpForProc) |
---|
40 | EXAMPLE: example changechar; shows an example |
---|
41 | { |
---|
42 | if( size(#)==0 ) { def @r=basering; } |
---|
43 | if( size(#)==1 ) { def @r=#[1]; } |
---|
44 | setring @r; |
---|
45 | ideal i = ideal(@r); int @q = size(i); |
---|
46 | if( @q!=0 ) |
---|
47 | { string @s = "@newr1"; } |
---|
48 | else |
---|
49 | { string @s = newr; } |
---|
50 | string @newring = @s+"=("+c+"),("+varstr(@r)+"),("+ordstr(@r)+");"; |
---|
51 | execute("ring "+@newring); |
---|
52 | if( @q!=0 ) |
---|
53 | { |
---|
54 | map phi = @r,maxideal(1); |
---|
55 | ideal i = phi(i); |
---|
56 | attrib(i,"isSB",1); //*** attrib funktioniert ? |
---|
57 | execute("qring "+newr+"=i;"); |
---|
58 | } |
---|
59 | export(`newr`); |
---|
60 | keepring(`newr`); |
---|
61 | if (voice==2) { "// basering is now",newr; } |
---|
62 | return(); |
---|
63 | } |
---|
64 | example |
---|
65 | { "EXAMPLE:"; echo = 2; |
---|
66 | ring r=0,(x,y,u,v),(dp(2),ds); |
---|
67 | changechar("R","2,A"); R;""; |
---|
68 | changechar("R1","32003",R); R1; |
---|
69 | kill R,R1; |
---|
70 | } |
---|
71 | /////////////////////////////////////////////////////////////////////////////// |
---|
72 | |
---|
73 | proc changeord (string newr, string o, list #) |
---|
74 | USAGE: changeord(newr,o[,r]); newr,o=strings, r=ring/qring |
---|
75 | CREATE: create a new ring with name `newr` and make it the basering if r is |
---|
76 | an existing ring/qring [default: r=basering]. |
---|
77 | The new ring differs from the old ring only in the ordering. If, say, |
---|
78 | (newr,o) = ("R","wp(2,3),dp") and the ring r exists and has >=3 |
---|
79 | variables, the new basering will have name R and ordering wp(2,3),dp. |
---|
80 | RETURN: No return value |
---|
81 | NOTE: This proc uses 'execute' or calls a procedure using 'execute'. |
---|
82 | If you use it in your own proc, let the local names of your proc |
---|
83 | start with @ (see the file HelpForProc) |
---|
84 | EXAMPLE: example changeord; shows an example |
---|
85 | { |
---|
86 | if( size(#)==0 ) { def @r=basering; } |
---|
87 | if( size(#)==1 ) { def @r=#[1]; } |
---|
88 | setring @r; |
---|
89 | ideal i = ideal(@r); int @q = size(i); |
---|
90 | if( @q!=0 ) |
---|
91 | { string @s = "@newr1"; } |
---|
92 | else |
---|
93 | { string @s = newr; } |
---|
94 | string @newring = @s+"=("+charstr(@r)+"),("+varstr(@r)+"),("+o+");"; |
---|
95 | execute("ring "+@newring); |
---|
96 | if( @q!=0 ) |
---|
97 | { |
---|
98 | map phi = @r,maxideal(1); |
---|
99 | ideal i = phi(i); |
---|
100 | attrib(i,"isSB",1); //*** attrib funktioniert ? |
---|
101 | execute("qring "+newr+"=i;"); |
---|
102 | } |
---|
103 | export(`newr`); |
---|
104 | keepring(`newr`); |
---|
105 | if (voice==2) { "// basering is now",newr; } |
---|
106 | return(); |
---|
107 | } |
---|
108 | example |
---|
109 | { "EXAMPLE:"; echo = 2; |
---|
110 | ring r=0,(x,y,u,v),(dp(2),ds); |
---|
111 | changeord("R","wp(2,3),dp"); R; ""; |
---|
112 | ideal i = x^2,y^2-u^3,v; |
---|
113 | qring Q = std(i); |
---|
114 | changeord("Q'","lp",Q); Q'; |
---|
115 | kill R,Q,Q'; |
---|
116 | } |
---|
117 | /////////////////////////////////////////////////////////////////////////////// |
---|
118 | |
---|
119 | proc changevar (string newr, string vars, list #) |
---|
120 | USAGE: changevar(newr,vars[,r]); newr,vars=strings, r=ring/qring |
---|
121 | CREATE: creates a new ring with name `newr` and makes it the basering if r |
---|
122 | is an existing ring/qring [default: r=basering]. |
---|
123 | The new ring differs from the old ring only in the variables. If, |
---|
124 | say, (newr,vars) = ("R","t()") and the ring r exists and has n |
---|
125 | variables, the new basering will have name R and variables |
---|
126 | t(1),...,t(n). |
---|
127 | If vars = "a,b,c,d", the new ring will have the variables a,b,c,d. |
---|
128 | RETURN: No return value |
---|
129 | NOTE: This procedure is useful in connection with the procedure ringtensor, |
---|
130 | when a conflict between variable names must be avoided. |
---|
131 | This proc uses 'execute' or calls a procedure using 'execute'. |
---|
132 | If you use it in your own proc, let the local names of your proc |
---|
133 | start with @ (see the file HelpForProc) |
---|
134 | EXAMPLE: example changevar; shows an example |
---|
135 | { |
---|
136 | if( size(#)==0 ) { def @r=basering; } |
---|
137 | if( size(#)==1 ) { def @r=#[1]; } |
---|
138 | setring @r; |
---|
139 | ideal i = ideal(@r); int @q = size(i); |
---|
140 | if( @q!=0 ) |
---|
141 | { string @s = "@newr1"; } |
---|
142 | else |
---|
143 | { string @s = newr; } |
---|
144 | string @newring = @s+"=("+charstr(@r)+"),("; |
---|
145 | if( vars[size(vars)-1]=="(" and vars[size(vars)]==")" ) |
---|
146 | { |
---|
147 | @newring = @newring+vars[1,size(vars)-2]+"(1.."+string(nvars(@r))+")"; |
---|
148 | } |
---|
149 | else { @newring = @newring+vars; } |
---|
150 | @newring = @newring+"),("+ordstr(@r)+");"; |
---|
151 | execute("ring "+@newring); |
---|
152 | if( @q!=0 ) |
---|
153 | { |
---|
154 | map phi = @r,maxideal(1); |
---|
155 | ideal i = phi(i); |
---|
156 | attrib(i,"isSB",1); //*** attrib funktioniert ? |
---|
157 | execute("qring "+newr+"=i;"); |
---|
158 | } |
---|
159 | export(`newr`); |
---|
160 | keepring(`newr`); |
---|
161 | if (voice==2) { "// basering is now",newr; } |
---|
162 | return(); |
---|
163 | } |
---|
164 | example |
---|
165 | { "EXAMPLE:"; echo = 2; |
---|
166 | ring r=0,(x,y,u,v),(dp(2),ds); |
---|
167 | ideal i = x^2,y^2-u^3,v; |
---|
168 | qring Q = std(i); |
---|
169 | setring(r); |
---|
170 | changevar("R","A()"); R; ""; |
---|
171 | changevar("Q'","a,b,c,d",Q); Q'; |
---|
172 | kill R,Q,Q'; |
---|
173 | } |
---|
174 | /////////////////////////////////////////////////////////////////////////////// |
---|
175 | |
---|
176 | proc copyring (string newr,list #) |
---|
177 | USAGE: copyring(newr[,r]); newr=string, r=ring/qring |
---|
178 | CREATE: create a new ring with name `newr` and make it the basering if r is |
---|
179 | an existing ring [default: r=basering]. |
---|
180 | The new ring is a copy of r but with a new name R1 if, say, newr="R1" |
---|
181 | RETURN: No return value |
---|
182 | NOTE: This proc uses 'execute' or calls a procedure using 'execute'. |
---|
183 | If you use it in your own proc, let the local names of your proc |
---|
184 | start with @ (see the file HelpForProc) |
---|
185 | EXAMPLE: example copyring; shows an example |
---|
186 | { |
---|
187 | if( size(#)==0 ) { def @r=basering; } |
---|
188 | if( size(#)==1 ) { def @r=#[1]; } |
---|
189 | string @o=ordstr(@r); |
---|
190 | changeord(newr,@o,@r); |
---|
191 | keepring(basering); |
---|
192 | if (voice==2) { "// basering is now",newr; } |
---|
193 | return(); |
---|
194 | } |
---|
195 | example |
---|
196 | { "EXAMPLE:"; echo = 2; |
---|
197 | ring r=0,(x,y,u,v),(dp(2),ds); |
---|
198 | copyring("R"); R;""; |
---|
199 | copyring("R1",r); R1; |
---|
200 | kill R,R1; |
---|
201 | } |
---|
202 | /////////////////////////////////////////////////////////////////////////////// |
---|
203 | |
---|
204 | proc defring (string s1, string s2, int n, string s3, string s4) |
---|
205 | USAGE: defring(s1,s2,n,s3,s4); s1..s4=strings, n=integer |
---|
206 | CREATE: Define a ring with name 's1', characteristic 's2', ordering 's4' and |
---|
207 | n variables with names derived from s3 and make it the basering. |
---|
208 | If s3 is a single letter, say s3="a", and if n<=26 then a and the |
---|
209 | following n-1 letters from the alphabeth (cyclic order) are taken as |
---|
210 | variables. If n>26 or if s3 is a single letter followed by (, say |
---|
211 | s3="T(", the variables are T(1),...,T(n). |
---|
212 | RETURN: No return value |
---|
213 | NOTE: This proc is useful for defining a ring in a procedure. |
---|
214 | This proc uses 'execute' or calls a procedure using 'execute'. |
---|
215 | If you use it in your own proc, let the local names of your proc |
---|
216 | start with @ (see the file HelpForProc) |
---|
217 | EXAMPLE: example defring; shows an example |
---|
218 | { |
---|
219 | string @newring = "ring "+s1+"=("+s2+"),("; |
---|
220 | if( n>26 or s3[2]=="(" ) { string @v = s3[1]+"(1.."+string(n)+")"; } |
---|
221 | else { string @v = A_Z(s3,n); } |
---|
222 | @newring=@newring+@v+"),("+s4+");"; |
---|
223 | execute(@newring); |
---|
224 | export(basering); |
---|
225 | keepring(`s1`); |
---|
226 | if (voice==2) { "// basering is now:",s1; } |
---|
227 | return(); |
---|
228 | } |
---|
229 | example |
---|
230 | { "EXAMPLE:"; echo = 2; |
---|
231 | defring("r","0",5,"u","ls"); r; ""; |
---|
232 | defring("R","2,A",10,"x(","dp(3),ws(1,2,3),ds"); R; |
---|
233 | kill r,R; |
---|
234 | } |
---|
235 | /////////////////////////////////////////////////////////////////////////////// |
---|
236 | |
---|
237 | proc defrings (int n, list #) |
---|
238 | USAGE: defrings(n,[p]); n,p integers |
---|
239 | CREATE: Defines a ring with name Sn, characteristic p, ordering ds and n |
---|
240 | variables x,y,z,a,b,...if n<=26 (resp. x(1..n) if n>26) and makes it |
---|
241 | the basering (default: p=32003) |
---|
242 | RETURN: No return value |
---|
243 | EXAMPLE: example defrings; shows an example |
---|
244 | { |
---|
245 | int p; |
---|
246 | if (size(#)==0) { p=32003; } |
---|
247 | else { p=#[1]; } |
---|
248 | if (n >26) |
---|
249 | { |
---|
250 | string s="ring S"+string(n)+"="+string(p)+",x(1.."+string(n)+"),ds;"; |
---|
251 | } |
---|
252 | else |
---|
253 | { |
---|
254 | string s="ring S"+string(n)+"="+string(p)+",("+A_Z("x",n)+"),ds;"; |
---|
255 | } |
---|
256 | execute(s); |
---|
257 | export basering; |
---|
258 | execute("keepring S"+string(n)+";"); |
---|
259 | if (voice==2) { "// basering is now:",s; } |
---|
260 | } |
---|
261 | example |
---|
262 | { "EXAMPLE:"; echo = 2; |
---|
263 | defrings(5,0); S5; ""; |
---|
264 | defrings(30); S30; |
---|
265 | kill S5, S30; |
---|
266 | } |
---|
267 | /////////////////////////////////////////////////////////////////////////////// |
---|
268 | |
---|
269 | proc defringp (int n,list #) |
---|
270 | USAGE: defringp(n,[p]); n,p=integers |
---|
271 | CREATE: defines a ring with name Pn, characteristic p, ordering dp and n |
---|
272 | variables x,y,z,a,b,...if n<=26 (resp. x(1..n) if n>26) and makes it |
---|
273 | the basering (default: p=32003) |
---|
274 | RETURN: No return value |
---|
275 | EXAMPLE: example defringp; shows an example |
---|
276 | { |
---|
277 | int p; |
---|
278 | if (size(#)==0) { p=32003; } |
---|
279 | else { p=#[1]; } |
---|
280 | if (n >26) |
---|
281 | { |
---|
282 | string s="ring P"+string(n)+"="+string(p)+",x(1.."+string(n)+"),dp;"; |
---|
283 | } |
---|
284 | else |
---|
285 | { |
---|
286 | string s="ring P"+string(n)+"="+string(p)+",("+A_Z("x",n)+"),dp;"; |
---|
287 | } |
---|
288 | execute(s); |
---|
289 | export basering; |
---|
290 | execute("keepring P"+string(n)+";"); |
---|
291 | //the next comment is only shown if defringp is not called by another proc |
---|
292 | if (voice==2) { "// basering is now:",s; } |
---|
293 | } |
---|
294 | example |
---|
295 | { "EXAMPLE:"; echo = 2; |
---|
296 | defringp(5,0); P5; ""; |
---|
297 | defringp(30); P30; |
---|
298 | kill P5, P30; |
---|
299 | } |
---|
300 | /////////////////////////////////////////////////////////////////////////////// |
---|
301 | |
---|
302 | proc extendring (string na, int n, string va, string o, list #) |
---|
303 | USAGE: extendring(na,n,va,o[,i,r]); na,va,o=strings (name, new vars, |
---|
304 | ordering of the new ring), n,i=integers, r=ring |
---|
305 | CREATE: Define a ring with name `na` which extends the ring r by adding n new |
---|
306 | variables in front of [after, if i!=0] the old variables and make it |
---|
307 | the basering [default: (i,r)=(0,basering)] |
---|
308 | -- The characteristic is the characteristic of r |
---|
309 | -- The new vars are derived from va. If va is a single letter, say |
---|
310 | va="T", and if n<=26 then T and the following n-1 letters from |
---|
311 | T..Z..T (resp. T(1..n) if n>26) are taken as additional variables. |
---|
312 | If va is a single letter followed by (, say va="x(", the new |
---|
313 | variables are x(1),...,x(n). |
---|
314 | If va is a string that contains a comma (e.g. "x,z,u,w"), the |
---|
315 | comma-separated symbols are taken as new variables |
---|
316 | -- The ordering is the ordering given by `o` [any allowed ordstr] |
---|
317 | RETURN: No return value |
---|
318 | NOTE: This proc is useful for adding deformation parameters. |
---|
319 | This proc uses 'execute' or calls a procedure using 'execute'. |
---|
320 | If you use it in your own proc, let the local names of your proc |
---|
321 | start with @ (see the file HelpForProc) |
---|
322 | EXAMPLE: example extendring; shows an example |
---|
323 | { |
---|
324 | //--------------- initialization and place c/C of ordering properly ----------- |
---|
325 | string @v,@newring; |
---|
326 | int @i; |
---|
327 | if( size(#)==0 ) { #[1]=0; def @r=basering; } |
---|
328 | else |
---|
329 | { |
---|
330 | if( size(#)==1 ) { @i=#[1]; def @r=basering; } |
---|
331 | if( size(#)==2 ) { @i=#[1]; def @r=#[2]; } |
---|
332 | } |
---|
333 | //------------------------ prepare string of new ring ------------------------- |
---|
334 | @newring = "ring "+na+"=("+charstr(@r)+"),("; |
---|
335 | if( find(va,",") != 0 ) |
---|
336 | { @v = va; } |
---|
337 | else |
---|
338 | { |
---|
339 | if( n>26 or va[2]=="(" ) |
---|
340 | { @v = va[1]+"(1.."+string(n)+")"; } |
---|
341 | else |
---|
342 | { @v = A_Z(va,n); } |
---|
343 | } |
---|
344 | |
---|
345 | if( @i==0 ) |
---|
346 | { |
---|
347 | @v=@v+","+varstr(@r); |
---|
348 | } |
---|
349 | else |
---|
350 | { |
---|
351 | @v=varstr(@r)+","+@v; |
---|
352 | } |
---|
353 | @newring=@newring+@v+"),("+o+");"; |
---|
354 | //---------------------------- execute and export ----------------------------- |
---|
355 | execute(@newring); |
---|
356 | export(basering); |
---|
357 | keepring(basering); |
---|
358 | if (voice==2) { "// basering is now",basering; } |
---|
359 | return(); |
---|
360 | } |
---|
361 | example |
---|
362 | { "EXAMPLE:"; echo = 2; |
---|
363 | ring r=0,(x,y,z),ds; |
---|
364 | show(r);""; |
---|
365 | extendring("R0",2,"u","ds"); |
---|
366 | show(R0); ""; |
---|
367 | extendring("R1",2,"a,w","ds(2),dp"); |
---|
368 | show(R1); ""; |
---|
369 | extendring("R2",5,"b","dp"); |
---|
370 | show(R2); ""; |
---|
371 | extendring("R3",4,"T()","c,dp",1,r); |
---|
372 | show(R3);""; |
---|
373 | kill R0,R1,R2,R3; |
---|
374 | } |
---|
375 | /////////////////////////////////////////////////////////////////////////////// |
---|
376 | |
---|
377 | proc extendring1 (string na, int n, string va, string o, list #) |
---|
378 | USAGE: extendring1(na,n,va,o[iv,i,r]); na,va,o=strings, |
---|
379 | n,i=integers, r=ring, iv=intvec of positive integers or iv=0 |
---|
380 | CREATE: Define a ring with name `na` which extends the ring r by adding n new |
---|
381 | variables in front of [after, if i!=0] the old variables and make it |
---|
382 | the basering [default: (i,r)=(0,basering)] |
---|
383 | -- The characteristic is the characteristic of r |
---|
384 | -- The new vars are derived from va. If va is a single letter, say |
---|
385 | va="T", and if n<=26 then T and the following n-1 letters from |
---|
386 | T..Z..T (resp. T(1..n) if n>26) are taken as additional variables. |
---|
387 | If va is a single letter followed by (, say va="x(", the new |
---|
388 | variables are x(1),...,x(n). |
---|
389 | If va is a string that contains a comma (e.g. "x,z,u,w"), the |
---|
390 | comma-separated symbols are taken as new variables |
---|
391 | -- The ordering is the product ordering between the ordering of r and |
---|
392 | an ordering derived from `o` [and iv] |
---|
393 | - If o contains a 'c' or a 'C' in front resp. at the end this is |
---|
394 | taken for the whole ordering in front resp. at the end. If o does |
---|
395 | not contain a 'c' or a 'C' the same rule applies to ordstr(r). |
---|
396 | - If no intvec iv is given, or if iv=0, o may be any allowed ordstr, |
---|
397 | like "ds" or "dp(2),wp(1,2,3),Ds(2)" or "ds(a),dp(b),ls" if a and b |
---|
398 | are globally (!) defined integers and if a+b+1<=n |
---|
399 | If, however, a and b are local to a proc calling extendring1, the |
---|
400 | intvec iv must be used to let extendring1 know the values of a, b |
---|
401 | - If an intvec iv !=0 is given, iv[1],iv[2],... is taken for the 1st, |
---|
402 | 2nd,... block of o, if o contains no substring "w" or "W" i.e. no |
---|
403 | weighted ordering (in the above case o="ds,dp,ls" and iv=a,b). |
---|
404 | If o contains a weighted ordering (only one (!) weighted block is |
---|
405 | allowed) iv[1] is taken as size for the weight-vector, the next |
---|
406 | iv[1] values of iv are taken as weights and the remaining values of |
---|
407 | iv as block-size for the remaining non-weighted blocks. |
---|
408 | e.g. o="dp,ws,Dp,ds", iv=3,2,3,4,2,5 creates the ordering |
---|
409 | dp(2),ws(2,3,4),Dp(5),ds |
---|
410 | RETURN: No return value |
---|
411 | NOTE: This proc is useful for adding deformation parameters. |
---|
412 | This proc uses 'execute' or calls a procedure using 'execute'. |
---|
413 | If you use it in your own proc, let the local names of your proc |
---|
414 | start with @ (see the file HelpForProc) |
---|
415 | EXAMPLE: example extendring1; shows an example |
---|
416 | { |
---|
417 | //--------------- initialization and place c/C of ordering properly ----------- |
---|
418 | string @o1,@o2,@ro,@wstr,@v,@newring; |
---|
419 | int @i,@w,@ii,@k; |
---|
420 | intvec @iv,@iw; |
---|
421 | if( find(o,"c")+find(o,"C") != 0) |
---|
422 | { |
---|
423 | @k=1; |
---|
424 | if( o[1]=="c" or o[1]=="C" ) { @o1=o[1,2]; o=o[3..size(o)]; } |
---|
425 | else { @o2=o[size(o)-1,2]; o=o[1..size(o)-2]; } |
---|
426 | } |
---|
427 | if( size(#)==0 ) { #[1]=0; } |
---|
428 | if( typeof(#[1])!="intvec" ) |
---|
429 | { |
---|
430 | if( size(#)==1 ) { @i=#[1]; def @r=basering; } |
---|
431 | if( size(#)==2 ) { @i=#[1]; def @r=#[2]; } |
---|
432 | if( o[size(o)]!=")" and find(o,",")==0 ) { o=o+"("+string(n)+")"; } |
---|
433 | } |
---|
434 | else |
---|
435 | { |
---|
436 | @iv=#[1]; |
---|
437 | if( size(#)==2 ) { @i=#[2]; def @r=basering; } |
---|
438 | if( size(#)==3 ) { @i=#[2]; def @r=#[3]; } |
---|
439 | if( @iv==0 && o[size(o)]!=")" && find(o,",")==0 ) {o=o+"("+string(n)+")";} |
---|
440 | } |
---|
441 | @ro=ordstr(@r); |
---|
442 | if( @ro[1]=="c" or @ro[1]=="C" ) |
---|
443 | { @v=@ro[1,2]; @ro=@ro[3..size(@ro)]; } |
---|
444 | else |
---|
445 | { @wstr=@ro[size(@ro)-1,2]; @ro=@ro[1..size(@ro)-2]; } |
---|
446 | if( @k==0) { @o1=@v; @o2=@wstr; } |
---|
447 | //----------------- prepare ordering if an intvec is given -------------------- |
---|
448 | if( typeof(#[1])=="intvec" and #[1]!=0 ) |
---|
449 | { |
---|
450 | @k=n; //@k counts no of vars not yet ordered |
---|
451 | @w=find(o,"w")+find(o,"W");o=o+" "; |
---|
452 | if( @w!=0 ) |
---|
453 | { |
---|
454 | @wstr=o[@w..@w+1]; |
---|
455 | o=o[1,@w-1]+"@"+o[@w+2,size(o)]; |
---|
456 | @iw=@iv[2..@iv[1]+1]; |
---|
457 | @wstr=@wstr+"("+string(@iw)+")"; |
---|
458 | @k=@k-@iv[1]; |
---|
459 | @iv=@iv[@iv[1]+2..size(@iv)]; |
---|
460 | @w=0; |
---|
461 | } |
---|
462 | for( @ii=1; @ii<=size(@iv); @ii=@ii+1 ) |
---|
463 | { |
---|
464 | if( find(o,",",@w+1)!=0 ) |
---|
465 | { |
---|
466 | @w=find(o,",",@w+1); |
---|
467 | if( o[@w-1]!="@" ) |
---|
468 | { |
---|
469 | o=o[1,@w-1]+"("+string(@iv[@ii])+")"+o[@w,size(o)]; |
---|
470 | @w=find(o,",",@w+1); |
---|
471 | @k=@k-@iv[@ii]; |
---|
472 | } |
---|
473 | else { @ii=@ii-1; } |
---|
474 | } |
---|
475 | } |
---|
476 | @w=find(o,"@"); |
---|
477 | if( @w!=0 ) { o=o[1,@w-1] + @wstr + o[@w+1,size(o)]; } |
---|
478 | if( @k>0 and o[size(o)]!=")" ) { o=o+"("+string(@k)+")"; } |
---|
479 | } |
---|
480 | //------------------------ prepare string of new ring ------------------------- |
---|
481 | @newring = "ring "+na+"=("+charstr(@r)+"),("; |
---|
482 | if( find(va,",") != 0 ) |
---|
483 | { @v = va; } |
---|
484 | else |
---|
485 | { |
---|
486 | if( n>26 or va[2]=="(" ) |
---|
487 | { @v = va[1]+"(1.."+string(n)+")"; } |
---|
488 | else |
---|
489 | { @v = A_Z(va,n); } |
---|
490 | } |
---|
491 | |
---|
492 | if( @i==0 ) |
---|
493 | { |
---|
494 | @v=@v+","+varstr(@r); |
---|
495 | o=@o1+o+","+@ro+@o2; |
---|
496 | } |
---|
497 | else |
---|
498 | { |
---|
499 | @v=varstr(@r)+","+@v; |
---|
500 | o=@o1+@ro+","+o+@o2; |
---|
501 | } |
---|
502 | @newring=@newring+@v+"),("+o+");"; |
---|
503 | //---------------------------- execute and export ----------------------------- |
---|
504 | execute(@newring); |
---|
505 | export(basering); |
---|
506 | keepring(`na`); |
---|
507 | if (voice==2) { "// basering is now",na; } |
---|
508 | return(); |
---|
509 | } |
---|
510 | example |
---|
511 | { "EXAMPLE:"; echo = 2; |
---|
512 | ring r=0,(x,y,z),ds; |
---|
513 | show(r);""; |
---|
514 | extendring1("S",2,"u","ds"); |
---|
515 | setring r; |
---|
516 | show(S); ""; |
---|
517 | extendring1("R0",2,"a,w","ds"); |
---|
518 | show(R0); ""; |
---|
519 | //no intvec given, no blocksize given: blocksize is derived from no of vars |
---|
520 | int t=5; |
---|
521 | extendring1("R1",t,"a","dp"); //t global: "dp" -> "dp(5)" |
---|
522 | show(R1); ""; |
---|
523 | extendring1("R2",4,"T(","c,dp",1,r); //"dp" -> "c,..,dp(4)" |
---|
524 | show(R2);""; |
---|
525 | |
---|
526 | //no intvec given, blocksize given: given blocksize is used |
---|
527 | extendring1("R3",4,"T(","dp(2)",0,r); // "dp(2)" -> "dp(2)" |
---|
528 | show(R3);""; |
---|
529 | |
---|
530 | //intvec given: weights and blocksize is derived from given intvec |
---|
531 | //(no specification of a blocksize in the given ordstr is allowed!) |
---|
532 | //if intvec does not cover all given blocks, the last block is used for |
---|
533 | //the remaining variables, if intvec has too many components, the last |
---|
534 | //ones are ignored |
---|
535 | intvec v=3,2,3,4,1,3; |
---|
536 | extendring1("R4",10,"A","ds,ws,Dp,dp",v,0,r); |
---|
537 | //v covers 3 blocks: v[1] (=3) : no of components of ws |
---|
538 | //next v[1] values (=v[2..4]) give weights |
---|
539 | //remaining components of v are used for the remaining blocks |
---|
540 | show(R4); |
---|
541 | kill S,R0,R1,R2,R3,R4; |
---|
542 | } |
---|
543 | /////////////////////////////////////////////////////////////////////////////// |
---|
544 | |
---|
545 | proc fetchall (R, list #) |
---|
546 | USAGE: fetchall(R[,s]); R=ring/qring, s=string |
---|
547 | CREATE: fetch all objects of ring R (of type poly/ideal/vector/module/number/ |
---|
548 | matrix) into the basering. |
---|
549 | If no 3rd argument is present, the names are the same as in R. If, |
---|
550 | say, f is a poly in R and the 3rd argument is the string "R", then f |
---|
551 | is maped to f_R etc. |
---|
552 | RETURN: no return value |
---|
553 | NOTE: As fetch, this procedure maps the 1st, 2nd, ... variable of R to the |
---|
554 | 1st, 2nd, ... variable of the basering. |
---|
555 | The 3rd argument is useful in order to avoid conflicts of names, the |
---|
556 | empty string is allowed |
---|
557 | CAUTION: fetchall does not work inside a procedure |
---|
558 | //***at the moment it does not work if R contains a map |
---|
559 | EXAMPLE: example fetchall; shows an example |
---|
560 | { |
---|
561 | list @L@=names(R); |
---|
562 | int @ii@; string @s@; |
---|
563 | if( size(#) > 0 ) { @s@=@s@+"_"+#[1]; } |
---|
564 | for( @ii@=size(@L@); @ii@>0; @ii@=@ii@-1 ) |
---|
565 | { |
---|
566 | execute("def "+@L@[@ii@]+@s@+"=fetch(R,`@L@[@ii@]`);"); |
---|
567 | execute("export "+@L@[@ii@]+@s@+";"); |
---|
568 | } |
---|
569 | return(); |
---|
570 | } |
---|
571 | example |
---|
572 | { "EXAMPLE:"; |
---|
573 | "// This example is not executed since fetchall does not work in a procedure"; |
---|
574 | "// (and hence not in the example procedure). Just try the following commands:"; |
---|
575 | " ring R=0,(x,y,z),dp;"; |
---|
576 | " ideal j=x,y2,z2;"; |
---|
577 | " matrix M[2][3]=1,2,3,x,y,z;"; |
---|
578 | " j; print(M);"; |
---|
579 | " ring S=0,(a,b,c),ds;"; |
---|
580 | " fetchall(R); // map from R to S: x->a, y->b, z->c"; |
---|
581 | " names(S);"; |
---|
582 | " j; print(M);"; |
---|
583 | " fetchall(S,\"1\"); // identity map of S: copy objects, change names"; |
---|
584 | " names(S);"; |
---|
585 | " kill R,S;"; |
---|
586 | } |
---|
587 | /////////////////////////////////////////////////////////////////////////////// |
---|
588 | |
---|
589 | proc imapall (R, list #) |
---|
590 | USAGE: imapall(R[,s]); R=ring/qring, s=string |
---|
591 | CREATE: map all objects of ring R (of type poly/ideal/vector/module/number/ |
---|
592 | matrix) into the basering, by applying imap to all objects of R. |
---|
593 | If no 3rd argument is present, the names are the same as in R. If, |
---|
594 | say, f is a poly in R and the 3rd argument is the string "R", then f |
---|
595 | is maped to f_R etc. |
---|
596 | RETURN: no return value |
---|
597 | NOTE: As imap, this procedure maps the variables of R to the variables with |
---|
598 | the same name in the basering, the other variables are maped to 0. |
---|
599 | The 3rd argument is useful in order to avoid conflicts of names, the |
---|
600 | empty string is allowed |
---|
601 | CAUTION: imapall does not work inside a procedure |
---|
602 | //***at the moment it does not work if R contains a map |
---|
603 | EXAMPLE: example imapall; shows an example |
---|
604 | { |
---|
605 | list @L@=names(R); |
---|
606 | int @ii@; string @s@; |
---|
607 | if( size(#) > 0 ) { @s@=@s@+"_"+#[1]; } |
---|
608 | for( @ii@=size(@L@); @ii@>0; @ii@=@ii@-1 ) |
---|
609 | { |
---|
610 | execute("def "+@L@[@ii@]+@s@+"=imap(R,`@L@[@ii@]`);"); |
---|
611 | execute("export "+@L@[@ii@]+@s@+";"); |
---|
612 | } |
---|
613 | return(); |
---|
614 | } |
---|
615 | example |
---|
616 | { "EXAMPLE:"; |
---|
617 | "// This example is not executed since imapall does not work in a procedure"; |
---|
618 | "// (and hence not in the example procedure). Just try the following commands:"; |
---|
619 | " ring R=0,(x,y,z,u),dp;"; |
---|
620 | " ideal j=x,y,z,u2+ux+z;"; |
---|
621 | " matrix M[2][3]=1,2,3,x,y,uz;"; |
---|
622 | " j; print(M);"; |
---|
623 | " ring S=0,(a,b,c,x,z,y),ds;"; |
---|
624 | " imapall(R); // map from R to S: x->x, y->y, z->z, u->0"; |
---|
625 | " names(S);"; |
---|
626 | " j; print(M);"; |
---|
627 | " imapall(S,\"1\"); // identity map of S: copy objects, change names"; |
---|
628 | " names(S);"; |
---|
629 | " kill R,S;"; |
---|
630 | } |
---|
631 | /////////////////////////////////////////////////////////////////////////////// |
---|
632 | |
---|
633 | proc mapall (R, ideal i, list #) |
---|
634 | USAGE: mapall(R,i[,s]); R=ring/qring, i=ideal of basering, s=string |
---|
635 | CREATE: map all objects of ring R (of type poly/ideal/vector/module/number/ |
---|
636 | matrix, map) into the basering, by mapping the jth variable of R to |
---|
637 | the jth generator of the ideal i. If no 3rd argument is present, the |
---|
638 | names are the same as in R. If, say, f is a poly in R and the 3rd |
---|
639 | argument is the string "R", then f is maped to f_R etc. |
---|
640 | RETURN: no return value |
---|
641 | NOTE: This procedure has the same effect as defining a map, say psi, by |
---|
642 | map psi=R,i; and then applying psi to all objects of R. In particular, |
---|
643 | maps from R to some ring S are composed with psi, creating thus a map |
---|
644 | from the basering to S. |
---|
645 | mapall may be combined with copyring to change vars for all objects. |
---|
646 | The 3rd argument is useful in order to avoid conflicts of names, the |
---|
647 | empty string is allowed |
---|
648 | CAUTION: mapall does not work inside a procedure |
---|
649 | EXAMPLE: example mapall; shows an example |
---|
650 | { |
---|
651 | list @L@=names(R); map @psi@ = R,i; |
---|
652 | int @ii@; string @s@; |
---|
653 | if( size(#) > 0 ) { @s@=@s@+"_"+#[1]; } |
---|
654 | for( @ii@=size(@L@); @ii@>0; @ii@=@ii@-1 ) |
---|
655 | { |
---|
656 | execute("def "+@L@[@ii@]+@s@+"=@psi@(`@L@[@ii@]`);"); |
---|
657 | execute("export "+@L@[@ii@]+@s@+";"); |
---|
658 | } |
---|
659 | return(); |
---|
660 | } |
---|
661 | example |
---|
662 | { "EXAMPLE:"; |
---|
663 | "// This example is not executed since mapall does not work in a procedure"; |
---|
664 | "// (and hence not in the example procedure). Just try the following commands:"; |
---|
665 | " ring R=0,(x,y,z),dp;"; |
---|
666 | " ideal j=x,y,z;"; |
---|
667 | " matrix M[2][3]=1,2,3,x,y,z;"; |
---|
668 | " map phi=R,x2,y2,z2; "; |
---|
669 | " ring S=0,(a,b,c),ds;"; |
---|
670 | " ideal i=c,a,b;"; |
---|
671 | " mapall(R,i); // map from R to S: x->c, y->a, z->b"; |
---|
672 | " names(S);"; |
---|
673 | " j; print(M); phi; // phi is a map from R to S: x->c2, y->a2, z->b2"; |
---|
674 | " ideal i1=a2,a+b,1;"; |
---|
675 | " mapall(R,i1,\"\"); // map from R to S: x->a2, y->a+b, z->1"; |
---|
676 | " names(S);"; |
---|
677 | " j_; print(M_); phi_;"; |
---|
678 | " changevar(\"T\",\"x()\",R); // change vars in R and call result T"; |
---|
679 | " mapall(R,maxideal(1)); // identity map from R to T"; |
---|
680 | " names(T);"; |
---|
681 | " j; print(M); phi;"; |
---|
682 | " kill R,S,T;"; |
---|
683 | } |
---|
684 | /////////////////////////////////////////////////////////////////////////////// |
---|
685 | |
---|
686 | proc ringtensor (string s, list #) |
---|
687 | USAGE: ringtensor(s,r1,r2,...); s=string, r1,r2,...=rings |
---|
688 | CREATE: A new base ring with name `s` if r1,r2,... are existing rings. |
---|
689 | If, say, s = "R" and the rings r1,r2,... exist, the new ring will |
---|
690 | have name R, variables from all rings r1,r2,... and as monomial |
---|
691 | ordering the block (product) ordering of r1,r2,... . Hence, R |
---|
692 | is the tensor product of the rings r1,r2,... with ordering matrix |
---|
693 | equal to the direct sum of the ordering matrices of r1,r2,... |
---|
694 | RETURN: no return value |
---|
695 | NOTE: The characteristic of the new ring will be that of r1. The names of |
---|
696 | variables in the rings r1,r2,... should differ (if a name, say x, |
---|
697 | occurs in r1 and r2, then, in the new ring r, x always refers to the |
---|
698 | variable with name x in r1, there is no access to x in r2). |
---|
699 | The procedure works also for quotient rings ri, if the characteristic |
---|
700 | of ri is compatible with the characteristic of r1 (i.e. if imap from |
---|
701 | ri to r1 is implemented) |
---|
702 | This proc uses 'execute' or calls a procedure using 'execute'. |
---|
703 | If you use it in your own proc, let the local names of your proc |
---|
704 | start with @ (see the file HelpForProc) |
---|
705 | EXAMPLE: example ringtensor; shows an example |
---|
706 | { |
---|
707 | int @ii,@q; |
---|
708 | int @n = size(#); |
---|
709 | string @vars,@order,@oi,@s1; |
---|
710 | //---- gather variables, orderings and ideals (of qrings) from given rings ---- |
---|
711 | for(@ii=1; @ii<=@n; @ii=@ii+1) |
---|
712 | { |
---|
713 | if( ordstr(#[@ii])[1]=="C" or ordstr(#[@ii])[1]=="c" ) |
---|
714 | { @oi=ordstr(#[@ii])[3,size(ordstr(#[@ii]))-2]; } |
---|
715 | else { @oi=ordstr(#[@ii])[1,size(ordstr(#[@ii]))-2]; } |
---|
716 | @vars = @vars+varstr(#[@ii])+","; |
---|
717 | @order= @order+@oi+","; |
---|
718 | def @r(@ii)=#[@ii]; |
---|
719 | setring @r(@ii); |
---|
720 | ideal i(@ii)=ideal(@r(@ii)); |
---|
721 | int @q(@ii)=size(i(@ii)); |
---|
722 | @q=@q+@q(@ii); |
---|
723 | } |
---|
724 | if( @q!=0 ) { @s1 = "@newr"; } // @q=0 iff none of the rings ri is a qring |
---|
725 | else { @s1 = s; } |
---|
726 | //------------------------------- create new ring ----------------------------- |
---|
727 | string @newring ="=("+charstr(#[1])+"),("+@vars[1,size(@vars)-1]+"),(" |
---|
728 | +@order[1,size(@order)-1]+");"; |
---|
729 | execute("ring "+@s1+@newring); |
---|
730 | //------ create ideal for new ring if one of the given rings is a qring ------- |
---|
731 | if( @q!=0 ) |
---|
732 | { |
---|
733 | ideal i; |
---|
734 | for(@ii=1; @ii<=@n; @ii=@ii+1) |
---|
735 | { |
---|
736 | if( @q(@ii)!=0 ) |
---|
737 | { |
---|
738 | i=i+imap(@r(@ii),i(@ii)); |
---|
739 | } |
---|
740 | } |
---|
741 | i=std(i); |
---|
742 | execute("qring "+s+"=i;"); |
---|
743 | } |
---|
744 | //----------------------- export and keep created ring ------------------------ |
---|
745 | export(`s`); |
---|
746 | keepring(`s`); |
---|
747 | if (voice==2) { "// basering is now",s; } |
---|
748 | return(); |
---|
749 | } |
---|
750 | example |
---|
751 | { "EXAMPLE:"; echo = 2; |
---|
752 | ring r=32003,(x,y,u,v),dp; |
---|
753 | ring s=0,(a,b,c),wp(1,2,3); |
---|
754 | ring t=0,x(1..5),(c,ls); |
---|
755 | ringtensor("R",r,s,t); |
---|
756 | type R;""; |
---|
757 | setring s; |
---|
758 | ideal i = a2+b3+c5; |
---|
759 | changevar("S","x,y,z"); //set S (change vars of s to x,y,z) the basering |
---|
760 | qring qS =std(fetch(s,i)); //create qring of S mod i (maped to S) |
---|
761 | changevar("T","d,e,f,g,h",t); //set T (change vars of t to d..h) the basering |
---|
762 | qring qT=std(d2+e2-f3); //create qring of T mod d2+e2-f3 |
---|
763 | ringtensor("Q",s,qS,t,qT); |
---|
764 | type Q; |
---|
765 | kill R,Q,S,T; |
---|
766 | } |
---|
767 | /////////////////////////////////////////////////////////////////////////////// |
---|
768 | |
---|
769 | proc substitute (id, vars, list #) |
---|
770 | USAGE: substitute(id,vars,li); id = object in basering which can be mapped, |
---|
771 | vars = ideal expression which must be a list of variables |
---|
772 | (not counting zeroes and costant factors), |
---|
773 | li = list of ideal expressions |
---|
774 | RETURN: id, with i-th entry of vars substituted by i-th polynomial of the |
---|
775 | ideal (say, conli) obtained by concatenatin of the lists in li; |
---|
776 | if conli has less polys than size(vars), the last element of conli |
---|
777 | substitutes the remaining variables in vars |
---|
778 | EXAMPLE: example substitute; shows an example |
---|
779 | { |
---|
780 | int ii,jj,k; |
---|
781 | def P = basering; |
---|
782 | int n = nvars(P); |
---|
783 | ideal va = simplify(vars,3); |
---|
784 | int sa = size(va); |
---|
785 | ideal all = #[1..size(#)]; |
---|
786 | int na = ncols(all); |
---|
787 | ideal m = maxideal(1); |
---|
788 | for( jj=1; jj<=sa; jj++) |
---|
789 | { |
---|
790 | if( size(va[jj]) > 1) |
---|
791 | { |
---|
792 | "// object to be substituted must be a variable"; |
---|
793 | return(); |
---|
794 | } |
---|
795 | for( ii=1; ii<=n; ii++ ) |
---|
796 | { |
---|
797 | if( va[jj]/var(ii) != 0 ) |
---|
798 | { |
---|
799 | if( jj <= na ) { m[ii] = all[jj]; } |
---|
800 | else { m[ii] = all[na]; } |
---|
801 | } |
---|
802 | } |
---|
803 | } |
---|
804 | map phi = P,m; |
---|
805 | return(phi(id)); |
---|
806 | } |
---|
807 | example |
---|
808 | { "EXAMPLE:"; echo=2; |
---|
809 | ring r=0,(a,b,t,s,u,v,x,y),ds; |
---|
810 | poly f=b+y+ax+sx+vy2+ux; |
---|
811 | ideal vars = a,y,b; |
---|
812 | ideal subs = t4,1,y+t; |
---|
813 | // the following commands all define the substitution: |
---|
814 | // a -> t4, y -> 1, b -> y+t |
---|
815 | substitute(f,vars,subs); |
---|
816 | substitute(f,vars,t4,1,y+t); |
---|
817 | substitute(f,ideal(a)+y+b,t4,1,y+t); |
---|
818 | // substitute all variables in vars by 1: |
---|
819 | substitute(f,vars,1); |
---|
820 | // substitute all variables by 1, except those in vars: |
---|
821 | substitute(f,substitute(maxideal(1),vars,0),1); |
---|
822 | } |
---|
823 | /////////////////////////////////////////////////////////////////////////////// |
---|
824 | |
---|
825 | proc swapvars (id,poly p1,poly p2) |
---|
826 | USAGE: swapvars(id,p1,p2); id = object in basering which can be mapped |
---|
827 | p1, p2 = variables which shall be interchanged |
---|
828 | RETURN: id, with p1 and p2 interchanged |
---|
829 | EXAMPLE: example swapvars; shows an example |
---|
830 | { |
---|
831 | def bR = basering; |
---|
832 | execute " ring @newR = ("+charstr(bR)+"),("+varstr(bR)+",@t),dp;"; |
---|
833 | def id = imap(bR,id); |
---|
834 | poly p1 = imap(bR,p1); |
---|
835 | poly p2 = imap(bR,p2); |
---|
836 | id = substitute(id,p2,@t); |
---|
837 | id = substitute(id,p1,p2); |
---|
838 | id = substitute(id,@t,p1); |
---|
839 | setring bR; |
---|
840 | id = imap(@newR,id); |
---|
841 | return(id); |
---|
842 | } |
---|
843 | } |
---|
844 | example |
---|
845 | { "EXAMPLE:"; echo=2; |
---|
846 | ring r; |
---|
847 | poly f = x5+y3+z2; |
---|
848 | swapvars(f,x,y); |
---|
849 | } |
---|
850 | /////////////////////////////////////////////////////////////////////////////// |
---|