| 1180 | | lres0[2*@n-1]=groebner(lres0[2*@n-1],"par2var"); |
| 1181 | | lres0[2*@n]=groebner(lres0[2*@n],"par2var"); |
| | 1177 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
| | 1178 | lres0[2*@n]=groebner(lres0[2*@n]); |
| | 1179 | } |
| | 1180 | } |
| | 1181 | } |
| | 1182 | primary=primary+lres0; |
| | 1183 | |
| | 1184 | //============================================================= |
| | 1185 | // if(npars(@P)>0) |
| | 1186 | // { |
| | 1187 | // @ri= "ring @Phelp =" |
| | 1188 | // +string(char(@P))+", |
| | 1189 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
| | 1190 | // } |
| | 1191 | // else |
| | 1192 | // { |
| | 1193 | // @ri= "ring @Phelp =" |
| | 1194 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
| | 1195 | // } |
| | 1196 | // execute(@ri); |
| | 1197 | // list @lvec; |
| | 1198 | // list @lr=imap(@P,lres0); |
| | 1199 | // ideal @lr1; |
| | 1200 | // |
| | 1201 | // if(size(@lr)==2) |
| | 1202 | // { |
| | 1203 | // @lr[2]=homog(@lr[2],@t); |
| | 1204 | // @lr1=std(@lr[2]); |
| | 1205 | // @lvec[2]=hilb(@lr1,1); |
| | 1206 | // } |
| | 1207 | // else |
| | 1208 | // { |
| | 1209 | // for(@n=1;@n<=size(@lr)/2;@n++) |
| | 1210 | // { |
| | 1211 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
| | 1212 | // { |
| | 1213 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
| | 1214 | // @lr1=std(@lr[2*@n-1]); |
| | 1215 | // @lvec[2*@n-1]=hilb(@lr1,1); |
| | 1216 | // @lvec[2*@n]=@lvec[2*@n-1]; |
| | 1217 | // } |
| | 1218 | // else |
| | 1219 | // { |
| | 1220 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
| | 1221 | // @lr1=std(@lr[2*@n-1]); |
| | 1222 | // @lvec[2*@n-1]=hilb(@lr1,1); |
| | 1223 | // @lr[2*@n]=homog(@lr[2*@n],@t); |
| | 1224 | // @lr1=std(@lr[2*@n]); |
| | 1225 | // @lvec[2*@n]=hilb(@lr1,1); |
| | 1226 | // |
| | 1227 | // } |
| | 1228 | // } |
| | 1229 | // } |
| | 1230 | // @ri= "ring @Phelp1 =" |
| | 1231 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
| | 1232 | // execute(@ri); |
| | 1233 | // list @lr=imap(@Phelp,@lr); |
| | 1234 | // |
| | 1235 | // kill @Phelp; |
| | 1236 | // if(size(@lr)==2) |
| | 1237 | // { |
| | 1238 | // @lr[2]=std(@lr[2],@lvec[2]); |
| | 1239 | // @lr[2]=subst(@lr[2],@t,1); |
| | 1240 | // } |
| | 1241 | // else |
| | 1242 | // { |
| | 1243 | // for(@n=1;@n<=size(@lr)/2;@n++) |
| | 1244 | // { |
| | 1245 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
| | 1246 | // { |
| | 1247 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
| | 1248 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
| | 1249 | // @lr[2*@n]=@lr[2*@n-1]; |
| | 1250 | // attrib(@lr[2*@n],"isSB",1); |
| | 1251 | // } |
| | 1252 | // else |
| | 1253 | // { |
| | 1254 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
| | 1255 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
| | 1256 | // @lr[2*@n]=std(@lr[2*@n],@lvec[2*@n]); |
| | 1257 | // @lr[2*@n]=subst(@lr[2*@n],@t,1); |
| | 1258 | // } |
| | 1259 | // } |
| | 1260 | // } |
| | 1261 | // kill @lvec; |
| | 1262 | // setring @P; |
| | 1263 | // lres0=imap(@Phelp1,@lr); |
| | 1264 | // kill @Phelp1; |
| | 1265 | // for(@n=1;@n<=size(lres0);@n++) |
| | 1266 | // { |
| | 1267 | // lres0[@n]=clearSB(lres0[@n]); |
| | 1268 | // attrib(lres0[@n],"isSB",1); |
| | 1269 | // } |
| | 1270 | // |
| | 1271 | // primary[2*@k-1]=lres0[1]; |
| | 1272 | // primary[2*@k]=lres0[2]; |
| | 1273 | // @s=size(primary)/2; |
| | 1274 | // for(@n=1;@n<=size(lres0)/2-1;@n++) |
| | 1275 | // { |
| | 1276 | // primary[2*@s+2*@n-1]=lres0[2*@n+1]; |
| | 1277 | // primary[2*@s+2*@n]=lres0[2*@n+2]; |
| | 1278 | // } |
| | 1279 | // @k--; |
| | 1280 | //============================================================= |
| | 1281 | } |
| | 1282 | } |
| | 1283 | return(primary); |
| | 1284 | } |
| | 1285 | example |
| | 1286 | { "EXAMPLE:"; echo = 2; |
| | 1287 | ring r = 0,(x,y,z),lp; |
| | 1288 | poly p = z2+1; |
| | 1289 | poly q = z4+2; |
| | 1290 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
| | 1291 | i=std(i); |
| | 1292 | list pr= zero_decomp(i,ideal(0),0); |
| | 1293 | pr; |
| | 1294 | } |
| | 1295 | /////////////////////////////////////////////////////////////////////////////// |
| | 1296 | proc extF(list l,list #) |
| | 1297 | { |
| | 1298 | //zero_dimensional primary decomposition after finite field extension |
| | 1299 | def R=basering; |
| | 1300 | int p=char(R); |
| | 1301 | |
| | 1302 | if((p==0)||(p>13)||(npars(R)>0)){return(l);} |
| | 1303 | |
| | 1304 | int ex=3; |
| | 1305 | if(size(#)>0){ex=#[1];} |
| | 1306 | |
| | 1307 | list peek,peek1; |
| | 1308 | while(size(l)>0) |
| | 1309 | { |
| | 1310 | if(size(l[2])==0) |
| | 1311 | { |
| | 1312 | peek[size(peek)+1]=l[1]; |
| | 1313 | } |
| | 1314 | else |
| | 1315 | { |
| | 1316 | peek1[size(peek1)+1]=l[1]; |
| | 1317 | peek1[size(peek1)+1]=l[2]; |
| | 1318 | } |
| | 1319 | l=delete(l,1); |
| | 1320 | l=delete(l,1); |
| | 1321 | } |
| | 1322 | if(size(peek)==0){return(peek1);} |
| | 1323 | |
| | 1324 | string gnir="ring RH=("+string(p)+"^"+string(ex)+",a),("+varstr(R)+"),lp;"; |
| | 1325 | execute(gnir); |
| | 1326 | string mp="minpoly="+string(minpoly)+";"; |
| | 1327 | gnir="ring RL=("+string(p)+",a),("+varstr(R)+"),lp;"; |
| | 1328 | execute(gnir); |
| | 1329 | execute(mp); |
| | 1330 | list L=imap(R,peek); |
| | 1331 | list pr, keep; |
| | 1332 | int i; |
| | 1333 | for(i=1;i<=size(L);i++) |
| | 1334 | { |
| | 1335 | attrib(L[i],"isSB",1); |
| | 1336 | pr=zero_decomp(L[i],0,0); |
| | 1337 | keep=keep+pr; |
| | 1338 | } |
| | 1339 | for(i=1;i<=size(keep);i++) |
| | 1340 | { |
| | 1341 | keep[i]=simplify(keep[i],1); |
| | 1342 | } |
| | 1343 | mp="poly pp="+string(minpoly)+";"; |
| | 1344 | |
| | 1345 | string gnir1="ring RS="+string(p)+",("+varstr(R)+",a),lp;"; |
| | 1346 | execute(gnir1); |
| | 1347 | execute(mp); |
| | 1348 | list L=imap(RL,keep); |
| | 1349 | |
| | 1350 | for(i=1;i<=size(L);i++) |
| | 1351 | { |
| | 1352 | L[i]=eliminate(L[i]+ideal(pp),a); |
| | 1353 | } |
| | 1354 | i=0; |
| | 1355 | int j; |
| | 1356 | while(i<size(L)/2-1) |
| | 1357 | { |
| | 1358 | i++; |
| | 1359 | j=i; |
| | 1360 | while(j<size(L)/2) |
| | 1361 | { |
| | 1362 | j++; |
| | 1363 | if(idealsEqual(L[2*i-1],L[2*j-1])) |
| | 1364 | { |
| | 1365 | L=delete(L,2*j-1); |
| | 1366 | L=delete(L,2*j-1); |
| | 1367 | j--; |
| | 1368 | } |
| | 1369 | } |
| | 1370 | } |
| | 1371 | setring R; |
| | 1372 | list re=imap(RS,L); |
| | 1373 | re=re+peek1; |
| | 1374 | |
| | 1375 | return(extF(re,ex+1)); |
| | 1376 | } |
| | 1377 | |
| | 1378 | /////////////////////////////////////////////////////////////////////////////// |
| | 1379 | proc zeroSp(ideal i) |
| | 1380 | { |
| | 1381 | //preparation for the separable closure |
| | 1382 | //decomposition into ideals of special type |
| | 1383 | //i.e. the minimal polynomials of every variable mod i are irreducible |
| | 1384 | //returns a list of 2 lists: rr=pe,qe |
| | 1385 | //the ideals in pe[l] are special, their special elements are in qe[l] |
| | 1386 | //pe[l] is a dp-Groebnerbasis |
| | 1387 | //the radical of the intersection of the pe[l] is equal to the radical of i |
| | 1388 | |
| | 1389 | def R=basering; |
| | 1390 | |
| | 1391 | //i has to be a reduced groebner basis |
| | 1392 | ideal F=finduni(i); |
| | 1393 | |
| | 1394 | int j,k,l,ready; |
| | 1395 | list fa; |
| | 1396 | fa[1]=factorize(F[1],1); |
| | 1397 | poly te,ti; |
| | 1398 | ideal tj; |
| | 1399 | //avoid factorization of the same polynomial |
| | 1400 | for(j=2;j<=size(F);j++) |
| | 1401 | { |
| | 1402 | for(k=1;k<=j-1;k++) |
| | 1403 | { |
| | 1404 | ti=F[k]; |
| | 1405 | te=subst(ti,var(k),var(j)); |
| | 1406 | if(te==F[j]) |
| | 1407 | { |
| | 1408 | tj=fa[k]; |
| | 1409 | fa[j]=subst(tj,var(k),var(j)); |
| | 1410 | ready=1; |
| | 1411 | break; |
| | 1412 | } |
| | 1413 | } |
| | 1414 | if(!ready) |
| | 1415 | { |
| | 1416 | fa[j]=factorize(F[j],1); |
| | 1417 | } |
| | 1418 | ready=0; |
| | 1419 | } |
| | 1420 | execute( "ring P=("+charstr(R)+"),("+varstr(R)+"),(C,dp);"); |
| | 1421 | ideal i=imap(R,i); |
| | 1422 | if(npars(basering)==0) |
| | 1423 | { |
| | 1424 | ideal J=fglm(R,i); |
| | 1425 | } |
| | 1426 | else |
| | 1427 | { |
| | 1428 | ideal J=groebner(i); |
| | 1429 | } |
| | 1430 | list fa=imap(R,fa); |
| | 1431 | list qe=J; //collects a dp-Groebnerbasis of the special ideals |
| | 1432 | list keep=ideal(0); //collects the special elements |
| | 1433 | |
| | 1434 | list re,em,ke; |
| | 1435 | ideal K,L; |
| | 1436 | |
| | 1437 | for(j=1;j<=nvars(basering);j++) |
| | 1438 | { |
| | 1439 | for(l=1;l<=size(qe);l++) |
| | 1440 | { |
| | 1441 | for(k=1;k<=size(fa[j]);k++) |
| | 1442 | { |
| | 1443 | L=std(qe[l],fa[j][k]); |
| | 1444 | K=keep[l],fa[j][k]; |
| | 1445 | if(deg(L[1])>0) |
| | 1446 | { |
| | 1447 | re[size(re)+1]=L; |
| | 1448 | ke[size(ke)+1]=K; |
| | 1449 | } |
| | 1450 | } |
| | 1451 | } |
| | 1452 | qe=re; |
| | 1453 | re=em; |
| | 1454 | keep=ke; |
| | 1455 | ke=em; |
| | 1456 | } |
| | 1457 | |
| | 1458 | setring R; |
| | 1459 | list qe=imap(P,keep); |
| | 1460 | list pe=imap(P,qe); |
| | 1461 | for(l=1;l<=size(qe);l++) |
| | 1462 | { |
| | 1463 | qe[l]=simplify(qe[l],2); |
| | 1464 | } |
| | 1465 | list rr=pe,qe; |
| | 1466 | return(rr); |
| | 1467 | } |
| | 1468 | /////////////////////////////////////////////////////////////////////////////// |
| | 1469 | |
| | 1470 | proc zeroSepClos(ideal I,ideal F) |
| | 1471 | { |
| | 1472 | //computes the separable closure of the special ideal I |
| | 1473 | //F is the set of special elements of I |
| | 1474 | //returns the separable closure sc(I) of I and an intvec v |
| | 1475 | //such that sc(I)=preimage(frobenius definde by v) |
| | 1476 | //i.e. var(i)----->var(i)^(p^v[i]) |
| | 1477 | |
| | 1478 | if(homog(I)==1){return(maxideal(1));} |
| | 1479 | |
| | 1480 | //assume F[i] irreducible in I and depending only on var(i) |
| | 1481 | |
| | 1482 | def R=basering; |
| | 1483 | int n=nvars(R); |
| | 1484 | int p=char(R); |
| | 1485 | intvec v; |
| | 1486 | v[n]=0; |
| | 1487 | int i,k; |
| | 1488 | list l; |
| | 1489 | |
| | 1490 | for(i=1;i<=n;i++) |
| | 1491 | { |
| | 1492 | l[i]=sep(F[i],i); |
| | 1493 | F[i]=l[i][1]; |
| | 1494 | if(l[i][2]>k){k=l[i][2];} |
| | 1495 | } |
| | 1496 | |
| | 1497 | if(k==0){return(list(I,v));} //the separable case |
| | 1498 | ideal m; |
| | 1499 | |
| | 1500 | for(i=1;i<=n;i++) |
| | 1501 | { |
| | 1502 | m[i]=var(i)^(p^l[i][2]); |
| | 1503 | v[i]=l[i][2]; |
| | 1504 | } |
| | 1505 | map phi=R,m; |
| | 1506 | ideal J=preimage(R,phi,I); |
| | 1507 | return(list(J,v)); |
| | 1508 | } |
| | 1509 | /////////////////////////////////////////////////////////////////////////////// |
| | 1510 | |
| | 1511 | proc insepDecomp(ideal i) |
| | 1512 | { |
| | 1513 | //decomposes i into special ideals |
| | 1514 | //computes the prime decomposition of the special ideals |
| | 1515 | //and transforms it back to a decomposition of i |
| | 1516 | |
| | 1517 | def R=basering; |
| | 1518 | list pr=zeroSp(i); |
| | 1519 | int l,k; |
| | 1520 | list re,wo,qr; |
| | 1521 | ideal m=maxideal(1); |
| | 1522 | ideal K; |
| | 1523 | map phi=R,m; |
| | 1524 | int p=char(R); |
| | 1525 | intvec op=option(get); |
| | 1526 | |
| | 1527 | for(l=1;l<=size(pr[1]);l++) |
| | 1528 | { |
| | 1529 | wo=zeroSepClos(pr[1][l],pr[2][l]); |
| | 1530 | for(k=1;k<=nvars(basering);k++) |
| | 1531 | { |
| | 1532 | m[k]=var(k)^(p^wo[2][k]); |
| | 1533 | } |
| | 1534 | phi=R,m; |
| | 1535 | qr=decomp(wo[1],2); |
| | 1536 | |
| | 1537 | option(redSB); |
| | 1538 | for(k=1;k<=size(qr)/2;k++) |
| | 1539 | { |
| | 1540 | K=qr[2*k]; |
| | 1541 | K=phi(K); |
| | 1542 | K=groebner(K); |
| | 1543 | re[size(re)+1]=zeroRad(K); |
| | 1544 | } |
| | 1545 | option(noredSB); |
| | 1546 | } |
| | 1547 | option(set,op); |
| | 1548 | return(re); |
| | 1549 | } |
| | 1550 | |
| | 1551 | |
| | 1552 | /////////////////////////////////////////////////////////////////////////////// |
| | 1553 | |
| | 1554 | static proc clearSB (ideal i,list #) |
| | 1555 | "USAGE: clearSB(i); i ideal which is SB ordered by monomial ordering |
| | 1556 | RETURN: ideal = minimal SB |
| | 1557 | NOTE: |
| | 1558 | EXAMPLE: example clearSB; shows an example |
| | 1559 | " |
| | 1560 | { |
| | 1561 | int k,j; |
| | 1562 | poly m; |
| | 1563 | int c=size(i); |
| | 1564 | |
| | 1565 | if(size(#)==0) |
| | 1566 | { |
| | 1567 | for(j=1;j<c;j++) |
| | 1568 | { |
| | 1569 | if(deg(i[j])==0) |
| | 1570 | { |
| | 1571 | i=ideal(1); |
| | 1572 | return(i); |
| | 1573 | } |
| | 1574 | if(deg(i[j])>0) |
| | 1575 | { |
| | 1576 | m=lead(i[j]); |
| | 1577 | for(k=j+1;k<=c;k++) |
| | 1578 | { |
| | 1579 | if(size(lead(i[k])/m)>0) |
| | 1580 | { |
| | 1581 | i[k]=0; |
| | 1582 | } |
| | 1583 | } |
| | 1584 | } |
| | 1585 | } |
| | 1586 | } |
| | 1587 | else |
| | 1588 | { |
| | 1589 | j=0; |
| | 1590 | while(j<c-1) |
| | 1591 | { |
| | 1592 | j++; |
| | 1593 | if(deg(i[j])==0) |
| | 1594 | { |
| | 1595 | i=ideal(1); |
| | 1596 | return(i); |
| | 1597 | } |
| | 1598 | if(deg(i[j])>0) |
| | 1599 | { |
| | 1600 | m=lead(i[j]); |
| | 1601 | for(k=j+1;k<=c;k++) |
| | 1602 | { |
| | 1603 | if(size(lead(i[k])/m)>0) |
| | 1604 | { |
| | 1605 | if((leadexp(m)!=leadexp(i[k]))||(#[j]<=#[k])) |
| | 1606 | { |
| | 1607 | i[k]=0; |
| | 1608 | } |
| | 1609 | else |
| | 1610 | { |
| | 1611 | i[j]=0; |
| | 1612 | break; |
| | 1613 | } |
| | 1614 | } |
| | 1615 | } |
| | 1616 | } |
| | 1617 | } |
| | 1618 | } |
| | 1619 | return(simplify(i,2)); |
| | 1620 | } |
| | 1621 | example |
| | 1622 | { "EXAMPLE:"; echo = 2; |
| | 1623 | ring r = (0,a,b),(x,y,z),dp; |
| | 1624 | ideal i=ax2+y,a2x+y,bx; |
| | 1625 | list l=1,2,1; |
| | 1626 | ideal j=clearSB(i,l); |
| | 1627 | j; |
| | 1628 | } |
| | 1629 | |
| | 1630 | /////////////////////////////////////////////////////////////////////////////// |
| | 1631 | static proc clearSBNeu (ideal i,list #) |
| | 1632 | "USAGE: clearSB(i); i ideal which is SB ordered by monomial ordering |
| | 1633 | RETURN: ideal = minimal SB |
| | 1634 | NOTE: |
| | 1635 | EXAMPLE: example clearSB; shows an example |
| | 1636 | " |
| | 1637 | { |
| | 1638 | int k,j; |
| | 1639 | intvec m,n,v,w; |
| | 1640 | int c=size(i); |
| | 1641 | w=leadexp(0); |
| | 1642 | v[size(i)]=0; |
| | 1643 | |
| | 1644 | j=0; |
| | 1645 | while(j<c-1) |
| | 1646 | { |
| | 1647 | j++; |
| | 1648 | if(deg(i[j])>=0) |
| | 1649 | { |
| | 1650 | m=leadexp(i[j]); |
| | 1651 | for(k=j+1;k<=c;k++) |
| | 1652 | { |
| | 1653 | n=leadexp(i[k]); |
| | 1654 | if(n!=w) |
| | 1655 | { |
| | 1656 | if(((m==n)&&(#[j]>#[k]))||((teilt(n,m))&&(n!=m))) |
| | 1657 | { |
| | 1658 | i[j]=0; |
| | 1659 | v[j]=1; |
| | 1660 | break; |
| | 1661 | } |
| | 1662 | if(((m==n)&&(#[j]<=#[k]))||((teilt(m,n))&&(n!=m))) |
| | 1663 | { |
| | 1664 | i[k]=0; |
| | 1665 | v[k]=1; |
| | 1666 | } |
| | 1667 | } |
| | 1668 | } |
| | 1669 | } |
| | 1670 | } |
| | 1671 | return(v); |
| | 1672 | } |
| | 1673 | |
| | 1674 | static proc teilt(intvec a, intvec b) |
| | 1675 | { |
| | 1676 | int i; |
| | 1677 | for(i=1;i<=size(a);i++) |
| | 1678 | { |
| | 1679 | if(a[i]>b[i]){return(0);} |
| | 1680 | } |
| | 1681 | return(1); |
| | 1682 | } |
| | 1683 | /////////////////////////////////////////////////////////////////////////////// |
| | 1684 | |
| | 1685 | static proc independSet (ideal j) |
| | 1686 | "USAGE: independentSet(i); i ideal |
| | 1687 | RETURN: list = new varstring with the independent set at the end, |
| | 1688 | ordstring with the corresponding block ordering, |
| | 1689 | the integer where the independent set starts in the varstring |
| | 1690 | NOTE: |
| | 1691 | EXAMPLE: example independentSet; shows an example |
| | 1692 | " |
| | 1693 | { |
| | 1694 | int n,k,di; |
| | 1695 | list resu,hilf; |
| | 1696 | string var1,var2; |
| | 1697 | list v=indepSet(j,1); |
| | 1698 | |
| | 1699 | for(n=1;n<=size(v);n++) |
| | 1700 | { |
| | 1701 | di=0; |
| | 1702 | var1=""; |
| | 1703 | var2=""; |
| | 1704 | for(k=1;k<=size(v[n]);k++) |
| | 1705 | { |
| | 1706 | if(v[n][k]!=0) |
| | 1707 | { |
| | 1708 | di++; |
| | 1709 | var2=var2+"var("+string(k)+"),"; |
| | 1710 | } |
| | 1711 | else |
| | 1712 | { |
| | 1713 | var1=var1+"var("+string(k)+"),"; |
| | 1714 | } |
| | 1715 | } |
| | 1716 | if(di>0) |
| | 1717 | { |
| | 1718 | var1=var1+var2; |
| | 1719 | var1=var1[1..size(var1)-1]; |
| | 1720 | hilf[1]=var1; |
| | 1721 | hilf[2]="lp"; |
| | 1722 | //"lp("+string(nvars(basering)-di)+"),dp("+string(di)+")"; |
| | 1723 | hilf[3]=di; |
| | 1724 | resu[n]=hilf; |
| | 1725 | } |
| | 1726 | else |
| | 1727 | { |
| | 1728 | resu[n]=varstr(basering),ordstr(basering),0; |
| | 1729 | } |
| | 1730 | } |
| | 1731 | return(resu); |
| | 1732 | } |
| | 1733 | example |
| | 1734 | { "EXAMPLE:"; echo = 2; |
| | 1735 | ring s1=(0,x,y),(a,b,c,d,e,f,g),lp; |
| | 1736 | ideal i=ea-fbg,fa+be,ec-fdg,fc+de; |
| | 1737 | i=std(i); |
| | 1738 | list l=independSet(i); |
| | 1739 | l; |
| | 1740 | i=i,g; |
| | 1741 | l=independSet(i); |
| | 1742 | l; |
| | 1743 | |
| | 1744 | ring s=0,(x,y,z),lp; |
| | 1745 | ideal i=z,yx; |
| | 1746 | list l=independSet(i); |
| | 1747 | l; |
| | 1748 | |
| | 1749 | |
| | 1750 | } |
| | 1751 | /////////////////////////////////////////////////////////////////////////////// |
| | 1752 | |
| | 1753 | static proc maxIndependSet (ideal j) |
| | 1754 | "USAGE: maxIndependentSet(i); i ideal |
| | 1755 | RETURN: list = new varstring with the maximal independent set at the end, |
| | 1756 | ordstring with the corresponding block ordering, |
| | 1757 | the integer where the independent set starts in the varstring |
| | 1758 | NOTE: |
| | 1759 | EXAMPLE: example maxIndependentSet; shows an example |
| | 1760 | " |
| | 1761 | { |
| | 1762 | int n,k,di; |
| | 1763 | list resu,hilf; |
| | 1764 | string var1,var2; |
| | 1765 | list v=indepSet(j,0); |
| | 1766 | |
| | 1767 | for(n=1;n<=size(v);n++) |
| | 1768 | { |
| | 1769 | di=0; |
| | 1770 | var1=""; |
| | 1771 | var2=""; |
| | 1772 | for(k=1;k<=size(v[n]);k++) |
| | 1773 | { |
| | 1774 | if(v[n][k]!=0) |
| | 1775 | { |
| | 1776 | di++; |
| | 1777 | var2=var2+"var("+string(k)+"),"; |
| | 1778 | } |
| | 1779 | else |
| | 1780 | { |
| | 1781 | var1=var1+"var("+string(k)+"),"; |
| | 1782 | } |
| | 1783 | } |
| | 1784 | if(di>0) |
| | 1785 | { |
| | 1786 | var1=var1+var2; |
| | 1787 | var1=var1[1..size(var1)-1]; |
| | 1788 | hilf[1]=var1; |
| | 1789 | hilf[2]="lp"; |
| | 1790 | hilf[3]=di; |
| | 1791 | resu[n]=hilf; |
| | 1792 | } |
| | 1793 | else |
| | 1794 | { |
| | 1795 | resu[n]=varstr(basering),ordstr(basering),0; |
| | 1796 | } |
| | 1797 | } |
| | 1798 | return(resu); |
| | 1799 | } |
| | 1800 | example |
| | 1801 | { "EXAMPLE:"; echo = 2; |
| | 1802 | ring s1=(0,x,y),(a,b,c,d,e,f,g),lp; |
| | 1803 | ideal i=ea-fbg,fa+be,ec-fdg,fc+de; |
| | 1804 | i=std(i); |
| | 1805 | list l=maxIndependSet(i); |
| | 1806 | l; |
| | 1807 | i=i,g; |
| | 1808 | l=maxIndependSet(i); |
| | 1809 | l; |
| | 1810 | |
| | 1811 | ring s=0,(x,y,z),lp; |
| | 1812 | ideal i=z,yx; |
| | 1813 | list l=maxIndependSet(i); |
| | 1814 | l; |
| | 1815 | |
| | 1816 | |
| | 1817 | } |
| | 1818 | |
| | 1819 | /////////////////////////////////////////////////////////////////////////////// |
| | 1820 | |
| | 1821 | static proc prepareQuotientring (int nnp) |
| | 1822 | "USAGE: prepareQuotientring(nnp); nnp int |
| | 1823 | RETURN: string = to define Kvar(nnp+1),...,var(nvars)[..rest ] |
| | 1824 | NOTE: |
| | 1825 | EXAMPLE: example independentSet; shows an example |
| | 1826 | " |
| | 1827 | { |
| | 1828 | ideal @ih,@jh; |
| | 1829 | int npar=npars(basering); |
| | 1830 | int @n; |
| | 1831 | |
| | 1832 | string quotring= "ring quring = ("+charstr(basering); |
| | 1833 | for(@n=nnp+1;@n<=nvars(basering);@n++) |
| | 1834 | { |
| | 1835 | quotring=quotring+",var("+string(@n)+")"; |
| | 1836 | @ih=@ih+var(@n); |
| | 1837 | } |
| | 1838 | |
| | 1839 | quotring=quotring+"),(var(1)"; |
| | 1840 | @jh=@jh+var(1); |
| | 1841 | for(@n=2;@n<=nnp;@n++) |
| | 1842 | { |
| | 1843 | quotring=quotring+",var("+string(@n)+")"; |
| | 1844 | @jh=@jh+var(@n); |
| | 1845 | } |
| | 1846 | quotring=quotring+"),(C,lp);"; |
| | 1847 | |
| | 1848 | return(quotring); |
| | 1849 | |
| | 1850 | } |
| | 1851 | example |
| | 1852 | { "EXAMPLE:"; echo = 2; |
| | 1853 | ring s1=(0,x),(a,b,c,d,e,f,g),lp; |
| | 1854 | def @Q=basering; |
| | 1855 | list l= prepareQuotientring(3); |
| | 1856 | l; |
| | 1857 | execute(l[1]); |
| | 1858 | execute(l[2]); |
| | 1859 | basering; |
| | 1860 | phi; |
| | 1861 | setring @Q; |
| | 1862 | |
| | 1863 | } |
| | 1864 | |
| | 1865 | /////////////////////////////////////////////////////////////////////////////// |
| | 1866 | static proc cleanPrimary(list l) |
| | 1867 | { |
| | 1868 | int i,j; |
| | 1869 | list lh; |
| | 1870 | for(i=1;i<=size(l)/2;i++) |
| | 1871 | { |
| | 1872 | if(deg(l[2*i-1][1])>0) |
| | 1873 | { |
| | 1874 | j++; |
| | 1875 | lh[j]=l[2*i-1]; |
| | 1876 | j++; |
| | 1877 | lh[j]=l[2*i]; |
| | 1878 | } |
| | 1879 | } |
| | 1880 | return(lh); |
| | 1881 | } |
| | 1882 | /////////////////////////////////////////////////////////////////////////////// |
| | 1883 | |
| | 1884 | |
| | 1885 | proc minAssPrimesold(ideal i, list #) |
| | 1886 | "USAGE: minAssPrimes(i); i ideal |
| | 1887 | minAssPrimes(i,1); i ideal (to use also the factorizing Groebner) |
| | 1888 | RETURN: list = the minimal associated prime ideals of i |
| | 1889 | EXAMPLE: example minAssPrimes; shows an example |
| | 1890 | " |
| | 1891 | { |
| | 1892 | def @P=basering; |
| | 1893 | if(size(i)==0){return(list(ideal(0)));} |
| | 1894 | list qr=simplifyIdeal(i); |
| | 1895 | map phi=@P,qr[2]; |
| | 1896 | i=qr[1]; |
| | 1897 | |
| | 1898 | execute ("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
| | 1899 | +ordstr(basering)+");"); |
| | 1900 | |
| | 1901 | |
| | 1902 | ideal i=fetch(@P,i); |
| | 1903 | if(size(#)==0) |
| | 1904 | { |
| | 1905 | int @wr; |
| | 1906 | list tluser,@res; |
| | 1907 | list primary=decomp(i,2); |
| | 1908 | |
| | 1909 | @res[1]=primary; |
| | 1910 | |
| | 1911 | tluser=union(@res); |
| | 1912 | setring @P; |
| | 1913 | list @res=imap(gnir,tluser); |
| | 1914 | return(phi(@res)); |
| | 1915 | } |
| | 1916 | list @res,empty; |
| | 1917 | ideal ser; |
| | 1918 | option(redSB); |
| | 1919 | list @pr=facstd(i); |
| | 1920 | //if(size(@pr)==1) |
| | 1921 | // { |
| | 1922 | // attrib(@pr[1],"isSB",1); |
| | 1923 | // if((dim(@pr[1])==0)&&(homog(@pr[1])==1)) |
| | 1924 | // { |
| | 1925 | // setring @P; |
| | 1926 | // list @res=maxideal(1); |
| | 1927 | // return(phi(@res)); |
| | 1928 | // } |
| | 1929 | // if(dim(@pr[1])>1) |
| | 1930 | // { |
| | 1931 | // setring @P; |
| | 1932 | // // kill gnir; |
| | 1933 | // execute ("ring gnir1 = ("+charstr(basering)+"), |
| | 1934 | // ("+varstr(basering)+"),(C,lp);"); |
| | 1935 | // ideal i=fetch(@P,i); |
| | 1936 | // list @pr=facstd(i); |
| | 1937 | // // ideal ser; |
| | 1938 | // setring gnir; |
| | 1939 | // @pr=fetch(gnir1,@pr); |
| | 1940 | // kill gnir1; |
| | 1941 | // } |
| | 1942 | // } |
| | 1943 | option(noredSB); |
| | 1944 | int j,k,odim,ndim,count; |
| | 1945 | attrib(@pr[1],"isSB",1); |
| | 1946 | if(#[1]==77) |
| | 1947 | { |
| | 1948 | odim=dim(@pr[1]); |
| | 1949 | count=1; |
| | 1950 | intvec pos; |
| | 1951 | pos[size(@pr)]=0; |
| | 1952 | for(j=2;j<=size(@pr);j++) |
| | 1953 | { |
| | 1954 | attrib(@pr[j],"isSB",1); |
| | 1955 | ndim=dim(@pr[j]); |
| | 1956 | if(ndim>odim) |
| | 1957 | { |
| | 1958 | for(k=count;k<=j-1;k++) |
| | 1959 | { |
| | 1960 | pos[k]=1; |
| | 1961 | } |
| | 1962 | count=j; |
| | 1963 | odim=ndim; |
| | 1964 | } |
| | 1965 | if(ndim<odim) |
| | 1966 | { |
| | 1967 | pos[j]=1; |
| | 1968 | } |
| | 1969 | } |
| | 1970 | for(j=1;j<=size(@pr);j++) |
| | 1971 | { |
| | 1972 | if(pos[j]!=1) |
| | 1973 | { |
| | 1974 | @res[j]=decomp(@pr[j],2); |
| | 1975 | } |
| | 1976 | else |
| | 1977 | { |
| | 1978 | @res[j]=empty; |
| | 1979 | } |
| | 1980 | } |
| | 1981 | } |
| | 1982 | else |
| | 1983 | { |
| | 1984 | ser=ideal(1); |
| | 1985 | for(j=1;j<=size(@pr);j++) |
| | 1986 | { |
| | 1987 | //@pr[j]; |
| | 1988 | //pause(); |
| | 1989 | @res[j]=decomp(@pr[j],2); |
| | 1990 | // @res[j]=decomp(@pr[j],2,@pr[j],ser); |
| | 1991 | // for(k=1;k<=size(@res[j]);k++) |
| | 1992 | // { |
| | 1993 | // ser=intersect(ser,@res[j][k]); |
| | 1994 | // } |
| | 1995 | } |
| | 1996 | } |
| | 1997 | |
| | 1998 | @res=union(@res); |
| | 1999 | setring @P; |
| | 2000 | list @res=imap(gnir,@res); |
| | 2001 | return(phi(@res)); |
| | 2002 | } |
| | 2003 | example |
| | 2004 | { "EXAMPLE:"; echo = 2; |
| | 2005 | ring r = 32003,(x,y,z),lp; |
| | 2006 | poly p = z2+1; |
| | 2007 | poly q = z4+2; |
| | 2008 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
| | 2009 | list pr= minAssPrimes(i); pr; |
| | 2010 | |
| | 2011 | minAssPrimes(i,1); |
| | 2012 | } |
| | 2013 | |
| | 2014 | static proc primT(ideal i) |
| | 2015 | { |
| | 2016 | //assumes that all generators of i are irreducible |
| | 2017 | //i is standard basis |
| | 2018 | |
| | 2019 | attrib(i,"isSB",1); |
| | 2020 | int j=size(i); |
| | 2021 | int k; |
| | 2022 | while(j>0) |
| | 2023 | { |
| | 2024 | if(deg(i[j])>1){break;} |
| | 2025 | j--; |
| | 2026 | } |
| | 2027 | if(j==0){return(1);} |
| | 2028 | if(deg(i[j])==vdim(i)){return(1);} |
| | 2029 | return(0); |
| | 2030 | } |
| | 2031 | |
| | 2032 | static proc minAssPrimes(ideal i, list #) |
| | 2033 | "USAGE: minAssPrimes(i); i ideal |
| | 2034 | Optional parameters in list #: (can be entered in any order) |
| | 2035 | 0, "facstd" -> uses facstd to first decompose the ideal |
| | 2036 | 1, "noFacstd" -> does not use facstd (default) |
| | 2037 | "SL" -> the new algorithm is used (default) |
| | 2038 | "GTZ" -> the old algorithm is used |
| | 2039 | RETURN: list = the minimal associated prime ideals of i |
| | 2040 | EXAMPLE: example minAssPrimes; shows an example |
| | 2041 | " |
| | 2042 | { |
| | 2043 | if(size(i) == 0){return(list(ideal(0)));} |
| | 2044 | string algorithm; // Algorithm to be used |
| | 2045 | string facstdOption; // To uses proc facstd |
| | 2046 | int j; // Counter |
| | 2047 | def P0 = basering; |
| | 2048 | list Pl=ringlist(P0); |
| | 2049 | intvec dp_w; |
| | 2050 | for(j=nvars(P0);j>0;j--) {dp_w[j]=1;} |
| | 2051 | Pl[3]=list(list("dp",dp_w),list("C",0)); |
| | 2052 | def P=ring(Pl); |
| | 2053 | setring P; |
| | 2054 | ideal i=imap(P0,i); |
| | 2055 | |
| | 2056 | // Set input parameters |
| | 2057 | algorithm = "SL"; // Default: SL algorithm |
| | 2058 | facstdOption = "noFacstd"; // Default: facstd is not used |
| | 2059 | if(size(#) > 0) |
| | 2060 | { |
| | 2061 | int valid; |
| | 2062 | for(j = 1; j <= size(#); j++) |
| | 2063 | { |
| | 2064 | valid = 0; |
| | 2065 | if((typeof(#[j]) == "int") or (typeof(#[j]) == "number")) |
| | 2066 | { |
| | 2067 | if (#[j] == 0) {facstdOption = "noFacstd"; valid = 1;} // If #[j] == 0, facstd is not used. |
| | 2068 | if (#[j] == 1) {facstdOption = "facstd"; valid = 1;} // If #[j] == 1, facstd is used. |
| | 2069 | } |
| | 2070 | if(typeof(#[j]) == "string") |
| | 2071 | { |
| | 2072 | if(#[j] == "GTZ" || #[j] == "SL") |
| | 2073 | { |
| | 2074 | algorithm = #[j]; |
| | 2075 | valid = 1; |
| | 2076 | } |
| | 2077 | if(#[j] == "noFacstd" || #[j] == "facstd") |
| | 2078 | { |
| | 2079 | facstdOption = #[j]; |
| | 2080 | valid = 1; |
| | 2081 | } |
| | 2082 | } |
| | 2083 | if(valid == 0) |
| | 2084 | { |
| | 2085 | dbprint(1, "Warning! The following input parameter was not recognized:", #[j]); |
| | 2086 | } |
| | 2087 | } |
| | 2088 | } |
| | 2089 | |
| | 2090 | list q = simplifyIdeal(i); |
| | 2091 | list re = maxideal(1); |
| | 2092 | int a, k; |
| | 2093 | intvec op = option(get); |
| | 2094 | map phi = P,q[2]; |
| | 2095 | |
| | 2096 | list result; |
| | 2097 | |
| | 2098 | if(npars(P) == 0){option(redSB);} |
| | 2099 | |
| | 2100 | if(attrib(i,"isSB")!=1) |
| | 2101 | { |
| | 2102 | i=groebner(q[1]); |
| | 2103 | } |
| | 2104 | else |
| | 2105 | { |
| | 2106 | for(j=1;j<=nvars(basering);j++) |
| | 2107 | { |
| | 2108 | if(q[2][j]!=var(j)){k=1;break;} |
| | 2109 | } |
| | 2110 | if(k) |
| | 2111 | { |
| | 2112 | i=groebner(q[1]); |
| | 2113 | } |
| | 2114 | } |
| | 2115 | |
| | 2116 | if(dim(i) == -1){setring P0;return(ideal(1));} |
| | 2117 | if((dim(i) == 0) && (npars(P) == 0)) |
| | 2118 | { |
| | 2119 | int di = vdim(i); |
| | 2120 | execute ("ring gnir=("+charstr(P)+"),("+varstr(P)+"),lp;"); |
| | 2121 | ideal J = std(imap(P,i)); |
| | 2122 | attrib(J, "isSB", 1); |
| | 2123 | if(vdim(J) != di) |
| | 2124 | { |
| | 2125 | J = fglm(P, i); |
| | 2126 | } |
| | 2127 | list pr = triangMH(J, 2); |
| | 2128 | list qr, re; |
| | 2129 | |
| | 2130 | for(k = 1; k <= size(pr); k++) |
| | 2131 | { |
| | 2132 | if(primT(pr[k])) |
| | 2133 | { |
| | 2134 | re[size(re) + 1] = pr[k]; |
| | 2135 | } |
| | 2136 | else |
| | 2137 | { |
| | 2138 | attrib(pr[k], "isSB", 1); |
| | 2139 | // Lines changed |
| | 2140 | if (algorithm == "GTZ") |
| | 2141 | { |
| | 2142 | qr = decomp(pr[k], 2); |
| | 2143 | } |
| | 2144 | else |
| | 2145 | { |
| | 2146 | qr = minAssSL(pr[k]); |
| | 2147 | } |
| | 2148 | for(j = 1; j <= size(qr) / 2; j++) |
| | 2149 | { |
| | 2150 | re[size(re) + 1] = qr[2 * j]; |
| | 2151 | } |
| | 2152 | } |
| | 2153 | } |
| | 2154 | setring P; |
| | 2155 | re = imap(gnir, re); |
| | 2156 | re=phi(re); |
| | 2157 | option(set, op); |
| | 2158 | setring(P0); |
| | 2159 | list re=imap(P,re); |
| | 2160 | return(re); |
| | 2161 | } |
| | 2162 | |
| | 2163 | // Lines changed |
| | 2164 | if ((facstdOption == "noFacstd") || (dim(i) == 0)) |
| | 2165 | { |
| | 2166 | if (algorithm == "GTZ") |
| | 2167 | { |
| | 2168 | re[1] = decomp(i, 2); |
| | 2169 | } |
| | 2170 | else |
| | 2171 | { |
| | 2172 | re[1] = minAssSL(i); |
| | 2173 | } |
| | 2174 | re = union(re); |
| | 2175 | option(set, op); |
| | 2176 | re=phi(re); |
| | 2177 | setring(P0); |
| | 2178 | list re=imap(P,re); |
| | 2179 | return(re); |
| | 2180 | } |
| | 2181 | q = facstd(i); |
| | 2182 | |
| | 2183 | /* |
| | 2184 | if((size(q) == 1) && (dim(i) > 1)) |
| | 2185 | { |
| | 2186 | execute ("ring gnir=("+charstr(P)+"),("+varstr(P)+"),lp;"); |
| | 2187 | list p = facstd(fetch(P, i)); |
| | 2188 | if(size(p) > 1) |
| | 2189 | { |
| | 2190 | a = 1; |
| | 2191 | setring P; |
| | 2192 | q = fetch(gnir,p); |
| | 2193 | } |
| | 2194 | else |
| | 2195 | { |
| | 2196 | setring P; |
| | 2197 | } |
| | 2198 | kill gnir; |
| | 2199 | } |
| | 2200 | */ |
| | 2201 | option(set,op); |
| | 2202 | // Debug |
| | 2203 | dbprint(printlevel - voice, "Components returned by facstd", size(q), q); |
| | 2204 | for(j = 1; j <= size(q); j++) |
| | 2205 | { |
| | 2206 | if(a == 0){attrib(q[j], "isSB", 1);} |
| | 2207 | // Debug |
| | 2208 | dbprint(printlevel - voice, "We compute the decomp of component", j); |
| | 2209 | // Lines changed |
| | 2210 | if (algorithm == "GTZ") |
| | 2211 | { |
| | 2212 | re[j] = decomp(q[j], 2); |
| | 2213 | } |
| | 2214 | else |
| | 2215 | { |
| | 2216 | re[j] = minAssSL(q[j]); |
| | 2217 | } |
| | 2218 | // Debug |
| | 2219 | dbprint(printlevel - voice, "Number of components obtained for this component:", size(re[j]) / 2); |
| | 2220 | dbprint(printlevel - voice, "re[j]:", re[j]); |
| | 2221 | } |
| | 2222 | re = union(re); |
| | 2223 | re=phi(re); |
| | 2224 | setring(P0); |
| | 2225 | list re=imap(P,re); |
| | 2226 | return(re); |
| | 2227 | } |
| | 2228 | example |
| | 2229 | { "EXAMPLE:"; echo = 2; |
| | 2230 | ring r = 32003,(x,y,z),lp; |
| | 2231 | poly p = z2+1; |
| | 2232 | poly q = z4+2; |
| | 2233 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
| | 2234 | list pr= minAssPrimes(i); pr; |
| | 2235 | |
| | 2236 | minAssPrimes(i,1); |
| | 2237 | } |
| | 2238 | |
| | 2239 | static proc union(list li) |
| | 2240 | { |
| | 2241 | int i,j,k; |
| | 2242 | |
| | 2243 | def P=basering; |
| | 2244 | |
| | 2245 | execute("ring ir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
| | 2246 | list l=fetch(P,li); |
| | 2247 | list @erg; |
| | 2248 | |
| | 2249 | for(k=1;k<=size(l);k++) |
| | 2250 | { |
| | 2251 | for(j=1;j<=size(l[k])/2;j++) |
| | 2252 | { |
| | 2253 | if(deg(l[k][2*j][1])!=0) |
| | 2254 | { |
| | 2255 | i++; |
| | 2256 | @erg[i]=l[k][2*j]; |
| | 2257 | } |
| | 2258 | } |
| | 2259 | } |
| | 2260 | |
| | 2261 | list @wos; |
| | 2262 | i=0; |
| | 2263 | ideal i1,i2; |
| | 2264 | while(i<size(@erg)-1) |
| | 2265 | { |
| | 2266 | i++; |
| | 2267 | k=i+1; |
| | 2268 | i1=lead(@erg[i]); |
| | 2269 | attrib(i1,"isSB",1); |
| | 2270 | attrib(@erg[i],"isSB",1); |
| | 2271 | |
| | 2272 | while(k<=size(@erg)) |
| | 2273 | { |
| | 2274 | if(deg(@erg[i][1])==0) |
| | 2275 | { |
| | 2276 | break; |
| | 2277 | } |
| | 2278 | i2=lead(@erg[k]); |
| | 2279 | attrib(@erg[k],"isSB",1); |
| | 2280 | attrib(i2,"isSB",1); |
| | 2281 | |
| | 2282 | if(size(reduce(i1,i2,1))==0) |
| | 2283 | { |
| | 2284 | if(size(reduce(@erg[i],@erg[k],1))==0) |
| | 2285 | { |
| | 2286 | @erg[k]=ideal(1); |
| | 2287 | i2=ideal(1); |
| | 2288 | } |
| | 2289 | } |
| | 2290 | if(size(reduce(i2,i1,1))==0) |
| | 2291 | { |
| | 2292 | if(size(reduce(@erg[k],@erg[i],1))==0) |
| | 2293 | { |
| | 2294 | break; |
| | 2295 | } |
| | 2296 | } |
| | 2297 | k++; |
| | 2298 | if(k>size(@erg)) |
| | 2299 | { |
| | 2300 | @wos[size(@wos)+1]=@erg[i]; |
| | 2301 | } |
| | 2302 | } |
| | 2303 | } |
| | 2304 | if(deg(@erg[size(@erg)][1])!=0) |
| | 2305 | { |
| | 2306 | @wos[size(@wos)+1]=@erg[size(@erg)]; |
| | 2307 | } |
| | 2308 | setring P; |
| | 2309 | list @ser=fetch(ir,@wos); |
| | 2310 | return(@ser); |
| | 2311 | } |
| | 2312 | /////////////////////////////////////////////////////////////////////////////// |
| | 2313 | proc equidim(ideal i,list #) |
| | 2314 | "USAGE: equidim(i) or equidim(i,1) ; i ideal |
| | 2315 | RETURN: list of equidimensional ideals a[1],...,a[s] with: |
| | 2316 | - a[s] the equidimensional locus of i, i.e. the intersection |
| | 2317 | of the primary ideals of dimension of i |
| | 2318 | - a[1],...,a[s-1] the lower dimensional equidimensional loci. |
| | 2319 | NOTE: An embedded component q (primary ideal) of i can be replaced in the |
| | 2320 | decomposition by a primary ideal q1 with the same radical as q. @* |
| | 2321 | @code{equidim(i,1)} uses the algorithm of Eisenbud/Huneke/Vasconcelos. |
| | 2322 | |
| | 2323 | EXAMPLE:example equidim; shows an example |
| | 2324 | " |
| | 2325 | { |
| | 2326 | if(attrib(basering,"global")!=1) |
| | 2327 | { |
| | 2328 | ERROR( |
| | 2329 | "// Not implemented for this ordering, please change to global ordering." |
| | 2330 | ); |
| | 2331 | } |
| | 2332 | intvec op ; |
| | 2333 | def P = basering; |
| | 2334 | list eq; |
| | 2335 | intvec w; |
| | 2336 | int n,m; |
| | 2337 | int g=size(i); |
| | 2338 | int a=attrib(i,"isSB"); |
| | 2339 | int homo=homog(i); |
| | 2340 | if(size(#)!=0) |
| | 2341 | { |
| | 2342 | m=1; |
| | 2343 | } |
| | 2344 | |
| | 2345 | if(((homo==1)||(a==1))&&(find(ordstr(basering),"l")==0) |
| | 2346 | &&(find(ordstr(basering),"s")==0)) |
| | 2347 | { |
| | 2348 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
| | 2349 | +ordstr(basering)+");"); |
| | 2350 | ideal i=imap(P,i); |
| | 2351 | ideal j=i; |
| | 2352 | if(a==1) |
| | 2353 | { |
| | 2354 | attrib(j,"isSB",1); |
| | 2355 | } |
| | 2356 | else |
| | 2357 | { |
| | 2358 | j=groebner(i); |
| | 2359 | } |
| | 2360 | } |
| | 2361 | else |
| | 2362 | { |
| | 2363 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),dp;"); |
| | 2364 | ideal i=imap(P,i); |
| | 2365 | ideal j=groebner(i); |
| | 2366 | } |
| | 2367 | if(homo==1) |
| | 2368 | { |
| | 2369 | for(n=1;n<=nvars(basering);n++) |
| | 2370 | { |
| | 2371 | w[n]=ord(var(n)); |
| | 2372 | } |
| | 2373 | intvec hil=hilb(j,1,w); |
| | 2374 | } |
| | 2375 | |
| | 2376 | if ((dim(j)==-1)||(size(j)==0)||(nvars(basering)==1) |
| | 2377 | ||(dim(j)==0)||(dim(j)+g==nvars(basering))) |
| | 2378 | { |
| | 2379 | setring P; |
| | 2380 | eq[1]=i; |
| | 2381 | return(eq); |
| | 2382 | } |
| | 2383 | |
| | 2384 | if(m==0) |
| | 2385 | { |
| | 2386 | ideal k=equidimMax(j); |
| | 2387 | } |
| | 2388 | else |
| | 2389 | { |
| | 2390 | ideal k=equidimMaxEHV(j); |
| | 2391 | } |
| | 2392 | if(size(reduce(k,j,1))==0) |
| | 2393 | { |
| | 2394 | setring P; |
| | 2395 | eq[1]=i; |
| | 2396 | kill gnir; |
| | 2397 | return(eq); |
| | 2398 | } |
| | 2399 | op=option(get); |
| | 2400 | option(returnSB); |
| | 2401 | j=quotient(j,k); |
| | 2402 | option(set,op); |
| | 2403 | |
| | 2404 | list equi=equidim(j); |
| | 2405 | if(deg(equi[size(equi)][1])<=0) |
| | 2406 | { |
| | 2407 | equi[size(equi)]=k; |
| | 2408 | } |
| | 2409 | else |
| | 2410 | { |
| | 2411 | equi[size(equi)+1]=k; |
| | 2412 | } |
| | 2413 | setring P; |
| | 2414 | eq=imap(gnir,equi); |
| | 2415 | kill gnir; |
| | 2416 | return(eq); |
| | 2417 | } |
| | 2418 | example |
| | 2419 | { "EXAMPLE:"; echo = 2; |
| | 2420 | ring r = 32003,(x,y,z),dp; |
| | 2421 | ideal i = intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
| | 2422 | equidim(i); |
| | 2423 | } |
| | 2424 | |
| | 2425 | /////////////////////////////////////////////////////////////////////////////// |
| | 2426 | proc equidimMax(ideal i) |
| | 2427 | "USAGE: equidimMax(i); i ideal |
| | 2428 | RETURN: ideal of equidimensional locus (of maximal dimension) of i. |
| | 2429 | EXAMPLE: example equidimMax; shows an example |
| | 2430 | " |
| | 2431 | { |
| | 2432 | if(attrib(basering,"global")!=1) |
| | 2433 | { |
| | 2434 | ERROR( |
| | 2435 | "// Not implemented for this ordering, please change to global ordering." |
| | 2436 | ); |
| | 2437 | } |
| | 2438 | def P = basering; |
| | 2439 | ideal eq; |
| | 2440 | intvec w; |
| | 2441 | int n; |
| | 2442 | int g=size(i); |
| | 2443 | int a=attrib(i,"isSB"); |
| | 2444 | int homo=homog(i); |
| | 2445 | |
| | 2446 | if(((homo==1)||(a==1))&&(find(ordstr(basering),"l")==0) |
| | 2447 | &&(find(ordstr(basering),"s")==0)) |
| | 2448 | { |
| | 2449 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
| | 2450 | +ordstr(basering)+");"); |
| | 2451 | ideal i=imap(P,i); |
| | 2452 | ideal j=i; |
| | 2453 | if(a==1) |
| | 2454 | { |
| | 2455 | attrib(j,"isSB",1); |
| | 2456 | } |
| | 2457 | else |
| | 2458 | { |
| | 2459 | j=groebner(i); |
| | 2460 | } |
| | 2461 | } |
| | 2462 | else |
| | 2463 | { |
| | 2464 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),dp;"); |
| | 2465 | ideal i=imap(P,i); |
| | 2466 | ideal j=groebner(i); |
| | 2467 | } |
| | 2468 | list indep; |
| | 2469 | ideal equ,equi; |
| | 2470 | if(homo==1) |
| | 2471 | { |
| | 2472 | for(n=1;n<=nvars(basering);n++) |
| | 2473 | { |
| | 2474 | w[n]=ord(var(n)); |
| | 2475 | } |
| | 2476 | intvec hil=hilb(j,1,w); |
| | 2477 | } |
| | 2478 | if ((dim(j)==-1)||(size(j)==0)||(nvars(basering)==1) |
| | 2479 | ||(dim(j)==0)||(dim(j)+g==nvars(basering))) |
| | 2480 | { |
| | 2481 | setring P; |
| | 2482 | return(i); |
| | 2483 | } |
| | 2484 | |
| | 2485 | indep=maxIndependSet(j); |
| | 2486 | |
| | 2487 | execute("ring gnir1 = ("+charstr(basering)+"),("+indep[1][1]+"),(" |
| | 2488 | +indep[1][2]+");"); |
| | 2489 | if(homo==1) |
| | 2490 | { |
| | 2491 | ideal j=std(imap(gnir,j),hil,w); |
| | 2492 | } |
| | 2493 | else |
| | 2494 | { |
| | 2495 | ideal j=groebner(imap(gnir,j)); |
| | 2496 | } |
| | 2497 | string quotring=prepareQuotientring(nvars(basering)-indep[1][3]); |
| | 2498 | execute(quotring); |
| | 2499 | ideal j=imap(gnir1,j); |
| | 2500 | kill gnir1; |
| | 2501 | j=clearSB(j); |
| | 2502 | ideal h; |
| | 2503 | for(n=1;n<=size(j);n++) |
| | 2504 | { |
| | 2505 | h[n]=leadcoef(j[n]); |
| | 2506 | } |
| | 2507 | setring gnir; |
| | 2508 | ideal h=imap(quring,h); |
| | 2509 | kill quring; |
| | 2510 | |
| | 2511 | list l=minSat(j,h); |
| | 2512 | |
| | 2513 | if(deg(l[2])>0) |
| | 2514 | { |
| | 2515 | equ=l[1]; |
| | 2516 | attrib(equ,"isSB",1); |
| | 2517 | j=std(j,l[2]); |
| | 2518 | |
| | 2519 | if(dim(equ)==dim(j)) |
| | 2520 | { |
| | 2521 | equi=equidimMax(j); |
| | 2522 | equ=interred(intersect(equ,equi)); |
| | 2523 | } |
| | 2524 | } |
| | 2525 | else |
| | 2526 | { |
| | 2527 | equ=i; |
| | 2528 | } |
| | 2529 | |
| | 2530 | setring P; |
| | 2531 | eq=imap(gnir,equ); |
| | 2532 | kill gnir; |
| | 2533 | return(eq); |
| | 2534 | } |
| | 2535 | example |
| | 2536 | { "EXAMPLE:"; echo = 2; |
| | 2537 | ring r = 32003,(x,y,z),dp; |
| | 2538 | ideal i = intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
| | 2539 | equidimMax(i); |
| | 2540 | } |
| | 2541 | /////////////////////////////////////////////////////////////////////////////// |
| | 2542 | static proc islp() |
| | 2543 | { |
| | 2544 | string s=ordstr(basering); |
| | 2545 | int n=find(s,"lp"); |
| | 2546 | if(!n){return(0);} |
| | 2547 | int k=find(s,","); |
| | 2548 | string t=s[k+1..size(s)]; |
| | 2549 | int l=find(t,","); |
| | 2550 | t=s[1..k-1]; |
| | 2551 | int m=find(t,","); |
| | 2552 | if(l+m){return(0);} |
| | 2553 | return(1); |
| | 2554 | } |
| | 2555 | /////////////////////////////////////////////////////////////////////////////// |
| | 2556 | |
| | 2557 | proc algeDeco(ideal i, int w) |
| | 2558 | { |
| | 2559 | //reduces primery decomposition over algebraic extensions to |
| | 2560 | //the other cases |
| | 2561 | def R=basering; |
| | 2562 | int n=nvars(R); |
| | 2563 | |
| | 2564 | //---Anfang Provisorium |
| | 2565 | if((size(i)==2) && (w==2)) |
| | 2566 | { |
| | 2567 | option(redSB); |
| | 2568 | ideal J=std(i); |
| | 2569 | option(noredSB); |
| | 2570 | if((size(J)==2)&&(deg(J[1])==1)) |
| | 2571 | { |
| | 2572 | ideal keep; |
| | 2573 | poly f; |
| | 2574 | int j; |
| | 2575 | for(j=1;j<=nvars(basering);j++) |
| | 2576 | { |
| | 2577 | f=J[2]; |
| | 2578 | while((f/var(j))*var(j)-f==0) |
| | 2579 | { |
| | 2580 | f=f/var(j); |
| | 2581 | keep=keep,var(j); |
| | 2582 | } |
| | 2583 | J[2]=f; |
| | 2584 | } |
| | 2585 | ideal K=factorize(J[2],1); |
| | 2586 | if(deg(K[1])==0){K=0;} |
| | 2587 | K=K+std(keep); |
| | 2588 | ideal L; |
| | 2589 | list resu; |
| | 2590 | for(j=1;j<=size(K);j++) |
| | 2591 | { |
| | 2592 | L=J[1],K[j]; |
| | 2593 | resu[j]=L; |
| | 2594 | } |
| | 2595 | return(resu); |
| | 2596 | } |
| | 2597 | } |
| | 2598 | //---Ende Provisorium |
| | 2599 | string mp="poly p="+string(minpoly)+";"; |
| | 2600 | string gnir="ring RH="+string(char(R))+",("+varstr(R)+","+string(par(1)) |
| | 2601 | +"),dp;"; |
| | 2602 | execute(gnir); |
| | 2603 | execute(mp); |
| | 2604 | ideal i=imap(R,i); |
| | 2605 | ideal I=subst(i,var(nvars(basering)),0); |
| | 2606 | int j; |
| | 2607 | for(j=1;j<=ncols(i);j++) |
| | 2608 | { |
| | 2609 | if(i[j]!=I[j]){break;} |
| | 2610 | } |
| | 2611 | if((j>ncols(i))&&(deg(p)==1)) |
| | 2612 | { |
| | 2613 | setring R; |
| | 2614 | kill RH; |
| | 2615 | kill gnir; |
| | 2616 | string gnir="ring RH="+string(char(R))+",("+varstr(R)+"),dp;"; |
| | 2617 | execute(gnir); |
| | 2618 | ideal i=imap(R,i); |
| | 2619 | ideal J; |
| | 2620 | } |
| | 2621 | else |
| | 2622 | { |
| | 2623 | i=i,p; |
| | 2624 | } |
| | 2625 | list pr; |
| | 2626 | |
| | 2627 | if(w==0) |
| | 2628 | { |
| | 2629 | pr=decomp(i); |
| | 2630 | } |
| | 2631 | if(w==1) |
| | 2632 | { |
| | 2633 | pr=prim_dec(i,1); |
| | 2634 | pr=reconvList(pr); |
| | 2635 | } |
| | 2636 | if(w==2) |
| | 2637 | { |
| | 2638 | pr=minAssPrimes(i); |
| | 2639 | } |
| | 2640 | if(n<nvars(basering)) |
| | 2641 | { |
| | 2642 | gnir="ring RS="+string(char(R))+",("+varstr(RH) |
| | 2643 | +"),(dp("+string(n)+"),lp);"; |
| | 2644 | execute(gnir); |
| | 2645 | list pr=imap(RH,pr); |
| | 2646 | ideal K; |
| | 2647 | for(j=1;j<=size(pr);j++) |
| | 2648 | { |
| | 2649 | K=groebner(pr[j]); |
| | 2650 | K=K[2..size(K)]; |
| | 2651 | pr[j]=K; |
| | 2652 | } |
| | 2653 | setring R; |
| | 2654 | list pr=imap(RS,pr); |
| | 2655 | } |
| | 2656 | else |
| | 2657 | { |
| | 2658 | setring R; |
| | 2659 | list pr=imap(RH,pr); |
| | 2660 | } |
| | 2661 | list re; |
| | 2662 | if(w==2) |
| | 2663 | { |
| | 2664 | re=pr; |
| | 2665 | } |
| | 2666 | else |
| | 2667 | { |
| | 2668 | re=convList(pr); |
| | 2669 | } |
| | 2670 | return(re); |
| | 2671 | } |
| | 2672 | |
| | 2673 | /////////////////////////////////////////////////////////////////////////////// |
| | 2674 | static proc decomp(ideal i,list #) |
| | 2675 | "USAGE: decomp(i); i ideal (for primary decomposition) (resp. |
| | 2676 | decomp(i,1); (for the associated primes of dimension of i) ) |
| | 2677 | decomp(i,2); (for the minimal associated primes) ) |
| | 2678 | decomp(i,3); (for the absolute primary decomposition) ) |
| | 2679 | RETURN: list = list of primary ideals and their associated primes |
| | 2680 | (at even positions in the list) |
| | 2681 | (resp. a list of the minimal associated primes) |
| | 2682 | NOTE: Algorithm of Gianni/Trager/Zacharias |
| | 2683 | EXAMPLE: example decomp; shows an example |
| | 2684 | " |
| | 2685 | { |
| | 2686 | intvec op,@vv; |
| | 2687 | def @P = basering; |
| | 2688 | list primary,indep,ltras; |
| | 2689 | intvec @vh,isat,@w; |
| | 2690 | int @wr,@k,@n,@m,@n1,@n2,@n3,homo,seri,keepdi,abspri,ab,nn; |
| | 2691 | ideal peek=i; |
| | 2692 | ideal ser,tras; |
| | 2693 | int isS=(attrib(i,"isSB")==1); |
| | 2694 | |
| | 2695 | |
| | 2696 | if(size(#)>0) |
| | 2697 | { |
| | 2698 | if((#[1]==1)||(#[1]==2)||(#[1]==3)) |
| | 2699 | { |
| | 2700 | @wr=#[1]; |
| | 2701 | if(@wr==3){abspri=1;@wr=0;} |
| | 2702 | if(size(#)>1) |
| | 2703 | { |
| | 2704 | seri=1; |
| | 2705 | peek=#[2]; |
| | 2706 | ser=#[3]; |
| | 2707 | } |
| | 2708 | } |
| | 2709 | else |
| | 2710 | { |
| | 2711 | seri=1; |
| | 2712 | peek=#[1]; |
| | 2713 | ser=#[2]; |
| | 2714 | } |
| | 2715 | } |
| | 2716 | if(abspri) |
| | 2717 | { |
| | 2718 | list absprimary,abskeep,absprimarytmp,abskeeptmp; |
| | 2719 | } |
| | 2720 | homo=homog(i); |
| | 2721 | if(homo==1) |
| | 2722 | { |
| | 2723 | if(attrib(i,"isSB")!=1) |
| | 2724 | { |
| | 2725 | //ltras=mstd(i); |
| | 2726 | tras=groebner(i); |
| | 2727 | ltras=tras,tras; |
| | 2728 | attrib(ltras[1],"isSB",1); |
| | 2729 | } |
| | 2730 | else |
| | 2731 | { |
| | 2732 | ltras=i,i; |
| | 2733 | attrib(ltras[1],"isSB",1); |
| | 2734 | } |
| | 2735 | tras=ltras[1]; |
| | 2736 | attrib(tras,"isSB",1); |
| | 2737 | if(dim(tras)==0) |
| | 2738 | { |
| | 2739 | primary[1]=ltras[2]; |
| | 2740 | primary[2]=maxideal(1); |
| | 2741 | if(@wr>0) |
| | 2742 | { |
| | 2743 | list l; |
| | 2744 | l[1]=maxideal(1); |
| | 2745 | l[2]=maxideal(1); |
| | 2746 | return(l); |
| | 2747 | } |
| | 2748 | return(primary); |
| | 2749 | } |
| | 2750 | for(@n=1;@n<=nvars(basering);@n++) |
| | 2751 | { |
| | 2752 | @w[@n]=ord(var(@n)); |
| | 2753 | } |
| | 2754 | intvec @hilb=hilb(tras,1,@w); |
| | 2755 | intvec keephilb=@hilb; |
| | 2756 | } |
| | 2757 | |
| | 2758 | //---------------------------------------------------------------- |
| | 2759 | //i is the zero-ideal |
| | 2760 | //---------------------------------------------------------------- |
| | 2761 | |
| | 2762 | if(size(i)==0) |
| | 2763 | { |
| | 2764 | primary=i,i; |
| | 2765 | return(primary); |
| | 2766 | } |
| | 2767 | |
| | 2768 | //---------------------------------------------------------------- |
| | 2769 | //pass to the lexicographical ordering and compute a standardbasis |
| | 2770 | //---------------------------------------------------------------- |
| | 2771 | |
| | 2772 | int lp=islp(); |
| | 2773 | |
| | 2774 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
| | 2775 | op=option(get); |
| | 2776 | option(redSB); |
| | 2777 | |
| | 2778 | ideal ser=fetch(@P,ser); |
| | 2779 | |
| | 2780 | if(homo==1) |
| | 2781 | { |
| | 2782 | if(!lp) |
| | 2783 | { |
| | 2784 | ideal @j=std(fetch(@P,i),@hilb,@w); |
| | 2785 | } |
| | 2786 | else |
| | 2787 | { |
| | 2788 | ideal @j=fetch(@P,tras); |
| | 2789 | attrib(@j,"isSB",1); |
| | 2790 | } |
| | 2791 | } |
| | 2792 | else |
| | 2793 | { |
| | 2794 | if(lp&&isS) |
| | 2795 | { |
| | 2796 | ideal @j=fetch(@P,i); |
| | 2797 | attrib(@j,"isSB",1); |
| | 2798 | } |
| | 2799 | else |
| | 2800 | { |
| | 2801 | ideal @j=groebner(fetch(@P,i)); |
| | 2802 | } |
| | 2803 | } |
| | 2804 | option(set,op); |
| | 2805 | if(seri==1) |
| | 2806 | { |
| | 2807 | ideal peek=fetch(@P,peek); |
| | 2808 | attrib(peek,"isSB",1); |
| | 2809 | } |
| | 2810 | else |
| | 2811 | { |
| | 2812 | ideal peek=@j; |
| | 2813 | } |
| | 2814 | if((size(ser)==0)&&(!abspri)) |
| | 2815 | { |
| | 2816 | ideal fried; |
| | 2817 | @n=size(@j); |
| | 2818 | for(@k=1;@k<=@n;@k++) |
| | 2819 | { |
| | 2820 | if(deg(lead(@j[@k]))==1) |
| | 2821 | { |
| | 2822 | fried[size(fried)+1]=@j[@k]; |
| | 2823 | @j[@k]=0; |
| | 2824 | } |
| | 2825 | } |
| | 2826 | if(size(fried)==nvars(basering)) |
| | 2827 | { |
| | 2828 | setring @P; |
| | 2829 | primary[1]=i; |
| | 2830 | primary[2]=i; |
| | 2831 | return(primary); |
| | 2832 | } |
| | 2833 | if(size(fried)>0) |
| | 2834 | { |
| | 2835 | string newva; |
| | 2836 | string newma; |
| | 2837 | for(@k=1;@k<=nvars(basering);@k++) |
| | 2838 | { |
| | 2839 | @n1=0; |
| | 2840 | for(@n=1;@n<=size(fried);@n++) |
| | 2841 | { |
| | 2842 | if(leadmonom(fried[@n])==var(@k)) |
| | 2843 | { |
| | 2844 | @n1=1; |
| | 2845 | break; |
| | 2846 | } |
| | 2847 | } |
| | 2848 | if(@n1==0) |
| | 2849 | { |
| | 2850 | newva=newva+string(var(@k))+","; |
| | 2851 | newma=newma+string(var(@k))+","; |
| | 2852 | } |
| | 2853 | else |
| | 2854 | { |
| | 2855 | newma=newma+string(0)+","; |
| | 2856 | } |
| | 2857 | } |
| | 2858 | newva[size(newva)]=")"; |
| | 2859 | newma[size(newma)]=";"; |
| | 2860 | execute("ring @deirf=("+charstr(gnir)+"),("+newva+",lp;"); |
| | 2861 | execute("map @kappa=gnir,"+newma); |
| | 2862 | ideal @j= @kappa(@j); |
| | 2863 | @j=simplify(@j,2); |
| | 2864 | attrib(@j,"isSB",1); |
| | 2865 | list pr=decomp(@j); |
| | 2866 | setring gnir; |
| | 2867 | list pr=imap(@deirf,pr); |
| | 2868 | for(@k=1;@k<=size(pr);@k++) |
| | 2869 | { |
| | 2870 | @j=pr[@k]+fried; |
| | 2871 | pr[@k]=@j; |
| | 2872 | } |
| | 2873 | setring @P; |
| | 2874 | return(imap(gnir,pr)); |
| | 2875 | } |
| | 2876 | } |
| | 2877 | //---------------------------------------------------------------- |
| | 2878 | //j is the ring |
| | 2879 | //---------------------------------------------------------------- |
| | 2880 | |
| | 2881 | if (dim(@j)==-1) |
| | 2882 | { |
| | 2883 | setring @P; |
| | 2884 | primary=ideal(1),ideal(1); |
| | 2885 | return(primary); |
| | 2886 | } |
| | 2887 | |
| | 2888 | //---------------------------------------------------------------- |
| | 2889 | // the case of one variable |
| | 2890 | //---------------------------------------------------------------- |
| | 2891 | |
| | 2892 | if(nvars(basering)==1) |
| | 2893 | { |
| | 2894 | |
| | 2895 | list fac=factor(@j[1]); |
| | 2896 | list gprimary; |
| | 2897 | for(@k=1;@k<=size(fac[1]);@k++) |
| | 2898 | { |
| | 2899 | if(@wr==0) |
| | 2900 | { |
| | 2901 | gprimary[2*@k-1]=ideal(fac[1][@k]^fac[2][@k]); |
| | 2902 | gprimary[2*@k]=ideal(fac[1][@k]); |
| | 2903 | } |
| | 2904 | else |
| | 2905 | { |
| | 2906 | gprimary[2*@k-1]=ideal(fac[1][@k]); |
| | 2907 | gprimary[2*@k]=ideal(fac[1][@k]); |
| | 2908 | } |
| | 2909 | } |
| | 2910 | setring @P; |
| | 2911 | primary=fetch(gnir,gprimary); |
| | 2912 | |
| | 2913 | //HIER |
| | 2914 | if(abspri) |
| | 2915 | { |
| | 2916 | list resu,tempo; |
| | 2917 | string absotto; |
| | 2918 | for(ab=1;ab<=size(primary)/2;ab++) |
| | 2919 | { |
| | 2920 | absotto= absFactorize(primary[2*ab][1],77); |
| | 2921 | tempo=primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
| | 2922 | resu[ab]=tempo; |
| | 2923 | } |
| | 2924 | primary=resu; |
| | 2925 | } |
| | 2926 | return(primary); |
| | 2927 | } |
| | 2928 | |
| | 2929 | //------------------------------------------------------------------ |
| | 2930 | //the zero-dimensional case |
| | 2931 | //------------------------------------------------------------------ |
| | 2932 | if (dim(@j)==0) |
| | 2933 | { |
| | 2934 | op=option(get); |
| | 2935 | option(redSB); |
| | 2936 | list gprimary= zero_decomp(@j,ser,@wr); |
| | 2937 | |
| | 2938 | setring @P; |
| | 2939 | primary=fetch(gnir,gprimary); |
| | 2940 | |
| | 2941 | if(size(ser)>0) |
| | 2942 | { |
| | 2943 | primary=cleanPrimary(primary); |
| | 2944 | } |
| | 2945 | //HIER |
| | 2946 | if(abspri) |
| | 2947 | { |
| | 2948 | list resu,tempo; |
| | 2949 | string absotto; |
| | 2950 | for(ab=1;ab<=size(primary)/2;ab++) |
| | 2951 | { |
| | 2952 | absotto= absFactorize(primary[2*ab][1],77); |
| | 2953 | tempo=primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
| | 2954 | resu[ab]=tempo; |
| | 2955 | } |
| | 2956 | primary=resu; |
| | 2957 | } |
| | 2958 | return(primary); |
| | 2959 | } |
| | 2960 | |
| | 2961 | poly @gs,@gh,@p; |
| | 2962 | string @va,quotring; |
| | 2963 | list quprimary,htprimary,collectprimary,lsau,lnew,allindep,restindep; |
| | 2964 | ideal @h; |
| | 2965 | int jdim=dim(@j); |
| | 2966 | list fett; |
| | 2967 | int lauf,di,newtest; |
| | 2968 | //------------------------------------------------------------------ |
| | 2969 | //search for a maximal independent set indep,i.e. |
| | 2970 | //look for subring such that the intersection with the ideal is zero |
| | 2971 | //j intersected with K[var(indep[3]+1),...,var(nvar] is zero, |
| | 2972 | //indep[1] is the new varstring and indep[2] the string for block-ordering |
| | 2973 | //------------------------------------------------------------------ |
| | 2974 | if(@wr!=1) |
| | 2975 | { |
| | 2976 | allindep=independSet(@j); |
| | 2977 | for(@m=1;@m<=size(allindep);@m++) |
| | 2978 | { |
| | 2979 | if(allindep[@m][3]==jdim) |
| | 2980 | { |
| | 2981 | di++; |
| | 2982 | indep[di]=allindep[@m]; |
| | 2983 | } |
| | 2984 | else |
| | 2985 | { |
| | 2986 | lauf++; |
| | 2987 | restindep[lauf]=allindep[@m]; |
| | 2988 | } |
| | 2989 | } |
| | 2990 | } |
| | 2991 | else |
| | 2992 | { |
| | 2993 | indep=maxIndependSet(@j); |
| | 2994 | } |
| | 2995 | |
| | 2996 | ideal jkeep=@j; |
| | 2997 | if(ordstr(@P)[1]=="w") |
| | 2998 | { |
| | 2999 | execute("ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),("+ordstr(@P)+");"); |
| | 3000 | } |
| | 3001 | else |
| | 3002 | { |
| | 3003 | execute( "ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),(C,dp);"); |
| | 3004 | } |
| | 3005 | |
| | 3006 | if(homo==1) |
| | 3007 | { |
| | 3008 | if((ordstr(@P)[3]=="d")||(ordstr(@P)[1]=="d")||(ordstr(@P)[1]=="w") |
| | 3009 | ||(ordstr(@P)[3]=="w")) |
| | 3010 | { |
| | 3011 | ideal jwork=imap(@P,tras); |
| | 3012 | attrib(jwork,"isSB",1); |
| | 3013 | } |
| | 3014 | else |
| | 3015 | { |
| | 3016 | ideal jwork=std(imap(gnir,@j),@hilb,@w); |
| | 3017 | } |
| | 3018 | |
| | 3019 | } |
| | 3020 | else |
| | 3021 | { |
| | 3022 | ideal jwork=groebner(imap(gnir,@j)); |
| | 3023 | } |
| | 3024 | list hquprimary; |
| | 3025 | poly @p,@q; |
| | 3026 | ideal @h,fac,ser; |
| | 3027 | ideal @Ptest=1; |
| | 3028 | di=dim(jwork); |
| | 3029 | keepdi=di; |
| | 3030 | |
| | 3031 | setring gnir; |
| | 3032 | for(@m=1;@m<=size(indep);@m++) |
| | 3033 | { |
| | 3034 | isat=0; |
| | 3035 | @n2=0; |
| | 3036 | if((indep[@m][1]==varstr(basering))&&(@m==1)) |
| | 3037 | //this is the good case, nothing to do, just to have the same notations |
| | 3038 | //change the ring |
| | 3039 | { |
| | 3040 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
| | 3041 | +ordstr(basering)+");"); |
| | 3042 | ideal @j=fetch(gnir,@j); |
| | 3043 | attrib(@j,"isSB",1); |
| | 3044 | ideal ser=fetch(gnir,ser); |
| | 3045 | |
| | 3046 | } |
| | 3047 | else |
| | 3048 | { |
| | 3049 | @va=string(maxideal(1)); |
| | 3050 | if(@m==1) |
| | 3051 | { |
| | 3052 | @j=fetch(@P,i); |
| | 3053 | } |
| | 3054 | execute("ring gnir1 = ("+charstr(basering)+"),("+indep[@m][1]+"),(" |
| | 3055 | +indep[@m][2]+");"); |
| | 3056 | execute("map phi=gnir,"+@va+";"); |
| | 3057 | op=option(get); |
| | 3058 | option(redSB); |
| | 3059 | if(homo==1) |
| | 3060 | { |
| | 3061 | ideal @j=std(phi(@j),@hilb,@w); |
| | 3062 | } |
| | 3063 | else |
| | 3064 | { |
| | 3065 | ideal @j=groebner(phi(@j)); |
| | 3066 | } |
| | 3067 | ideal ser=phi(ser); |
| | 3068 | |
| | 3069 | option(set,op); |
| | 3070 | } |
| | 3071 | if((deg(@j[1])==0)||(dim(@j)<jdim)) |
| | 3072 | { |
| | 3073 | setring gnir; |
| | 3074 | break; |
| | 3075 | } |
| | 3076 | for (lauf=1;lauf<=size(@j);lauf++) |
| | 3077 | { |
| | 3078 | fett[lauf]=size(@j[lauf]); |
| | 3079 | } |
| | 3080 | //------------------------------------------------------------------------ |
| | 3081 | //we have now the following situation: |
| | 3082 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
| | 3083 | //to this quotientring, j is their still a standardbasis, the |
| | 3084 | //leading coefficients of the polynomials there (polynomials in |
| | 3085 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
| | 3086 | //we need their ggt, gh, because of the following: let |
| | 3087 | //(j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..the rest..] |
| | 3088 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
| | 3089 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
| | 3090 | |
| | 3091 | //------------------------------------------------------------------------ |
| | 3092 | |
| | 3093 | //arrangement for quotientring K(var(nnp+1),..,var(nva))[..the rest..] and |
| | 3094 | //map phi:K[var(1),...,var(nva)] --->K(var(nnpr+1),..,var(nva))[..rest..] |
| | 3095 | //------------------------------------------------------------------------ |
| | 3096 | |
| | 3097 | quotring=prepareQuotientring(nvars(basering)-indep[@m][3]); |
| | 3098 | |
| | 3099 | //--------------------------------------------------------------------- |
| | 3100 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
| | 3101 | //--------------------------------------------------------------------- |
| | 3102 | |
| | 3103 | ideal @jj=lead(@j); //!! vorn vereinbaren |
| | 3104 | execute(quotring); |
| | 3105 | |
| | 3106 | ideal @jj=imap(gnir1,@jj); |
| | 3107 | @vv=clearSBNeu(@jj,fett); //!! vorn vereinbaren |
| | 3108 | setring gnir1; |
| | 3109 | @k=size(@j); |
| | 3110 | for (lauf=1;lauf<=@k;lauf++) |
| | 3111 | { |
| | 3112 | if(@vv[lauf]==1) |
| | 3113 | { |
| | 3114 | @j[lauf]=0; |
| | 3115 | } |
| | 3116 | } |
| | 3117 | @j=simplify(@j,2); |
| | 3118 | setring quring; |
| | 3119 | // @j considered in the quotientring |
| | 3120 | ideal @j=imap(gnir1,@j); |
| | 3121 | |
| | 3122 | ideal ser=imap(gnir1,ser); |
| | 3123 | |
| | 3124 | kill gnir1; |
| | 3125 | |
| | 3126 | //j is a standardbasis in the quotientring but usually not minimal |
| | 3127 | //here it becomes minimal |
| | 3128 | |
| | 3129 | attrib(@j,"isSB",1); |
| | 3130 | |
| | 3131 | //we need later ggt(h[1],...)=gh for saturation |
| | 3132 | ideal @h; |
| | 3133 | if(deg(@j[1])>0) |
| | 3134 | { |
| | 3135 | for(@n=1;@n<=size(@j);@n++) |
| | 3136 | { |
| | 3137 | @h[@n]=leadcoef(@j[@n]); |
| | 3138 | } |
| | 3139 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..the rest..] |
| | 3140 | op=option(get); |
| | 3141 | option(redSB); |
| | 3142 | |
| | 3143 | list uprimary= zero_decomp(@j,ser,@wr); |
| | 3144 | //HIER |
| | 3145 | if(abspri) |
| | 3146 | { |
| | 3147 | ideal II; |
| | 3148 | ideal jmap; |
| | 3149 | map sigma; |
| | 3150 | nn=nvars(basering); |
| | 3151 | map invsigma=basering,maxideal(1); |
| | 3152 | for(ab=1;ab<=size(uprimary)/2;ab++) |
| | 3153 | { |
| | 3154 | II=uprimary[2*ab]; |
| | 3155 | attrib(II,"isSB",1); |
| | 3156 | if(deg(II[1])!=vdim(II)) |
| | 3157 | { |
| | 3158 | jmap=randomLast(50); |
| | 3159 | sigma=basering,jmap; |
| | 3160 | jmap[nn]=2*var(nn)-jmap[nn]; |
| | 3161 | invsigma=basering,jmap; |
| | 3162 | II=groebner(sigma(II)); |
| | 3163 | } |
| | 3164 | absprimarytmp[ab]= absFactorize(II[1],77); |
| | 3165 | II=var(nn); |
| | 3166 | abskeeptmp[ab]=string(invsigma(II)); |
| | 3167 | invsigma=basering,maxideal(1); |
| | 3168 | } |
| | 3169 | } |
| | 3170 | option(set,op); |
| | 3171 | } |
| | 3172 | else |
| | 3173 | { |
| | 3174 | list uprimary; |
| | 3175 | uprimary[1]=ideal(1); |
| | 3176 | uprimary[2]=ideal(1); |
| | 3177 | } |
| | 3178 | //we need the intersection of the ideals in the list quprimary with the |
| | 3179 | //polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
| | 3180 | //but fi polynomials, then the intersection of q with the polynomialring |
| | 3181 | //is the saturation of the ideal generated by f1,...,fr with respect to |
| | 3182 | //h which is the lcm of the leading coefficients of the fi considered in |
| | 3183 | //in the quotientring: this is coded in saturn |
| | 3184 | |
| | 3185 | list saturn; |
| | 3186 | ideal hpl; |
| | 3187 | |
| | 3188 | for(@n=1;@n<=size(uprimary);@n++) |
| | 3189 | { |
| | 3190 | uprimary[@n]=interred(uprimary[@n]); // temporary fix |
| | 3191 | hpl=0; |
| | 3192 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
| | 3193 | { |
| | 3194 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
| | 3195 | } |
| | 3196 | saturn[@n]=hpl; |
| | 3197 | } |
| | 3198 | |
| | 3199 | //-------------------------------------------------------------------- |
| | 3200 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
| | 3201 | //back to the polynomialring |
| | 3202 | //--------------------------------------------------------------------- |
| | 3203 | setring gnir; |
| | 3204 | |
| | 3205 | collectprimary=imap(quring,uprimary); |
| | 3206 | lsau=imap(quring,saturn); |
| | 3207 | @h=imap(quring,@h); |
| | 3208 | |
| | 3209 | kill quring; |
| | 3210 | |
| | 3211 | @n2=size(quprimary); |
| | 3212 | @n3=@n2; |
| | 3213 | |
| | 3214 | for(@n1=1;@n1<=size(collectprimary)/2;@n1++) |
| | 3215 | { |
| | 3216 | if(deg(collectprimary[2*@n1][1])>0) |
| | 3217 | { |
| | 3218 | @n2++; |
| | 3219 | quprimary[@n2]=collectprimary[2*@n1-1]; |
| | 3220 | lnew[@n2]=lsau[2*@n1-1]; |
| | 3221 | @n2++; |
| | 3222 | lnew[@n2]=lsau[2*@n1]; |
| | 3223 | quprimary[@n2]=collectprimary[2*@n1]; |
| | 3224 | if(abspri) |
| | 3225 | { |
| | 3226 | absprimary[@n2/2]=absprimarytmp[@n1]; |
| | 3227 | abskeep[@n2/2]=abskeeptmp[@n1]; |
| | 3228 | } |
| | 3229 | } |
| | 3230 | } |
| | 3231 | //here the intersection with the polynomialring |
| | 3232 | //mentioned above is really computed |
| | 3233 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
| | 3234 | { |
| | 3235 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
| | 3236 | { |
| | 3237 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
| | 3238 | quprimary[2*@n]=quprimary[2*@n-1]; |
| | 3239 | } |
| | 3240 | else |
| | 3241 | { |
| | 3242 | if(@wr==0) |
| | 3243 | { |
| | 3244 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
| | 3245 | } |
| | 3246 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
| | 3247 | } |
| | 3248 | } |
| | 3249 | |
| | 3250 | if(size(@h)>0) |
| | 3251 | { |
| | 3252 | //--------------------------------------------------------------- |
| | 3253 | //we change to @Phelp to have the ordering dp for saturation |
| | 3254 | //--------------------------------------------------------------- |
| | 3255 | setring @Phelp; |
| | 3256 | @h=imap(gnir,@h); |
| | 3257 | if(@wr!=1) |
| | 3258 | { |
| | 3259 | if(defined(@LL)){kill @LL;} |
| | 3260 | list @LL=minSat(jwork,@h); |
| | 3261 | @Ptest=intersect(@Ptest,@LL[1]); |
| | 3262 | @q=@LL[2]; |
| | 3263 | } |
| | 3264 | else |
| | 3265 | { |
| | 3266 | fac=ideal(0); |
| | 3267 | for(lauf=1;lauf<=ncols(@h);lauf++) |
| | 3268 | { |
| | 3269 | if(deg(@h[lauf])>0) |
| | 3270 | { |
| | 3271 | fac=fac+factorize(@h[lauf],1); |
| | 3272 | } |
| | 3273 | } |
| | 3274 | fac=simplify(fac,6); |
| | 3275 | @q=1; |
| | 3276 | for(lauf=1;lauf<=size(fac);lauf++) |
| | 3277 | { |
| | 3278 | @q=@q*fac[lauf]; |
| | 3279 | } |
| | 3280 | } |
| | 3281 | jwork=std(jwork,@q); |
| | 3282 | keepdi=dim(jwork); |
| | 3283 | if(keepdi<di) |
| | 3284 | { |
| | 3285 | setring gnir; |
| | 3286 | @j=imap(@Phelp,jwork); |
| | 3287 | break; |
| | 3288 | } |
| | 3289 | if(homo==1) |
| | 3290 | { |
| | 3291 | @hilb=hilb(jwork,1,@w); |
| | 3292 | } |
| | 3293 | |
| | 3294 | setring gnir; |
| | 3295 | @j=imap(@Phelp,jwork); |
| | 3296 | } |
| | 3297 | } |
| | 3298 | |
| | 3299 | if((size(quprimary)==0)&&(@wr==1)) |
| | 3300 | { |
| | 3301 | @j=ideal(1); |
| | 3302 | quprimary[1]=ideal(1); |
| | 3303 | quprimary[2]=ideal(1); |
| | 3304 | } |
| | 3305 | if((size(quprimary)==0)) |
| | 3306 | { |
| | 3307 | keepdi=di-1; |
| | 3308 | quprimary[1]=ideal(1); |
| | 3309 | quprimary[2]=ideal(1); |
| | 3310 | } |
| | 3311 | //--------------------------------------------------------------- |
| | 3312 | //notice that j=sat(j,gh) intersected with (j,gh^n) |
| | 3313 | //we finished with sat(j,gh) and have to start with (j,gh^n) |
| | 3314 | //--------------------------------------------------------------- |
| | 3315 | if((deg(@j[1])!=0)&&(@wr!=1)) |
| | 3316 | { |
| | 3317 | if(size(quprimary)>0) |
| | 3318 | { |
| | 3319 | setring @Phelp; |
| | 3320 | ser=imap(gnir,ser); |
| | 3321 | hquprimary=imap(gnir,quprimary); |
| | 3322 | if(@wr==0) |
| | 3323 | { |
| | 3324 | //HIER STATT DURCHSCHNITT SATURIEREN! |
| | 3325 | ideal htest=@Ptest; |
| | 3326 | } |
| | 3327 | else |
| | 3328 | { |
| | 3329 | ideal htest=hquprimary[2]; |
| | 3330 | |
| | 3331 | for (@n1=2;@n1<=size(hquprimary)/2;@n1++) |
| | 3332 | { |
| | 3333 | htest=intersect(htest,hquprimary[2*@n1]); |
| | 3334 | } |
| | 3335 | } |
| | 3336 | |
| | 3337 | if(size(ser)>0) |
| | 3338 | { |
| | 3339 | ser=intersect(htest,ser); |
| | 3340 | } |
| | 3341 | else |
| | 3342 | { |
| | 3343 | ser=htest; |
| | 3344 | } |
| | 3345 | setring gnir; |
| | 3346 | ser=imap(@Phelp,ser); |
| | 3347 | } |
| | 3348 | if(size(reduce(ser,peek,1))!=0) |
| | 3349 | { |
| | 3350 | for(@m=1;@m<=size(restindep);@m++) |
| | 3351 | { |
| | 3352 | // if(restindep[@m][3]>=keepdi) |
| | 3353 | // { |
| | 3354 | isat=0; |
| | 3355 | @n2=0; |
| | 3356 | |
| | 3357 | if(restindep[@m][1]==varstr(basering)) |
| | 3358 | //the good case, nothing to do, just to have the same notations |
| | 3359 | //change the ring |
| | 3360 | { |
| | 3361 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
| | 3362 | varstr(basering)+"),("+ordstr(basering)+");"); |
| | 3363 | ideal @j=fetch(gnir,jkeep); |
| | 3364 | attrib(@j,"isSB",1); |
| | 3365 | } |
| | 3366 | else |
| | 3367 | { |
| | 3368 | @va=string(maxideal(1)); |
| | 3369 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
| | 3370 | restindep[@m][1]+"),(" +restindep[@m][2]+");"); |
| | 3371 | execute("map phi=gnir,"+@va+";"); |
| | 3372 | op=option(get); |
| | 3373 | option(redSB); |
| | 3374 | if(homo==1) |
| | 3375 | { |
| | 3376 | ideal @j=std(phi(jkeep),keephilb,@w); |
| | 3377 | } |
| | 3378 | else |
| | 3379 | { |
| | 3380 | ideal @j=groebner(phi(jkeep)); |
| | 3381 | } |
| | 3382 | ideal ser=phi(ser); |
| | 3383 | option(set,op); |
| | 3384 | } |
| | 3385 | |
| | 3386 | for (lauf=1;lauf<=size(@j);lauf++) |
| | 3387 | { |
| | 3388 | fett[lauf]=size(@j[lauf]); |
| | 3389 | } |
| | 3390 | //------------------------------------------------------------------ |
| | 3391 | //we have now the following situation: |
| | 3392 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may |
| | 3393 | //pass to this quotientring, j is their still a standardbasis, the |
| | 3394 | //leading coefficients of the polynomials there (polynomials in |
| | 3395 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
| | 3396 | //we need their ggt, gh, because of the following: |
| | 3397 | //let (j:gh^n)=(j:gh^infinity) then |
| | 3398 | //j*K(var(nnp+1),..,var(nva))[..the rest..] |
| | 3399 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
| | 3400 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
| | 3401 | |
| | 3402 | //------------------------------------------------------------------ |
| | 3403 | |
| | 3404 | //the arrangement for the quotientring |
| | 3405 | // K(var(nnp+1),..,var(nva))[..the rest..] |
| | 3406 | //and the map phi:K[var(1),...,var(nva)] ----> |
| | 3407 | //--->K(var(nnpr+1),..,var(nva))[..the rest..] |
| | 3408 | //------------------------------------------------------------------ |
| | 3409 | |
| | 3410 | quotring=prepareQuotientring(nvars(basering)-restindep[@m][3]); |
| | 3411 | |
| | 3412 | //------------------------------------------------------------------ |
| | 3413 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
| | 3414 | //------------------------------------------------------------------ |
| | 3415 | |
| | 3416 | execute(quotring); |
| | 3417 | |
| | 3418 | // @j considered in the quotientring |
| | 3419 | ideal @j=imap(gnir1,@j); |
| | 3420 | ideal ser=imap(gnir1,ser); |
| | 3421 | |
| | 3422 | kill gnir1; |
| | 3423 | |
| | 3424 | //j is a standardbasis in the quotientring but usually not minimal |
| | 3425 | //here it becomes minimal |
| | 3426 | @j=clearSB(@j,fett); |
| | 3427 | attrib(@j,"isSB",1); |
| | 3428 | |
| | 3429 | //we need later ggt(h[1],...)=gh for saturation |
| | 3430 | ideal @h; |
| | 3431 | |
| | 3432 | for(@n=1;@n<=size(@j);@n++) |
| | 3433 | { |
| | 3434 | @h[@n]=leadcoef(@j[@n]); |
| | 3435 | } |
| | 3436 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..rest..] |
| | 3437 | |
| | 3438 | op=option(get); |
| | 3439 | option(redSB); |
| | 3440 | list uprimary= zero_decomp(@j,ser,@wr); |
| | 3441 | //HIER |
| | 3442 | if(abspri) |
| | 3443 | { |
| | 3444 | ideal II; |
| | 3445 | ideal jmap; |
| | 3446 | map sigma; |
| | 3447 | nn=nvars(basering); |
| | 3448 | map invsigma=basering,maxideal(1); |
| | 3449 | for(ab=1;ab<=size(uprimary)/2;ab++) |
| | 3450 | { |
| | 3451 | II=uprimary[2*ab]; |
| | 3452 | attrib(II,"isSB",1); |
| | 3453 | if(deg(II[1])!=vdim(II)) |
| | 3454 | { |
| | 3455 | jmap=randomLast(50); |
| | 3456 | sigma=basering,jmap; |
| | 3457 | jmap[nn]=2*var(nn)-jmap[nn]; |
| | 3458 | invsigma=basering,jmap; |
| | 3459 | II=groebner(sigma(II)); |
| | 3460 | } |
| | 3461 | absprimarytmp[ab]= absFactorize(II[1],77); |
| | 3462 | II=var(nn); |
| | 3463 | abskeeptmp[ab]=string(invsigma(II)); |
| | 3464 | invsigma=basering,maxideal(1); |
| | 3465 | } |
| | 3466 | } |
| | 3467 | option(set,op); |
| | 3468 | |
| | 3469 | //we need the intersection of the ideals in the list quprimary with |
| | 3470 | //the polynomialring, i.e. let q=(f1,...,fr) in the quotientring |
| | 3471 | //such an ideal but fi polynomials, then the intersection of q with |
| | 3472 | //the polynomialring is the saturation of the ideal generated by |
| | 3473 | //f1,...,fr with respect toh which is the lcm of the leading |
| | 3474 | //coefficients of the fi considered in the quotientring: |
| | 3475 | //this is coded in saturn |
| | 3476 | |
| | 3477 | list saturn; |
| | 3478 | ideal hpl; |
| | 3479 | |
| | 3480 | for(@n=1;@n<=size(uprimary);@n++) |
| | 3481 | { |
| | 3482 | hpl=0; |
| | 3483 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
| | 3484 | { |
| | 3485 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
| | 3486 | } |
| | 3487 | saturn[@n]=hpl; |
| | 3488 | } |
| | 3489 | //------------------------------------------------------------------ |
| | 3490 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
| | 3491 | //back to the polynomialring |
| | 3492 | //------------------------------------------------------------------ |
| | 3493 | setring gnir; |
| | 3494 | collectprimary=imap(quring,uprimary); |
| | 3495 | lsau=imap(quring,saturn); |
| | 3496 | @h=imap(quring,@h); |
| | 3497 | |
| | 3498 | kill quring; |
| | 3499 | |
| | 3500 | |
| | 3501 | @n2=size(quprimary); |
| | 3502 | @n3=@n2; |
| | 3503 | |
| | 3504 | for(@n1=1;@n1<=size(collectprimary)/2;@n1++) |
| | 3505 | { |
| | 3506 | if(deg(collectprimary[2*@n1][1])>0) |
| | 3507 | { |
| | 3508 | @n2++; |
| | 3509 | quprimary[@n2]=collectprimary[2*@n1-1]; |
| | 3510 | lnew[@n2]=lsau[2*@n1-1]; |
| | 3511 | @n2++; |
| | 3512 | lnew[@n2]=lsau[2*@n1]; |
| | 3513 | quprimary[@n2]=collectprimary[2*@n1]; |
| | 3514 | if(abspri) |
| | 3515 | { |
| | 3516 | absprimary[@n2/2]=absprimarytmp[@n1]; |
| | 3517 | abskeep[@n2/2]=abskeeptmp[@n1]; |
| | 3518 | } |
| | 3519 | } |
| | 3520 | } |
| | 3521 | |
| | 3522 | |
| | 3523 | //here the intersection with the polynomialring |
| | 3524 | //mentioned above is really computed |
| | 3525 | |
| | 3526 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
| | 3527 | { |
| | 3528 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
| | 3529 | { |
| | 3530 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
| | 3531 | quprimary[2*@n]=quprimary[2*@n-1]; |
| | 3532 | } |
| | 3533 | else |
| | 3534 | { |
| | 3535 | if(@wr==0) |
| | 3536 | { |
| | 3537 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
| | 3538 | } |
| | 3539 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
| | 3540 | } |
| | 3541 | } |
| | 3542 | if(@n2>=@n3+2) |
| | 3543 | { |
| | 3544 | setring @Phelp; |
| | 3545 | ser=imap(gnir,ser); |
| | 3546 | hquprimary=imap(gnir,quprimary); |
| | 3547 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
| | 3548 | { |
| | 3549 | if(@wr==0) |
| | 3550 | { |
| | 3551 | ser=intersect(ser,hquprimary[2*@n-1]); |
| | 3552 | } |
| | 3553 | else |
| | 3554 | { |
| | 3555 | ser=intersect(ser,hquprimary[2*@n]); |
| | 3556 | } |
| | 3557 | } |
| | 3558 | setring gnir; |
| | 3559 | ser=imap(@Phelp,ser); |
| | 3560 | } |
| | 3561 | |
| | 3562 | // } |
| | 3563 | } |
| | 3564 | //HIER |
| | 3565 | if(abspri) |
| | 3566 | { |
| | 3567 | list resu,tempo; |
| | 3568 | for(ab=1;ab<=size(quprimary)/2;ab++) |
| | 3569 | { |
| | 3570 | if (deg(quprimary[2*ab][1])!=0) |
| | 3571 | { |
| | 3572 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
| | 3573 | absprimary[ab],abskeep[ab]; |
| | 3574 | resu[ab]=tempo; |
| | 3575 | } |
| | 3576 | } |
| | 3577 | quprimary=resu; |
| | 3578 | @wr=3; |
| | 3579 | } |
| | 3580 | if(size(reduce(ser,peek,1))!=0) |
| | 3581 | { |
| | 3582 | if(@wr>0) |
| | 3583 | { |
| | 3584 | htprimary=decomp(@j,@wr,peek,ser); |
| | 3585 | } |
| | 3586 | else |
| | 3587 | { |
| | 3588 | htprimary=decomp(@j,peek,ser); |
| | 3589 | } |
| | 3590 | // here we collect now both results primary(sat(j,gh)) |
| | 3591 | // and primary(j,gh^n) |
| | 3592 | @n=size(quprimary); |
| | 3593 | for (@k=1;@k<=size(htprimary);@k++) |
| | 3594 | { |
| | 3595 | quprimary[@n+@k]=htprimary[@k]; |
| | 3596 | } |
| | 3597 | } |
| | 3598 | } |
| | 3599 | |
| | 3600 | } |
| | 3601 | else |
| | 3602 | { |
| | 3603 | if(abspri) |
| | 3604 | { |
| | 3605 | list resu,tempo; |
| | 3606 | for(ab=1;ab<=size(quprimary)/2;ab++) |
| | 3607 | { |
| | 3608 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
| | 3609 | absprimary[ab],abskeep[ab]; |
| | 3610 | resu[ab]=tempo; |
| | 3611 | } |
| | 3612 | quprimary=resu; |
| | 3613 | } |
| | 3614 | } |
| | 3615 | //--------------------------------------------------------------------------- |
| | 3616 | //back to the ring we started with |
| | 3617 | //the final result: primary |
| | 3618 | //--------------------------------------------------------------------------- |
| | 3619 | |
| | 3620 | setring @P; |
| | 3621 | primary=imap(gnir,quprimary); |
| | 3622 | if(!abspri) |
| | 3623 | { |
| | 3624 | primary=cleanPrimary(primary); |
| | 3625 | } |
| | 3626 | return(primary); |
| | 3627 | } |
| | 3628 | |
| | 3629 | |
| | 3630 | example |
| | 3631 | { "EXAMPLE:"; echo = 2; |
| | 3632 | ring r = 32003,(x,y,z),lp; |
| | 3633 | poly p = z2+1; |
| | 3634 | poly q = z4+2; |
| | 3635 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
| | 3636 | list pr= decomp(i); |
| | 3637 | pr; |
| | 3638 | testPrimary( pr, i); |
| | 3639 | } |
| | 3640 | |
| | 3641 | /////////////////////////////////////////////////////////////////////////////// |
| | 3642 | static proc powerCoeffs(poly f,int e) |
| | 3643 | //computes a polynomial with the same monomials as f but coefficients |
| | 3644 | //the p^e th power of the coefficients of f |
| | 3645 | { |
| | 3646 | int i; |
| | 3647 | poly g; |
| | 3648 | int ex=char(basering)^e; |
| | 3649 | for(i=1;i<=size(f);i++) |
| | 3650 | { |
| | 3651 | g=g+leadcoef(f[i])^ex*leadmonom(f[i]); |
| | 3652 | } |
| | 3653 | return(g); |
| | 3654 | } |
| | 3655 | /////////////////////////////////////////////////////////////////////////////// |
| | 3656 | |
| | 3657 | proc sep(poly f,int i, list #) |
| | 3658 | "USAGE: input: a polynomial f depending on the i-th variable and optional |
| | 3659 | an integer k considering the polynomial f defined over Fp(t1,...,tm) |
| | 3660 | as polynomial over Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
| | 3661 | RETURN: the separabel part of f as polynomial in Fp(t1,...,tm) |
| | 3662 | and an integer k to indicate that f should be considerd |
| | 3663 | as polynomial over Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
| | 3664 | EXAMPLE: example sep; shows an example |
| | 3665 | { |
| | 3666 | def R=basering; |
| | 3667 | int k; |
| | 3668 | if(size(#)>0){k=#[1];} |
| | 3669 | |
| | 3670 | |
| | 3671 | poly h=gcd(f,diff(f,var(i))); |
| | 3672 | if((reduce(f,std(h))!=0)||(reduce(diff(f,var(i)),std(h))!=0)) |
| | 3673 | { |
| | 3674 | ERROR("FEHLER IN GCD"); |
| | 3675 | } |
| | 3676 | poly g1=lift(h,f)[1][1]; // f/h |
| | 3677 | poly h1; |
| | 3678 | |
| | 3679 | while(h!=h1) |
| | 3680 | { |
| | 3681 | h1=h; |
| | 3682 | h=gcd(h,diff(h,var(i))); |
| | 3683 | } |
| | 3684 | |
| | 3685 | if(deg(h1)==0){return(list(g1,k));} //in characteristic 0 we return here |
| | 3686 | |
| | 3687 | k++; |
| | 3688 | |
| | 3689 | ideal ma=maxideal(1); |
| | 3690 | ma[i]=var(i)^char(R); |
| | 3691 | map phi=R,ma; |
| | 3692 | ideal hh=h; //this is technical because preimage works only for ideals |
| | 3693 | |
| | 3694 | poly u=preimage(R,phi,hh)[1]; //h=u(x(i)^p) |
| | 3695 | |
| | 3696 | list g2=sep(u,i,k); //we consider u(t(1)^(p^-1),...,t(m)^(p^-1)) |
| | 3697 | g1=powerCoeffs(g1,g2[2]-k+1); //to have g1 over the same field as g2[1] |
| | 3698 | |
| | 3699 | list g3=sep(g1*g2[1],i,g2[2]); |
| | 3700 | return(g3); |
| | 3701 | } |
| | 3702 | example |
| | 3703 | { "EXAMPLE:"; echo = 2; |
| | 3704 | ring R=(5,t,s),(x,y,z),dp; |
| | 3705 | poly f=(x^25-t*x^5+t)*(x^3+s); |
| | 3706 | sep(f,1); |
| | 3707 | } |
| | 3708 | |
| | 3709 | /////////////////////////////////////////////////////////////////////////////// |
| | 3710 | proc zeroRad(ideal I,list #) |
| | 3711 | "USAGE: zeroRad(I) , I a zero-dimensional ideal |
| | 3712 | RETURN: the radical of I |
| | 3713 | NOTE: Algorithm of Kemper |
| | 3714 | EXAMPLE: example zeroRad; shows an example |
| | 3715 | { |
| | 3716 | if(homog(I)==1){return(maxideal(1));} |
| | 3717 | //I needs to be a reduced standard basis |
| | 3718 | def R=basering; |
| | 3719 | int m=npars(R); |
| | 3720 | int n=nvars(R); |
| | 3721 | int p=char(R); |
| | 3722 | int d=vdim(I); |
| | 3723 | int i,k; |
| | 3724 | list l; |
| | 3725 | if(((p==0)||(p>d))&&(d==deg(I[1]))) |
| | 3726 | { |
| | 3727 | intvec e=leadexp(I[1]); |
| | 3728 | for(i=1;i<=nvars(basering);i++) |
| | 3729 | { |
| | 3730 | if(e[i]!=0) break; |
| | 3731 | } |
| | 3732 | I[1]=sep(I[1],i)[1]; |
| | 3733 | return(interred(I)); |
| | 3734 | } |
| | 3735 | intvec op=option(get); |
| | 3736 | |
| | 3737 | option(redSB); |
| | 3738 | ideal F=finduni(I);//F[i] generates I intersected with K[var(i)] |
| | 3739 | |
| | 3740 | option(set,op); |
| | 3741 | if(size(#)>0){I=#[1];} |
| | 3742 | |
| | 3743 | for(i=1;i<=n;i++) |
| | 3744 | { |
| | 3745 | l[i]=sep(F[i],i); |
| | 3746 | F[i]=l[i][1]; |
| | 3747 | if(l[i][2]>k){k=l[i][2];} //computation of the maximal k |
| | 3748 | } |
| | 3749 | |
| | 3750 | if((k==0)||(m==0)){return(interred(I+F));} //the separable case |
| | 3751 | |
| | 3752 | for(i=1;i<=n;i++) //consider all polynomials over |
| | 3753 | { //Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
| | 3754 | F[i]=powerCoeffs(F[i],k-l[i][2]); |
| | 3755 | } |
| | 3756 | |
| | 3757 | string cR="ring @R="+string(p)+",("+parstr(R)+","+varstr(R)+"),dp;"; |
| | 3758 | execute(cR); |
| | 3759 | ideal F=imap(R,F); |
| | 3760 | |
| | 3761 | string nR="ring @S="+string(p)+",(y(1..m),"+varstr(R)+","+parstr(R)+"),dp;"; |
| | 3762 | execute(nR); |
| | 3763 | |
| | 3764 | ideal G=fetch(@R,F); //G[i](t(1)^(p^-k),...,t(m)^(p^-k),x(i))=sep(F[i]) |
| | 3765 | |
| | 3766 | ideal I=imap(R,I); |
| | 3767 | ideal J=I+G; |
| | 3768 | poly el=1; |
| | 3769 | k=p^k; |
| | 3770 | for(i=1;i<=m;i++) |
| | 3771 | { |
| | 3772 | J=J,var(i)^k-var(m+n+i); |
| | 3773 | el=el*y(i); |
| | 3774 | } |
| | 3775 | |
| | 3776 | J=eliminate(J,el); |
| | 3777 | setring R; |
| | 3778 | ideal J=imap(@S,J); |
| | 3779 | return(J); |
| | 3780 | } |
| | 3781 | example |
| | 3782 | { "EXAMPLE:"; echo = 2; |
| | 3783 | ring R=(5,t),(x,y),dp; |
| | 3784 | ideal I=x^5-t,y^5-t; |
| | 3785 | zeroRad(I); |
| | 3786 | } |
| | 3787 | |
| | 3788 | /////////////////////////////////////////////////////////////////////////////// |
| | 3789 | |
| | 3790 | proc radicalEHV(ideal i) |
| | 3791 | "USAGE: radicalEHV(i); i ideal. |
| | 3792 | RETURN: ideal, the radical of i. |
| | 3793 | NOTE: Uses the algorithm of Eisenbud/Huneke/Vasconcelos, which |
| | 3794 | reduces the computation to the complete intersection case, |
| | 3795 | by taking, in the general case, a generic linear combination |
| | 3796 | of the input. |
| | 3797 | Works only in characteristic 0 or p large. |
| | 3798 | EXAMPLE: example radicalEHV; shows an example |
| | 3799 | " |
| | 3800 | { |
| | 3801 | if(attrib(basering,"global")!=1) |
| | 3802 | { |
| | 3803 | ERROR( |
| | 3804 | "// Not implemented for this ordering, please change to global ordering." |
| | 3805 | ); |
| | 3806 | } |
| | 3807 | if((char(basering)<100)&&(char(basering)!=0)) |
| | 3808 | { |
| | 3809 | "WARNING: The characteristic is too small, the result may be wrong"; |
| | 3810 | } |
| | 3811 | ideal J,I,I0,radI0,L,radI1,I2,radI2; |
| | 3812 | int l,n; |
| | 3813 | intvec op=option(get); |
| | 3814 | matrix M; |
| | 3815 | |
| | 3816 | option(redSB); |
| | 3817 | list m=mstd(i); |
| | 3818 | I=m[2]; |
| | 3819 | option(set,op); |
| | 3820 | |
| | 3821 | int cod=nvars(basering)-dim(m[1]); |
| | 3822 | //-------------------complete intersection case:---------------------- |
| | 3823 | if(cod==size(m[2])) |
| | 3824 | { |
| | 3825 | J=minor(jacob(I),cod); |
| | 3826 | return(quotient(I,J)); |
| | 3827 | } |
| | 3828 | //-----first codim elements of I are a complete intersection:--------- |
| | 3829 | for(l=1;l<=cod;l++) |
| | 3830 | { |
| | 3831 | I0[l]=I[l]; |
| | 3832 | } |
| | 3833 | n=dim(std(I0))+cod-nvars(basering); |
| | 3834 | //-----last codim elements of I are a complete intersection:---------- |
| | 3835 | if(n!=0) |
| | 3836 | { |
| | 3837 | for(l=1;l<=cod;l++) |
| | 3838 | { |
| | 3839 | I0[l]=I[size(I)-l+1]; |
| | 3840 | } |
| | 3841 | n=dim(std(I0))+cod-nvars(basering); |
| | 3842 | } |
| | 3843 | //-----taking a generic linear combination of the input:-------------- |
| | 3844 | if(n!=0) |
| | 3845 | { |
| | 3846 | M=transpose(sparsetriag(size(m[2]),cod,95,1)); |
| | 3847 | I0=ideal(M*transpose(I)); |
| | 3848 | n=dim(std(I0))+cod-nvars(basering); |
| | 3849 | } |
| | 3850 | //-----taking a more generic linear combination of the input:--------- |
| | 3851 | if(n!=0) |
| | 3852 | { |
| | 3853 | M=transpose(sparsetriag(size(m[2]),cod,0,100)); |
| | 3854 | I0=ideal(M*transpose(I)); |
| | 3855 | n=dim(std(I0))+cod-nvars(basering); |
| | 3856 | } |
| | 3857 | if(n==0) |
| | 3858 | { |
| | 3859 | J=minor(jacob(I0),cod); |
| | 3860 | radI0=quotient(I0,J); |
| | 3861 | L=quotient(radI0,I); |
| | 3862 | radI1=quotient(radI0,L); |
| | 3863 | |
| | 3864 | if(size(reduce(radI1,m[1],1))==0) |
| | 3865 | { |
| | 3866 | return(I); |
| | 3867 | } |
| | 3868 | |
| | 3869 | I2=sat(I,radI1)[1]; |
| | 3870 | |
| | 3871 | if(deg(I2[1])<=0) |
| | 3872 | { |
| | 3873 | return(radI1); |
| | 3874 | } |
| | 3875 | return(intersect(radI1,radicalEHV(I2))); |
| | 3876 | } |
| | 3877 | //---------------------general case------------------------------------- |
| | 3878 | return(radical(I)); |
| | 3879 | } |
| | 3880 | example |
| | 3881 | { "EXAMPLE:"; echo = 2; |
| | 3882 | ring r = 0,(x,y,z),dp; |
| | 3883 | poly p = z2+1; |
| | 3884 | poly q = z3+2; |
| | 3885 | ideal i = p*q^2,y-z2; |
| | 3886 | ideal pr= radicalEHV(i); |
| | 3887 | pr; |
| | 3888 | } |
| | 3889 | |
| | 3890 | /////////////////////////////////////////////////////////////////////////////// |
| | 3891 | |
| | 3892 | proc Ann(module M) |
| | 3893 | "USAGE: Ann(M); M module |
| | 3894 | RETURN: ideal, the annihilator of coker(M) |
| | 3895 | NOTE: The output is the ideal of all elements a of the basering R such that |
| | 3896 | a * R^m is contained in M (m=number of rows of M). |
| | 3897 | EXAMPLE: example Ann; shows an example |
| | 3898 | " |
| | 3899 | { |
| | 3900 | M=prune(M); //to obtain a small embedding |
| | 3901 | ideal ann=quotient1(M,freemodule(nrows(M))); |
| | 3902 | return(ann); |
| | 3903 | } |
| | 3904 | example |
| | 3905 | { "EXAMPLE:"; echo = 2; |
| | 3906 | ring r = 0,(x,y,z),lp; |
| | 3907 | module M = x2-y2,z3; |
| | 3908 | Ann(M); |
| | 3909 | M = [1,x2],[y,x]; |
| | 3910 | Ann(M); |
| | 3911 | qring Q=std(xy-1); |
| | 3912 | module M=imap(r,M); |
| | 3913 | Ann(M); |
| | 3914 | } |
| | 3915 | |
| | 3916 | /////////////////////////////////////////////////////////////////////////////// |
| | 3917 | |
| | 3918 | //computes the equidimensional part of the ideal i of codimension e |
| | 3919 | static proc int_ass_primary_e(ideal i, int e) |
| | 3920 | { |
| | 3921 | if(homog(i)!=1) |
| | 3922 | { |
| | 3923 | i=std(i); |
| | 3924 | } |
| | 3925 | list re=sres(i,0); //the resolution |
| | 3926 | re=minres(re); //minimized resolution |
| | 3927 | ideal ann=AnnExt_R(e,re); |
| | 3928 | if(nvars(basering)-dim(std(ann))!=e) |
| | 3929 | { |
| | 3930 | return(ideal(1)); |
| | 3931 | } |
| | 3932 | return(ann); |
| | 3933 | } |
| | 3934 | |
| | 3935 | /////////////////////////////////////////////////////////////////////////////// |
| | 3936 | |
| | 3937 | //computes the annihilator of Ext^n(R/i,R) with given resolution re |
| | 3938 | //n is not necessarily the number of variables |
| | 3939 | static proc AnnExt_R(int n,list re) |
| | 3940 | { |
| | 3941 | if(n<nvars(basering)) |
| | 3942 | { |
| | 3943 | matrix f=transpose(re[n+1]); //Hom(_,R) |
| | 3944 | module k=nres(f,2)[2]; //the kernel |
| | 3945 | matrix g=transpose(re[n]); //the image of Hom(_,R) |
| | 3946 | |
| | 3947 | ideal ann=quotient1(g,k); //the anihilator |
| | 3948 | } |
| | 3949 | else |
| | 3950 | { |
| | 3951 | ideal ann=Ann(transpose(re[n])); |
| | 3952 | } |
| | 3953 | return(ann); |
| | 3954 | } |
| | 3955 | /////////////////////////////////////////////////////////////////////////////// |
| | 3956 | |
| | 3957 | static proc analyze(list pr) |
| | 3958 | { |
| | 3959 | int ii,jj; |
| | 3960 | for(ii=1;ii<=size(pr)/2;ii++) |
| | 3961 | { |
| | 3962 | dim(std(pr[2*ii])); |
| | 3963 | idealsEqual(pr[2*ii-1],pr[2*ii]); |
| | 3964 | "==========================="; |
| | 3965 | } |
| | 3966 | |
| | 3967 | for(ii=size(pr)/2;ii>1;ii--) |
| | 3968 | { |
| | 3969 | for(jj=1;jj<ii;jj++) |
| | 3970 | { |
| | 3971 | if(size(reduce(pr[2*jj],std(pr[2*ii],1)))==0) |
| | 3972 | { |
| | 3973 | "eingebette Komponente"; |
| | 3974 | jj; |
| | 3975 | ii; |
| | 3976 | } |
| | 3977 | } |
| | 3978 | } |
| | 3979 | } |
| | 3980 | |
| | 3981 | /////////////////////////////////////////////////////////////////////////////// |
| | 3982 | // |
| | 3983 | // Shimoyama-Yokoyama |
| | 3984 | // |
| | 3985 | /////////////////////////////////////////////////////////////////////////////// |
| | 3986 | |
| | 3987 | static proc simplifyIdeal(ideal i) |
| | 3988 | { |
| | 3989 | def r=basering; |
| | 3990 | |
| | 3991 | int j,k; |
| | 3992 | map phi; |
| | 3993 | poly p; |
| | 3994 | |
| | 3995 | ideal iwork=i; |
| | 3996 | ideal imap1=maxideal(1); |
| | 3997 | ideal imap2=maxideal(1); |
| | 3998 | |
| | 3999 | |
| | 4000 | for(j=1;j<=nvars(basering);j++) |
| | 4001 | { |
| | 4002 | for(k=1;k<=size(i);k++) |
| | 4003 | { |
| | 4004 | if(deg(iwork[k]/var(j))==0) |
| | 4005 | { |
| | 4006 | p=-1/leadcoef(iwork[k]/var(j))*iwork[k]; |
| | 4007 | imap1[j]=p+2*var(j); |
| | 4008 | phi=r,imap1; |
| | 4009 | iwork=phi(iwork); |
| | 4010 | iwork=subst(iwork,var(j),0); |
| | 4011 | iwork[k]=var(j); |
| | 4012 | imap1=maxideal(1); |
| | 4013 | imap2[j]=-p; |
| | 4014 | break; |
| | 4015 | } |
| | 4016 | } |
| | 4017 | } |
| | 4018 | return(iwork,imap2); |
| | 4019 | } |
| | 4020 | |
| | 4021 | |
| | 4022 | /////////////////////////////////////////////////////// |
| | 4023 | // ini_mod |
| | 4024 | // input: a polynomial p |
| | 4025 | // output: the initial term of p as needed |
| | 4026 | // in the context of characteristic sets |
| | 4027 | ////////////////////////////////////////////////////// |
| | 4028 | |
| | 4029 | static proc ini_mod(poly p) |
| | 4030 | { |
| | 4031 | if (p==0) |
| | 4032 | { |
| | 4033 | return(0); |
| | 4034 | } |
| | 4035 | int n; matrix m; |
| | 4036 | for( n=nvars(basering); n>0; n--) |
| | 4037 | { |
| | 4038 | m=coef(p,var(n)); |
| | 4039 | if(m[1,1]!=1) |
| | 4040 | { |
| | 4041 | p=m[2,1]; |
| | 4042 | break; |
| | 4043 | } |
| | 4044 | } |
| | 4045 | if(deg(p)==0) |
| | 4046 | { |
| | 4047 | p=0; |
| | 4048 | } |
| | 4049 | return(p); |
| | 4050 | } |
| | 4051 | /////////////////////////////////////////////////////// |
| | 4052 | // min_ass_prim_charsets |
| | 4053 | // input: generators of an ideal PS and an integer cho |
| | 4054 | // If cho=0, the given ordering of the variables is used. |
| | 4055 | // Otherwise, the system tries to find an "optimal ordering", |
| | 4056 | // which in some cases may considerably speed up the algorithm |
| | 4057 | // output: the minimal associated primes of PS |
| | 4058 | // algorithm: via characteriostic sets |
| | 4059 | ////////////////////////////////////////////////////// |
| | 4060 | |
| | 4061 | |
| | 4062 | static proc min_ass_prim_charsets (ideal PS, int cho) |
| | 4063 | { |
| | 4064 | if((cho<0) and (cho>1)) |
| | 4065 | { |
| | 4066 | ERROR("<int> must be 0 or 1"); |
| | 4067 | } |
| | 4068 | option(notWarnSB); |
| | 4069 | if(cho==0) |
| | 4070 | { |
| | 4071 | return(min_ass_prim_charsets0(PS)); |
| | 4072 | } |
| | 4073 | else |
| | 4074 | { |
| | 4075 | return(min_ass_prim_charsets1(PS)); |
| | 4076 | } |
| | 4077 | } |
| | 4078 | /////////////////////////////////////////////////////// |
| | 4079 | // min_ass_prim_charsets0 |
| | 4080 | // input: generators of an ideal PS |
| | 4081 | // output: the minimal associated primes of PS |
| | 4082 | // algorithm: via characteristic sets |
| | 4083 | // the given ordering of the variables is used |
| | 4084 | ////////////////////////////////////////////////////// |
| | 4085 | |
| | 4086 | |
| | 4087 | static proc min_ass_prim_charsets0 (ideal PS) |
| | 4088 | { |
| | 4089 | intvec op; |
| | 4090 | matrix m=char_series(PS); // We compute an irreducible |
| | 4091 | // characteristic series |
| | 4092 | int i,j,k; |
| | 4093 | list PSI; |
| | 4094 | list PHI; // the ideals given by the characteristic series |
| | 4095 | for(i=nrows(m);i>=1; i--) |
| | 4096 | { |
| | 4097 | PHI[i]=ideal(m[i,1..ncols(m)]); |
| | 4098 | } |
| | 4099 | // We compute the radical of each ideal in PHI |
| | 4100 | ideal I,JS,II; |
| | 4101 | int sizeJS, sizeII; |
| | 4102 | for(i=size(PHI);i>=1; i--) |
| | 4103 | { |
| | 4104 | I=0; |
| | 4105 | for(j=size(PHI[i]);j>0;j--) |
| | 4106 | { |
| | 4107 | I=I+ini_mod(PHI[i][j]); |
| | 4108 | } |
| | 4109 | JS=std(PHI[i]); |
| | 4110 | sizeJS=size(JS); |
| | 4111 | for(j=size(I);j>0;j--) |
| | 4112 | { |
| | 4113 | II=0; |
| | 4114 | sizeII=0; |
| | 4115 | k=0; |
| | 4116 | while(k<=sizeII) // successive saturation |
| | 4117 | { |
| | 4118 | op=option(get); |
| | 4119 | option(returnSB); |
| | 4120 | II=quotient(JS,I[j]); |
| | 4121 | option(set,op); |
| | 4122 | sizeII=size(II); |
| | 4123 | if(sizeII==sizeJS) |
| | 4124 | { |
| | 4125 | for(k=1;k<=sizeII;k++) |
| | 4126 | { |
| | 4127 | if(leadexp(II[k])!=leadexp(JS[k])) break; |
| | 4128 | } |
| | 4129 | } |
| | 4130 | JS=II; |
| | 4131 | sizeJS=sizeII; |
| | 4132 | } |
| | 4133 | } |
| | 4134 | PSI=insert(PSI,JS); |
| | 4135 | } |
| | 4136 | int sizePSI=size(PSI); |
| | 4137 | // We eliminate redundant ideals |
| | 4138 | for(i=1;i<sizePSI;i++) |
| | 4139 | { |
| | 4140 | for(j=i+1;j<=sizePSI;j++) |
| | 4141 | { |
| | 4142 | if(size(PSI[i])!=0) |
| | 4143 | { |
| | 4144 | if(size(PSI[j])!=0) |
| | 4145 | { |
| | 4146 | if(size(NF(PSI[i],PSI[j],1))==0) |
| | 4147 | { |
| | 4148 | PSI[j]=ideal(0); |
| | 4149 | } |
| | 4150 | else |
| | 4151 | { |
| | 4152 | if(size(NF(PSI[j],PSI[i],1))==0) |
| | 4153 | { |
| | 4154 | PSI[i]=ideal(0); |
| | 4155 | } |
| | 4156 | } |
| | 4157 | } |
| | 4158 | } |
| | 4159 | } |
| | 4160 | } |
| | 4161 | for(i=sizePSI;i>=1;i--) |
| | 4162 | { |
| | 4163 | if(size(PSI[i])==0) |
| | 4164 | { |
| | 4165 | PSI=delete(PSI,i); |
| | 4166 | } |
| | 4167 | } |
| | 4168 | return (PSI); |
| | 4169 | } |
| | 4170 | |
| | 4171 | /////////////////////////////////////////////////////// |
| | 4172 | // min_ass_prim_charsets1 |
| | 4173 | // input: generators of an ideal PS |
| | 4174 | // output: the minimal associated primes of PS |
| | 4175 | // algorithm: via characteristic sets |
| | 4176 | // input: generators of an ideal PS and an integer i |
| | 4177 | // The system tries to find an "optimal ordering" of |
| | 4178 | // the variables |
| | 4179 | ////////////////////////////////////////////////////// |
| | 4180 | |
| | 4181 | |
| | 4182 | static proc min_ass_prim_charsets1 (ideal PS) |
| | 4183 | { |
| | 4184 | intvec op; |
| | 4185 | def oldring=basering; |
| | 4186 | string n=system("neworder",PS); |
| | 4187 | execute("ring r=("+charstr(oldring)+"),("+n+"),dp;"); |
| | 4188 | ideal PS=imap(oldring,PS); |
| | 4189 | matrix m=char_series(PS); // We compute an irreducible |
| | 4190 | // characteristic series |
| | 4191 | int i,j,k; |
| | 4192 | ideal I; |
| | 4193 | list PSI; |
| | 4194 | list PHI; // the ideals given by the characteristic series |
| | 4195 | list ITPHI; // their initial terms |
| | 4196 | for(i=nrows(m);i>=1; i--) |
| | 4197 | { |
| | 4198 | PHI[i]=ideal(m[i,1..ncols(m)]); |
| | 4199 | I=0; |
| | 4200 | for(j=size(PHI[i]);j>0;j=j-1) |
| | 4201 | { |
| | 4202 | I=I,ini_mod(PHI[i][j]); |
| | 4203 | } |
| | 4204 | I=I[2..ncols(I)]; |
| | 4205 | ITPHI[i]=I; |
| | 4206 | } |
| | 4207 | setring oldring; |
| | 4208 | matrix m=imap(r,m); |
| | 4209 | list PHI=imap(r,PHI); |
| | 4210 | list ITPHI=imap(r,ITPHI); |
| | 4211 | // We compute the radical of each ideal in PHI |
| | 4212 | ideal I,JS,II; |
| | 4213 | int sizeJS, sizeII; |
| | 4214 | for(i=size(PHI);i>=1; i--) |
| | 4215 | { |
| | 4216 | I=0; |
| | 4217 | for(j=size(PHI[i]);j>0;j--) |
| | 4218 | { |
| | 4219 | I=I+ITPHI[i][j]; |
| | 4220 | } |
| | 4221 | JS=std(PHI[i]); |
| | 4222 | sizeJS=size(JS); |
| | 4223 | for(j=size(I);j>0;j--) |
| | 4224 | { |
| | 4225 | II=0; |
| | 4226 | sizeII=0; |
| | 4227 | k=0; |
| | 4228 | while(k<=sizeII) // successive iteration |
| | 4229 | { |
| | 4230 | op=option(get); |
| | 4231 | option(returnSB); |
| | 4232 | II=quotient(JS,I[j]); |
| | 4233 | option(set,op); |
| | 4234 | //std |
| | 4235 | // II=std(II); |
| | 4236 | sizeII=size(II); |
| | 4237 | if(sizeII==sizeJS) |
| | 4238 | { |
| | 4239 | for(k=1;k<=sizeII;k++) |
| | 4240 | { |
| | 4241 | if(leadexp(II[k])!=leadexp(JS[k])) break; |
| | 4242 | } |
| | 4243 | } |
| | 4244 | JS=II; |
| | 4245 | sizeJS=sizeII; |
| | 4246 | } |
| | 4247 | } |
| | 4248 | PSI=insert(PSI,JS); |
| | 4249 | } |
| | 4250 | int sizePSI=size(PSI); |
| | 4251 | // We eliminate redundant ideals |
| | 4252 | for(i=1;i<sizePSI;i++) |
| | 4253 | { |
| | 4254 | for(j=i+1;j<=sizePSI;j++) |
| | 4255 | { |
| | 4256 | if(size(PSI[i])!=0) |
| | 4257 | { |
| | 4258 | if(size(PSI[j])!=0) |
| | 4259 | { |
| | 4260 | if(size(NF(PSI[i],PSI[j],1))==0) |
| | 4261 | { |
| | 4262 | PSI[j]=ideal(0); |
| | 4263 | } |
| | 4264 | else |
| | 4265 | { |
| | 4266 | if(size(NF(PSI[j],PSI[i],1))==0) |
| | 4267 | { |
| | 4268 | PSI[i]=ideal(0); |
| | 4269 | } |
| | 4270 | } |
| | 4271 | } |
| | 4272 | } |
| | 4273 | } |
| | 4274 | } |
| | 4275 | for(i=sizePSI;i>=1;i--) |
| | 4276 | { |
| | 4277 | if(size(PSI[i])==0) |
| | 4278 | { |
| | 4279 | PSI=delete(PSI,i); |
| | 4280 | } |
| | 4281 | } |
| | 4282 | return (PSI); |
| | 4283 | } |
| | 4284 | |
| | 4285 | |
| | 4286 | ///////////////////////////////////////////////////// |
| | 4287 | // proc prim_dec |
| | 4288 | // input: generators of an ideal I and an integer choose |
| | 4289 | // If choose=0, min_ass_prim_charsets with the given |
| | 4290 | // ordering of the variables is used. |
| | 4291 | // If choose=1, min_ass_prim_charsets with the "optimized" |
| | 4292 | // ordering of the variables is used. |
| | 4293 | // If choose=2, minAssPrimes from primdec.lib is used |
| | 4294 | // If choose=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
| | 4295 | // output: a primary decomposition of I, i.e., a list |
| | 4296 | // of pairs consisting of a standard basis of a primary component |
| | 4297 | // of I and a standard basis of the corresponding associated prime. |
| | 4298 | // To compute the minimal associated primes of a given ideal |
| | 4299 | // min_ass_prim_l is called, i.e., the minimal associated primes |
| | 4300 | // are computed via characteristic sets. |
| | 4301 | // In the homogeneous case, the performance of the procedure |
| | 4302 | // will be improved if I is already given by a minimal set of |
| | 4303 | // generators. Apply minbase if necessary. |
| | 4304 | ////////////////////////////////////////////////////////// |
| | 4305 | |
| | 4306 | |
| | 4307 | static proc prim_dec(ideal I, int choose) |
| | 4308 | { |
| | 4309 | if((choose<0) or (choose>3)) |
| | 4310 | { |
| | 4311 | ERROR("ERROR: <int> must be 0 or 1 or 2 or 3"); |
| | 4312 | } |
| | 4313 | option(notWarnSB); |
| | 4314 | ideal H=1; // The intersection of the primary components |
| | 4315 | list U; // the leaves of the decomposition tree, i.e., |
| | 4316 | // pairs consisting of a primary component of I |
| | 4317 | // and the corresponding associated prime |
| | 4318 | list W; // the non-leaf vertices in the decomposition tree. |
| | 4319 | // every entry has 6 components: |
| | 4320 | // 1- the vertex itself , i.e., a standard bais of the |
| | 4321 | // given ideal I (type 1), or a standard basis of a |
| | 4322 | // pseudo-primary component arising from |
| | 4323 | // pseudo-primary decomposition (type 2), or a |
| | 4324 | // standard basis of a remaining component arising from |
| | 4325 | // pseudo-primary decomposition or extraction (type 3) |
| | 4326 | // 2- the type of the vertex as indicated above |
| | 4327 | // 3- the weighted_tree_depth of the vertex |
| | 4328 | // 4- the tester of the vertex |
| | 4329 | // 5- a standard basis of the associated prime |
| | 4330 | // of a vertex of type 2, or 0 otherwise |
| | 4331 | // 6- a list of pairs consisting of a standard |
| | 4332 | // basis of a minimal associated prime ideal |
| | 4333 | // of the father of the vertex and the |
| | 4334 | // irreducible factors of the "minimal |
| | 4335 | // divisor" of the seperator or extractor |
| | 4336 | // corresponding to the prime ideal |
| | 4337 | // as computed by the procedure minsat, |
| | 4338 | // if the vertex is of type 3, or |
| | 4339 | // the empty list otherwise |
| | 4340 | ideal SI=std(I); |
| | 4341 | if(SI[1]==1) // primdecSY(ideal(1)) |
| | 4342 | { |
| | 4343 | return(list()); |
| | 4344 | } |
| | 4345 | int ncolsSI=ncols(SI); |
| | 4346 | int ncolsH=1; |
| | 4347 | W[1]=list(I,1,0,poly(1),ideal(0),list()); // The root of the tree |
| | 4348 | int weighted_tree_depth; |
| | 4349 | int i,j; |
| | 4350 | int check; |
| | 4351 | list V; // current vertex |
| | 4352 | list VV; // new vertex |
| | 4353 | list QQ; |
| | 4354 | list WI; |
| | 4355 | ideal Qi,SQ,SRest,fac; |
| | 4356 | poly tester; |
| | 4357 | |
| | 4358 | while(1) |
| | 4359 | { |
| | 4360 | i=1; |
| | 4361 | while(1) |
| | 4362 | { |
| | 4363 | while(i<=size(W)) // find vertex V of smallest weighted tree-depth |
| | 4364 | { |
| | 4365 | if (W[i][3]<=weighted_tree_depth) break; |
| | 4366 | i++; |
| | 4367 | } |
| | 4368 | if (i<=size(W)) break; |
| | 4369 | i=1; |
| | 4370 | weighted_tree_depth++; |
| | 4371 | } |
| | 4372 | V=W[i]; |
| | 4373 | W=delete(W,i); // delete V from W |
| | 4374 | |
| | 4375 | // now proceed by type of vertex V |
| | 4376 | |
| | 4377 | if (V[2]==2) // extraction needed |
| | 4378 | { |
| | 4379 | SQ,SRest,fac=extraction(V[1],V[5]); |
| | 4380 | // standard basis of primary component, |
| | 4381 | // standard basis of remaining component, |
| | 4382 | // irreducible factors of |
| | 4383 | // the "minimal divisor" of the extractor |
| | 4384 | // as computed by the procedure minsat, |
| | 4385 | check=0; |
| | 4386 | for(j=1;j<=ncolsH;j++) |
| | 4387 | { |
| | 4388 | if (NF(H[j],SQ,1)!=0) // Q is not redundant |
| | 4389 | { |
| | 4390 | check=1; |
| | 4391 | break; |
| | 4392 | } |
| | 4393 | } |
| | 4394 | if(check==1) // Q is not redundant |
| | 4395 | { |
| | 4396 | QQ=list(); |
| | 4397 | QQ[1]=list(SQ,V[5]); // primary component, associated prime, |
| | 4398 | // i.e., standard bases thereof |
| | 4399 | U=U+QQ; |
| | 4400 | H=intersect(H,SQ); |
| | 4401 | H=std(H); |
| | 4402 | ncolsH=ncols(H); |
| | 4403 | check=0; |
| | 4404 | if(ncolsH==ncolsSI) |
| | 4405 | { |
| | 4406 | for(j=1;j<=ncolsSI;j++) |
| | 4407 | { |
| | 4408 | if(leadexp(H[j])!=leadexp(SI[j])) |
| | 4409 | { |
| | 4410 | check=1; |
| | 4411 | break; |
| | 4412 | } |
| | 4413 | } |
| | 4414 | } |
| | 4415 | else |
| | 4416 | { |
| | 4417 | check=1; |
| | 4418 | } |
| | 4419 | if(check==0) // H==I => U is a primary decomposition |
| | 4420 | { |
| | 4421 | return(U); |
| | 4422 | } |
| | 4423 | } |
| | 4424 | if (SRest[1]!=1) // the remaining component is not |
| | 4425 | // the whole ring |
| | 4426 | { |
| | 4427 | if (rad_con(V[4],SRest)==0) // the new vertex is not the |
| | 4428 | // root of a redundant subtree |
| | 4429 | { |
| | 4430 | VV[1]=SRest; // remaining component |
| | 4431 | VV[2]=3; // pseudoprimdec_special |
| | 4432 | VV[3]=V[3]+1; // weighted depth |
| | 4433 | VV[4]=V[4]; // the tester did not change |
| | 4434 | VV[5]=ideal(0); |
| | 4435 | VV[6]=list(list(V[5],fac)); |
| | 4436 | W=insert(W,VV,size(W)); |
| | 4437 | } |
| | 4438 | } |
| | 4439 | } |
| | 4440 | else |
| | 4441 | { |
| | 4442 | if (V[2]==3) // pseudo_prim_dec_special is needed |
| | 4443 | { |
| | 4444 | QQ,SRest=pseudo_prim_dec_special_charsets(V[1],V[6],choose); |
| | 4445 | // QQ = quadruples: |
| | 4446 | // standard basis of pseudo-primary component, |
| | 4447 | // standard basis of corresponding prime, |
| | 4448 | // seperator, irreducible factors of |
| | 4449 | // the "minimal divisor" of the seperator |
| | 4450 | // as computed by the procedure minsat, |
| | 4451 | // SRest=standard basis of remaining component |
| | 4452 | } |
| | 4453 | else // V is the root, pseudo_prim_dec is needed |
| | 4454 | { |
| | 4455 | QQ,SRest=pseudo_prim_dec_charsets(I,SI,choose); |
| | 4456 | // QQ = quadruples: |
| | 4457 | // standard basis of pseudo-primary component, |
| | 4458 | // standard basis of corresponding prime, |
| | 4459 | // seperator, irreducible factors of |
| | 4460 | // the "minimal divisor" of the seperator |
| | 4461 | // as computed by the procedure minsat, |
| | 4462 | // SRest=standard basis of remaining component |
| | 4463 | |
| | 4464 | } |
| | 4465 | //check |
| | 4466 | for(i=size(QQ);i>=1;i--) |
| | 4467 | //for(i=1;i<=size(QQ);i++) |
| | 4468 | { |
| | 4469 | tester=QQ[i][3]*V[4]; |
| | 4470 | Qi=QQ[i][2]; |
| | 4471 | if(NF(tester,Qi,1)!=0) // the new vertex is not the |
| | 4472 | // root of a redundant subtree |
| | 4473 | { |
| | 4474 | VV[1]=QQ[i][1]; |
| | 4475 | VV[2]=2; |
| | 4476 | VV[3]=V[3]+1; |
| | 4477 | VV[4]=tester; // the new tester as computed above |
| | 4478 | VV[5]=Qi; // QQ[i][2]; |
| | 4479 | VV[6]=list(); |
| | 4480 | W=insert(W,VV,size(W)); |
| | 4481 | } |
| | 4482 | } |
| | 4483 | if (SRest[1]!=1) // the remaining component is not |
| | 4484 | // the whole ring |
| | 4485 | { |
| | 4486 | if (rad_con(V[4],SRest)==0) // the vertex is not the root |
| | 4487 | // of a redundant subtree |
| | 4488 | { |
| | 4489 | VV[1]=SRest; |
| | 4490 | VV[2]=3; |
| | 4491 | VV[3]=V[3]+2; |
| | 4492 | VV[4]=V[4]; // the tester did not change |
| | 4493 | VV[5]=ideal(0); |
| | 4494 | WI=list(); |
| | 4495 | for(i=1;i<=size(QQ);i++) |
| | 4496 | { |
| | 4497 | WI=insert(WI,list(QQ[i][2],QQ[i][4])); |
| | 4498 | } |
| | 4499 | VV[6]=WI; |
| | 4500 | W=insert(W,VV,size(W)); |
| | 4501 | } |
| | 4502 | } |
| | 4503 | } |
| | 4504 | } |
| | 4505 | } |
| | 4506 | |
| | 4507 | ////////////////////////////////////////////////////////////////////////// |
| | 4508 | // proc pseudo_prim_dec_charsets |
| | 4509 | // input: Generators of an arbitrary ideal I, a standard basis SI of I, |
| | 4510 | // and an integer choo |
| | 4511 | // If choo=0, min_ass_prim_charsets with the given |
| | 4512 | // ordering of the variables is used. |
| | 4513 | // If choo=1, min_ass_prim_charsets with the "optimized" |
| | 4514 | // ordering of the variables is used. |
| | 4515 | // If choo=2, minAssPrimes from primdec.lib is used |
| | 4516 | // If choo=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
| | 4517 | // output: a pseudo primary decomposition of I, i.e., a list |
| | 4518 | // of pseudo primary components together with a standard basis of the |
| | 4519 | // remaining component. Each pseudo primary component is |
| | 4520 | // represented by a quadrupel: A standard basis of the component, |
| | 4521 | // a standard basis of the corresponding associated prime, the |
| | 4522 | // seperator of the component, and the irreducible factors of the |
| | 4523 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
| | 4524 | // calls proc pseudo_prim_dec_i |
| | 4525 | ////////////////////////////////////////////////////////////////////////// |
| | 4526 | |
| | 4527 | |
| | 4528 | static proc pseudo_prim_dec_charsets (ideal I, ideal SI, int choo) |
| | 4529 | { |
| | 4530 | list L; // The list of minimal associated primes, |
| | 4531 | // each one given by a standard basis |
| | 4532 | if((choo==0) or (choo==1)) |
| | 4533 | { |
| | 4534 | L=min_ass_prim_charsets(I,choo); |
| | 4535 | } |
| | 4536 | else |
| | 4537 | { |
| | 4538 | if(choo==2) |
| | 4539 | { |
| | 4540 | L=minAssPrimes(I); |
| | 4541 | } |
| | 4542 | else |
| | 4543 | { |
| | 4544 | L=minAssPrimes(I,1); |
| | 4545 | } |
| | 4546 | for(int i=size(L);i>=1;i--) |
| | 4547 | { |
| | 4548 | L[i]=std(L[i]); |
| | 4549 | } |
| | 4550 | } |
| | 4551 | return (pseudo_prim_dec_i(SI,L)); |
| | 4552 | } |
| | 4553 | |
| | 4554 | //////////////////////////////////////////////////////////////// |
| | 4555 | // proc pseudo_prim_dec_special_charsets |
| | 4556 | // input: a standard basis of an ideal I whose radical is the |
| | 4557 | // intersection of the radicals of ideals generated by one prime ideal |
| | 4558 | // P_i together with one polynomial f_i, the list V6 must be the list of |
| | 4559 | // pairs (standard basis of P_i, irreducible factors of f_i), |
| | 4560 | // and an integer choo |
| | 4561 | // If choo=0, min_ass_prim_charsets with the given |
| | 4562 | // ordering of the variables is used. |
| | 4563 | // If choo=1, min_ass_prim_charsets with the "optimized" |
| | 4564 | // ordering of the variables is used. |
| | 4565 | // If choo=2, minAssPrimes from primdec.lib is used |
| | 4566 | // If choo=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
| | 4567 | // output: a pseudo primary decomposition of I, i.e., a list |
| | 4568 | // of pseudo primary components together with a standard basis of the |
| | 4569 | // remaining component. Each pseudo primary component is |
| | 4570 | // represented by a quadrupel: A standard basis of the component, |
| | 4571 | // a standard basis of the corresponding associated prime, the |
| | 4572 | // seperator of the component, and the irreducible factors of the |
| | 4573 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
| | 4574 | // calls proc pseudo_prim_dec_i |
| | 4575 | //////////////////////////////////////////////////////////////// |
| | 4576 | |
| | 4577 | |
| | 4578 | static proc pseudo_prim_dec_special_charsets (ideal SI,list V6, int choo) |
| | 4579 | { |
| | 4580 | int i,j,l; |
| | 4581 | list m; |
| | 4582 | list L; |
| | 4583 | int sizeL; |
| | 4584 | ideal P,SP; ideal fac; |
| | 4585 | int dimSP; |
| | 4586 | for(l=size(V6);l>=1;l--) // creates a list of associated primes |
| | 4587 | // of I, possibly redundant |
| | 4588 | { |
| | 4589 | P=V6[l][1]; |
| | 4590 | fac=V6[l][2]; |
| | 4591 | for(i=ncols(fac);i>=1;i--) |
| | 4592 | { |
| | 4593 | SP=P+fac[i]; |
| | 4594 | SP=std(SP); |
| | 4595 | if(SP[1]!=1) |
| | 4596 | { |
| | 4597 | if((choo==0) or (choo==1)) |
| | 4598 | { |
| | 4599 | m=min_ass_prim_charsets(SP,choo); // a list of SB |
| | 4600 | } |
| | 4601 | else |
| | 4602 | { |
| | 4603 | if(choo==2) |
| | 4604 | { |
| | 4605 | m=minAssPrimes(SP); |
| | 4606 | } |
| | 4607 | else |
| | 4608 | { |
| | 4609 | m=minAssPrimes(SP,1); |
| | 4610 | } |
| | 4611 | for(j=size(m);j>=1;j=j-1) |
| | 4612 | { |
| | 4613 | m[j]=std(m[j]); |
| | 4614 | } |
| | 4615 | } |
| | 4616 | dimSP=dim(SP); |
| | 4617 | for(j=size(m);j>=1; j--) |
| | 4618 | { |
| | 4619 | if(dim(m[j])==dimSP) |
| | 4620 | { |
| | 4621 | L=insert(L,m[j],size(L)); |
| | 4622 | } |
| | 4623 | } |
| | 4624 | } |
| | 4625 | } |
| | 4626 | } |
| | 4627 | sizeL=size(L); |
| | 4628 | for(i=1;i<sizeL;i++) // get rid of redundant primes |
| | 4629 | { |
| | 4630 | for(j=i+1;j<=sizeL;j++) |
| | 4631 | { |
| | 4632 | if(size(L[i])!=0) |
| | 4633 | { |
| | 4634 | if(size(L[j])!=0) |
| | 4635 | { |
| | 4636 | if(size(NF(L[i],L[j],1))==0) |
| | 4637 | { |
| | 4638 | L[j]=ideal(0); |
| | 4639 | } |
| | 4640 | else |
| | 4641 | { |
| | 4642 | if(size(NF(L[j],L[i],1))==0) |
| | 4643 | { |
| | 4644 | L[i]=ideal(0); |
| | 4645 | } |
| | 4646 | } |
| | 4647 | } |
| | 4648 | } |
| | 4649 | } |
| | 4650 | } |
| | 4651 | for(i=sizeL;i>=1;i--) |
| | 4652 | { |
| | 4653 | if(size(L[i])==0) |
| | 4654 | { |
| | 4655 | L=delete(L,i); |
| | 4656 | } |
| | 4657 | } |
| | 4658 | return (pseudo_prim_dec_i(SI,L)); |
| | 4659 | } |
| | 4660 | |
| | 4661 | |
| | 4662 | //////////////////////////////////////////////////////////////// |
| | 4663 | // proc pseudo_prim_dec_i |
| | 4664 | // input: A standard basis of an arbitrary ideal I, and standard bases |
| | 4665 | // of the minimal associated primes of I |
| | 4666 | // output: a pseudo primary decomposition of I, i.e., a list |
| | 4667 | // of pseudo primary components together with a standard basis of the |
| | 4668 | // remaining component. Each pseudo primary component is |
| | 4669 | // represented by a quadrupel: A standard basis of the component Q_i, |
| | 4670 | // a standard basis of the corresponding associated prime P_i, the |
| | 4671 | // seperator of the component, and the irreducible factors of the |
| | 4672 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
| | 4673 | //////////////////////////////////////////////////////////////// |
| | 4674 | |
| | 4675 | |
| | 4676 | static proc pseudo_prim_dec_i (ideal SI, list L) |
| | 4677 | { |
| | 4678 | list Q; |
| | 4679 | if (size(L)==1) // one minimal associated prime only |
| | 4680 | // the ideal is already pseudo primary |
| | 4681 | { |
| | 4682 | Q=SI,L[1],1; |
| | 4683 | list QQ; |
| | 4684 | QQ[1]=Q; |
| | 4685 | return (QQ,ideal(1)); |
| | 4686 | } |
| | 4687 | |
| | 4688 | poly f0,f,g; |
| | 4689 | ideal fac; |
| | 4690 | int i,j,k,l; |
| | 4691 | ideal SQi; |
| | 4692 | ideal I'=SI; |
| | 4693 | list QP; |
| | 4694 | int sizeL=size(L); |
| | 4695 | for(i=1;i<=sizeL;i++) |
| | 4696 | { |
| | 4697 | fac=0; |
| | 4698 | for(j=1;j<=sizeL;j++) // compute the seperator sep_i |
| | 4699 | // of the i-th component |
| | 4700 | { |
| | 4701 | if (i!=j) // search g not in L[i], but L[j] |
| | 4702 | { |
| | 4703 | for(k=1;k<=ncols(L[j]);k++) |
| | 4704 | { |
| | 4705 | if(NF(L[j][k],L[i],1)!=0) |
| | 4706 | { |
| | 4707 | break; |
| | 4708 | } |
| | 4709 | } |
| | 4710 | fac=fac+L[j][k]; |
| | 4711 | } |
| | 4712 | } |
| | 4713 | // delete superfluous polynomials |
| | 4714 | fac=simplify(fac,8+2); |
| | 4715 | // saturation |
| | 4716 | SQi,f0,f,fac=minsat_ppd(SI,fac); |
| | 4717 | I'=I',f; |
| | 4718 | QP=SQi,L[i],f0,fac; |
| | 4719 | // the quadrupel: |
| | 4720 | // a standard basis of Q_i, |
| | 4721 | // a standard basis of P_i, |
| | 4722 | // sep_i, |
| | 4723 | // irreducible factors of |
| | 4724 | // the "minimal divisor" of the seperator |
| | 4725 | // as computed by the procedure minsat, |
| | 4726 | Q[i]=QP; |
| | 4727 | } |
| | 4728 | I'=std(I'); |
| | 4729 | return (Q, I'); |
| | 4730 | // I' = remaining component |
| | 4731 | } |
| | 4732 | |
| | 4733 | |
| | 4734 | //////////////////////////////////////////////////////////////// |
| | 4735 | // proc extraction |
| | 4736 | // input: A standard basis of a pseudo primary ideal I, and a standard |
| | 4737 | // basis of the unique minimal associated prime P of I |
| | 4738 | // output: an extraction of I, i.e., a standard basis of the primary |
| | 4739 | // component Q of I with associated prime P, a standard basis of the |
| | 4740 | // remaining component, and the irreducible factors of the |
| | 4741 | // "minimal divisor" of the extractor as computed by the procedure minsat |
| | 4742 | //////////////////////////////////////////////////////////////// |
| | 4743 | |
| | 4744 | |
| | 4745 | static proc extraction (ideal SI, ideal SP) |
| | 4746 | { |
| | 4747 | list indsets=indepSet(SP,0); |
| | 4748 | poly f; |
| | 4749 | if(size(indsets)!=0) //check, whether dim P != 0 |
| | 4750 | { |
| | 4751 | intvec v; // a maximal independent set of variables |
| | 4752 | // modulo P |
| | 4753 | string U; // the independent variables |
| | 4754 | string A; // the dependent variables |
| | 4755 | int j,k; |
| | 4756 | int a; // the size of A |
| | 4757 | int degf; |
| | 4758 | ideal g; |
| | 4759 | list polys; |
| | 4760 | int sizepolys; |
| | 4761 | list newpoly; |
| | 4762 | def R=basering; |
| | 4763 | //intvec hv=hilb(SI,1); |
| | 4764 | for (k=1;k<=size(indsets);k++) |
| | 4765 | { |
| | 4766 | v=indsets[k]; |
| | 4767 | for (j=1;j<=nvars(R);j++) |
| | 4768 | { |
| | 4769 | if (v[j]==1) |
| | 4770 | { |
| | 4771 | U=U+varstr(j)+","; |
| | 4772 | } |
| | 4773 | else |
| | 4774 | { |
| | 4775 | A=A+varstr(j)+","; |
| | 4776 | a++; |
| | 4777 | } |
| | 4778 | } |
| | 4779 | |
| | 4780 | U[size(U)]=")"; // we compute the extractor of I (w.r.t. U) |
| | 4781 | execute("ring RAU=("+charstr(basering)+"),("+A+U+",(dp("+string(a)+"),dp);"); |
| | 4782 | ideal I=imap(R,SI); |
| | 4783 | //I=std(I,hv); // the standard basis in (R[U])[A] |
| | 4784 | I=std(I); // the standard basis in (R[U])[A] |
| | 4785 | A[size(A)]=")"; |
| | 4786 | execute("ring Rloc=("+charstr(basering)+","+U+",("+A+",dp;"); |
| | 4787 | ideal I=imap(RAU,I); |
| | 4788 | //"std in lokalisierung:"+newline,I; |
| | 4789 | ideal h; |
| | 4790 | for(j=ncols(I);j>=1;j--) |
| | 4791 | { |
| | 4792 | h[j]=leadcoef(I[j]); // consider I in (R(U))[A] |
| | 4793 | } |
| | 4794 | setring R; |
| | 4795 | g=imap(Rloc,h); |
| | 4796 | kill RAU,Rloc; |
| | 4797 | U=""; |
| | 4798 | A=""; |
| | 4799 | a=0; |
| | 4800 | f=lcm(g); |
| | 4801 | newpoly[1]=f; |
| | 4802 | polys=polys+newpoly; |
| | 4803 | newpoly=list(); |
| | 4804 | } |
| | 4805 | f=polys[1]; |
| | 4806 | degf=deg(f); |
| | 4807 | sizepolys=size(polys); |
| | 4808 | for (k=2;k<=sizepolys;k++) |
| | 4809 | { |
| | 4810 | if (deg(polys[k])<degf) |
| | 4811 | { |
| | 4812 | f=polys[k]; |
| | 4813 | degf=deg(f); |
| | 4814 | } |
| | 4815 | } |
| | 4816 | } |
| | 4817 | else |
| | 4818 | { |
| | 4819 | f=1; |
| | 4820 | } |
| | 4821 | poly f0,h0; ideal SQ; ideal fac; |
| | 4822 | if(f!=1) |
| | 4823 | { |
| | 4824 | SQ,f0,h0,fac=minsat(SI,f); |
| | 4825 | return(SQ,std(SI+h0),fac); |
| | 4826 | // the tripel |
| | 4827 | // a standard basis of Q, |
| | 4828 | // a standard basis of remaining component, |
| | 4829 | // irreducible factors of |
| | 4830 | // the "minimal divisor" of the extractor |
| | 4831 | // as computed by the procedure minsat |
| | 4832 | } |
| | 4833 | else |
| | 4834 | { |
| | 4835 | return(SI,ideal(1),ideal(1)); |
| | 4836 | } |
| | 4837 | } |
| | 4838 | |
| | 4839 | ///////////////////////////////////////////////////// |
| | 4840 | // proc minsat |
| | 4841 | // input: a standard basis of an ideal I and a polynomial p |
| | 4842 | // output: a standard basis IS of the saturation of I w.r. to p, |
| | 4843 | // the maximal squarefree factor f0 of p, |
| | 4844 | // the "minimal divisor" f of f0 such that the saturation of |
| | 4845 | // I w.r. to f equals the saturation of I w.r. to f0 (which is IS), |
| | 4846 | // the irreducible factors of f |
| | 4847 | ////////////////////////////////////////////////////////// |
| | 4848 | |
| | 4849 | |
| | 4850 | static proc minsat(ideal SI, poly p) |
| | 4851 | { |
| | 4852 | ideal fac=factorize(p,1); //the irreducible factors of p |
| | 4853 | fac=sort(fac)[1]; |
| | 4854 | int i,k; |
| | 4855 | poly f0=1; |
| | 4856 | for(i=ncols(fac);i>=1;i--) |
| | 4857 | { |
| | 4858 | f0=f0*fac[i]; |
| | 4859 | } |
| | 4860 | poly f=1; |
| | 4861 | ideal iold; |
| | 4862 | list quotM; |
| | 4863 | quotM[1]=SI; |
| | 4864 | quotM[2]=fac; |
| | 4865 | quotM[3]=f0; |
| | 4866 | // we deal seperately with the first quotient; |
| | 4867 | // factors, which do not contribute to this one, |
| | 4868 | // are omitted |
| | 4869 | iold=quotM[1]; |
| | 4870 | quotM=minquot(quotM); |
| | 4871 | fac=quotM[2]; |
| | 4872 | if(quotM[3]==1) |
| | 4873 | { |
| | 4874 | return(quotM[1],f0,f,fac); |
| | 4875 | } |
| | 4876 | while(special_ideals_equal(iold,quotM[1])==0) |
| | 4877 | { |
| | 4878 | f=f*quotM[3]; |
| | 4879 | iold=quotM[1]; |
| | 4880 | quotM=minquot(quotM); |
| | 4881 | } |
| | 4882 | return(quotM[1],f0,f,fac); // the quadrupel ((I:p),f0,f, irr. factors of f) |
| | 4883 | } |
| | 4884 | |
| | 4885 | ///////////////////////////////////////////////////// |
| | 4886 | // proc minsat_ppd |
| | 4887 | // input: a standard basis of an ideal I and a polynomial p |
| | 4888 | // output: a standard basis IS of the saturation of I w.r. to p, |
| | 4889 | // the maximal squarefree factor f0 of p, |
| | 4890 | // the "minimal divisor" f of f0 such that the saturation of |
| | 4891 | // I w.r. to f equals the saturation of I w.r. to f0 (which is IS), |
| | 4892 | // the irreducible factors of f |
| | 4893 | ////////////////////////////////////////////////////////// |
| | 4894 | |
| | 4895 | |
| | 4896 | static proc minsat_ppd(ideal SI, ideal fac) |
| | 4897 | { |
| | 4898 | fac=sort(fac)[1]; |
| | 4899 | int i,k; |
| | 4900 | poly f0=1; |
| | 4901 | for(i=ncols(fac);i>=1;i--) |
| | 4902 | { |
| | 4903 | f0=f0*fac[i]; |
| | 4904 | } |
| | 4905 | poly f=1; |
| | 4906 | ideal iold; |
| | 4907 | list quotM; |
| | 4908 | quotM[1]=SI; |
| | 4909 | quotM[2]=fac; |
| | 4910 | quotM[3]=f0; |
| | 4911 | // we deal seperately with the first quotient; |
| | 4912 | // factors, which do not contribute to this one, |
| | 4913 | // are omitted |
| | 4914 | iold=quotM[1]; |
| | 4915 | quotM=minquot(quotM); |
| | 4916 | fac=quotM[2]; |
| | 4917 | if(quotM[3]==1) |
| | 4918 | { |
| | 4919 | return(quotM[1],f0,f,fac); |
| | 4920 | } |
| | 4921 | while(special_ideals_equal(iold,quotM[1])==0) |
| | 4922 | { |
| | 4923 | f=f*quotM[3]; |
| | 4924 | iold=quotM[1]; |
| | 4925 | quotM=minquot(quotM); |
| | 4926 | k++; |
| | 4927 | } |
| | 4928 | return(quotM[1],f0,f,fac); // the quadrupel ((I:p),f0,f, irr. factors of f) |
| | 4929 | } |
| | 4930 | ///////////////////////////////////////////////////////////////// |
| | 4931 | // proc minquot |
| | 4932 | // input: a list with 3 components: a standard basis |
| | 4933 | // of an ideal I, a set of irreducible polynomials, and |
| | 4934 | // there product f0 |
| | 4935 | // output: a standard basis of the ideal (I:f0), the irreducible |
| | 4936 | // factors of the "minimal divisor" f of f0 with (I:f0) = (I:f), |
| | 4937 | // the "minimal divisor" f |
| | 4938 | ///////////////////////////////////////////////////////////////// |
| | 4939 | |
| | 4940 | static proc minquot(list tsil) |
| | 4941 | { |
| | 4942 | intvec op; |
| | 4943 | int i,j,k,action; |
| | 4944 | ideal verg; |
| | 4945 | list l; |
| | 4946 | poly g; |
| | 4947 | ideal laedi=tsil[1]; |
| | 4948 | ideal fac=tsil[2]; |
| | 4949 | poly f=tsil[3]; |
| | 4950 | |
| | 4951 | //std |
| | 4952 | // ideal star=quotient(laedi,f); |
| | 4953 | // star=std(star); |
| | 4954 | op=option(get); |
| | 4955 | option(returnSB); |
| | 4956 | ideal star=quotient(laedi,f); |
| | 4957 | option(set,op); |
| | 4958 | if(special_ideals_equal(laedi,star)==1) |
| | 4959 | { |
| | 4960 | return(laedi,ideal(1),1); |
| | 4961 | } |
| | 4962 | action=1; |
| | 4963 | while(action==1) |
| | 4964 | { |
| | 4965 | if(size(fac)==1) |
| | 4966 | { |
| | 4967 | action=0; |
| | 4968 | break; |
| | 4969 | } |
| | 4970 | for(i=1;i<=size(fac);i++) |
| | 4971 | { |
| | 4972 | g=1; |
| | 4973 | for(j=1;j<=size(fac);j++) |
| | 4974 | { |
| | 4975 | if(i!=j) |
| | 4976 | { |
| | 4977 | g=g*fac[j]; |
| | 4978 | } |
| | 4979 | } |
| | 4980 | //std |
| | 4981 | // verg=quotient(laedi,g); |
| | 4982 | // verg=std(verg); |
| | 4983 | op=option(get); |
| | 4984 | option(returnSB); |
| | 4985 | verg=quotient(laedi,g); |
| | 4986 | option(set,op); |
| | 4987 | if(special_ideals_equal(verg,star)==1) |
| | 4988 | { |
| | 4989 | f=g; |
| | 4990 | fac[i]=0; |
| | 4991 | fac=simplify(fac,2); |
| | 4992 | break; |
| | 4993 | } |
| | 4994 | if(i==size(fac)) |
| | 4995 | { |
| | 4996 | action=0; |
| | 4997 | } |
| | 4998 | } |
| | 4999 | } |
| | 5000 | l=star,fac,f; |
| | 5001 | return(l); |
| | 5002 | } |
| | 5003 | ///////////////////////////////////////////////// |
| | 5004 | // proc special_ideals_equal |
| | 5005 | // input: standard bases of ideal k1 and k2 such that |
| | 5006 | // k1 is contained in k2, or k2 is contained ink1 |
| | 5007 | // output: 1, if k1 equals k2, 0 otherwise |
| | 5008 | ////////////////////////////////////////////////// |
| | 5009 | |
| | 5010 | static proc special_ideals_equal( ideal k1, ideal k2) |
| | 5011 | { |
| | 5012 | int j; |
| | 5013 | if(size(k1)==size(k2)) |
| | 5014 | { |
| | 5015 | for(j=1;j<=size(k1);j++) |
| | 5016 | { |
| | 5017 | if(leadexp(k1[j])!=leadexp(k2[j])) |
| | 5018 | { |
| | 5019 | return(0); |
| | 5020 | } |
| | 5021 | } |
| | 5022 | return(1); |
| | 5023 | } |
| | 5024 | return(0); |
| | 5025 | } |
| | 5026 | |
| | 5027 | |
| | 5028 | /////////////////////////////////////////////////////////////////////////////// |
| | 5029 | |
| | 5030 | static proc convList(list l) |
| | 5031 | { |
| | 5032 | int i; |
| | 5033 | list re,he; |
| | 5034 | for(i=1;i<=size(l)/2;i++) |
| | 5035 | { |
| | 5036 | he=l[2*i-1],l[2*i]; |
| | 5037 | re[i]=he; |
| | 5038 | } |
| | 5039 | return(re); |
| | 5040 | } |
| | 5041 | /////////////////////////////////////////////////////////////////////////////// |
| | 5042 | |
| | 5043 | static proc reconvList(list l) |
| | 5044 | { |
| | 5045 | int i; |
| | 5046 | list re; |
| | 5047 | for(i=1;i<=size(l);i++) |
| | 5048 | { |
| | 5049 | re[2*i-1]=l[i][1]; |
| | 5050 | re[2*i]=l[i][2]; |
| | 5051 | } |
| | 5052 | return(re); |
| | 5053 | } |
| | 5054 | |
| | 5055 | /////////////////////////////////////////////////////////////////////////////// |
| | 5056 | // |
| | 5057 | // The main procedures |
| | 5058 | // |
| | 5059 | /////////////////////////////////////////////////////////////////////////////// |
| | 5060 | |
| | 5061 | proc primdecGTZ(ideal i) |
| | 5062 | "USAGE: primdecGTZ(i); i ideal |
| | 5063 | RETURN: a list pr of primary ideals and their associated primes: |
| | 5064 | @format |
| | 5065 | pr[i][1] the i-th primary component, |
| | 5066 | pr[i][2] the i-th prime component. |
| | 5067 | @end format |
| | 5068 | NOTE: Algorithm of Gianni/Trager/Zacharias. |
| | 5069 | Designed for characteristic 0, works also in char k > 0, if it |
| | 5070 | terminates (may result in an infinite loop in small characteristic!) |
| | 5071 | EXAMPLE: example primdecGTZ; shows an example |
| | 5072 | " |
| | 5073 | { |
| | 5074 | if(attrib(basering,"global")!=1) |
| | 5075 | { |
| | 5076 | ERROR( |
| | 5077 | "// Not implemented for this ordering, please change to global ordering." |
| | 5078 | ); |
| | 5079 | } |
| | 5080 | if(minpoly!=0) |
| | 5081 | { |
| | 5082 | return(algeDeco(i,0)); |
| | 5083 | ERROR( |
| | 5084 | "// Not implemented yet for algebraic extensions.Simulate the ring extension by adding the minpoly to the ideal" |
| | 5085 | ); |
| | 5086 | } |
| | 5087 | return(convList(decomp(i))); |
| | 5088 | } |
| | 5089 | example |
| | 5090 | { "EXAMPLE:"; echo = 2; |
| | 5091 | ring r = 0,(x,y,z),lp; |
| | 5092 | poly p = z2+1; |
| | 5093 | poly q = z3+2; |
| | 5094 | ideal i = p*q^2,y-z2; |
| | 5095 | list pr = primdecGTZ(i); |
| | 5096 | pr; |
| | 5097 | } |
| | 5098 | /////////////////////////////////////////////////////////////////////////////// |
| | 5099 | |
| | 5100 | proc absPrimdecGTZ(ideal I) |
| | 5101 | "USAGE: absPrimdecGTZ(I); I ideal |
| | 5102 | ASSUME: Ground field has characteristic 0. |
| | 5103 | RETURN: a ring containing two lists: @code{absolute_primes} (the absolute |
| | 5104 | prime components of I) and @code{primary_decomp} (the output of |
| | 5105 | @code{primdecGTZ(I)}). |
| | 5106 | The list absolute_primes has to be interpreted as follows: |
| | 5107 | each entry describes a class of conjugated absolute primes, |
| | 5108 | @format |
| | 5109 | absolute_primes[i][1] the absolute prime component, |
| | 5110 | absolute_primes[i][2] the number of conjugates. |
| | 5111 | @end format |
| | 5112 | The first entry of @code{absolute_primes[i][1]} is the minimal |
| | 5113 | polynomial of a minimal finite field extension over which the |
| | 5114 | absolute prime component is defined. |
| | 5115 | NOTE: Algorithm of Gianni/Trager/Zacharias combined with the |
| | 5116 | @code{absFactorize} command. |
| | 5117 | SEE ALSO: primdecGTZ; absFactorize |
| | 5118 | EXAMPLE: example absPrimdecGTZ; shows an example |
| | 5119 | " |
| | 5120 | { |
| | 5121 | if (char(basering) != 0) |
| | 5122 | { |
| | 5123 | ERROR("primdec.lib::absPrimdecGTZ is only implemented for "+ |
| | 5124 | +"characteristic 0"); |
| | 5125 | } |
| | 5126 | |
| | 5127 | if(attrib(basering,"global")!=1) |
| | 5128 | { |
| | 5129 | ERROR( |
| | 5130 | "// Not implemented for this ordering, please change to global ordering." |
| | 5131 | ); |
| | 5132 | } |
| | 5133 | if(minpoly!=0) |
| | 5134 | { |
| | 5135 | //return(algeDeco(i,0)); |
| | 5136 | ERROR( |
| | 5137 | "// Not implemented yet for algebraic extensions.Simulate the ring extension by adding the minpoly to the ideal" |
| | 5138 | ); |
| | 5139 | } |
| | 5140 | def R=basering; |
| | 5141 | int n=nvars(R); |
| | 5142 | list L=decomp(I,3); |
| | 5143 | string newvar=L[1][3]; |
| | 5144 | int k=find(newvar,",",find(newvar,",")+1); |
| | 5145 | newvar=newvar[k+1..size(newvar)]; |
| | 5146 | list lR=ringlist(R); |
| | 5147 | int i,d; |
| | 5148 | intvec vv; |
| | 5149 | for(i=1;i<=n;i++){vv[i]=1;} |
| | 5150 | |
| | 5151 | list orst; |
| | 5152 | orst[1]=list("dp",vv); |
| | 5153 | orst[2]=list("dp",intvec(1)); |
| | 5154 | orst[3]=list("C",0); |
| | 5155 | lR[3]=orst; |
| | 5156 | lR[2][n+1] = newvar; |
| | 5157 | def Rz = ring(lR); |
| | 5158 | setring Rz; |
| | 5159 | list L=imap(R,L); |
| | 5160 | list absolute_primes,primary_decomp; |
| | 5161 | ideal I,M,N,K; |
| | 5162 | M=maxideal(1); |
| | 5163 | N=maxideal(1); |
| | 5164 | poly p,q,f,g; |
| | 5165 | map phi,psi; |
| | 5166 | for(i=1;i<=size(L);i++) |
| | 5167 | { |
| | 5168 | I=L[i][2]; |
| | 5169 | execute("K="+L[i][3]+";"); |
| | 5170 | p=K[1]; |
| | 5171 | q=K[2]; |
| | 5172 | execute("f="+L[i][4]+";"); |
| | 5173 | g=2*var(n)-f; |
| | 5174 | M[n]=f; |
| | 5175 | N[n]=g; |
| | 5176 | d=deg(p); |
| | 5177 | phi=Rz,M; |
| | 5178 | psi=Rz,N; |
| | 5179 | I=phi(I),p,q; |
| | 5180 | I=std(I); |
| | 5181 | absolute_primes[i]=list(psi(I),d); |
| | 5182 | primary_decomp[i]=list(L[i][1],L[i][2]); |
| | 5183 | } |
| | 5184 | export(primary_decomp); |
| | 5185 | export(absolute_primes); |
| | 5186 | setring R; |
| | 5187 | dbprint( printlevel-voice+3," |
| | 5188 | // 'absPrimdecGTZ' created a ring, in which two lists absolute_primes (the |
| | 5189 | // absolute prime components) and primary_decomp (the primary and prime |
| | 5190 | // components over the current basering) are stored. |
| | 5191 | // To access the list of absolute prime components, type (if the name S was |
| | 5192 | // assigned to the return value): |
| | 5193 | setring S; absolute_primes; "); |
| | 5194 | |
| | 5195 | return(Rz); |
| | 5196 | } |
| | 5197 | example |
| | 5198 | { "EXAMPLE:"; echo = 2; |
| | 5199 | ring r = 0,(x,y,z),lp; |
| | 5200 | poly p = z2+1; |
| | 5201 | poly q = z3+2; |
| | 5202 | ideal i = p*q^2,y-z2; |
| | 5203 | def S = absPrimdecGTZ(i); |
| | 5204 | setring S; |
| | 5205 | absolute_primes; |
| | 5206 | } |
| | 5207 | /////////////////////////////////////////////////////////////////////////////// |
| | 5208 | |
| | 5209 | proc primdecSY(ideal i, list #) |
| | 5210 | "USAGE: primdecSY(I, c); I ideal, c int (optional) |
| | 5211 | RETURN: a list pr of primary ideals and their associated primes: |
| | 5212 | @format |
| | 5213 | pr[i][1] the i-th primary component, |
| | 5214 | pr[i][2] the i-th prime component. |
| | 5215 | @end format |
| | 5216 | NOTE: Algorithm of Shimoyama/Yokoyama. |
| | 5217 | @format |
| | 5218 | if c=0, the given ordering of the variables is used, |
| | 5219 | if c=1, minAssChar tries to use an optimal ordering (default), |
| | 5220 | if c=2, minAssGTZ is used, |
| | 5221 | if c=3, minAssGTZ and facstd are used. |
| | 5222 | @end format |
| | 5223 | EXAMPLE: example primdecSY; shows an example |
| | 5224 | " |
| | 5225 | { |
| | 5226 | if(attrib(basering,"global")!=1) |
| | 5227 | { |
| | 5228 | ERROR( |
| | 5229 | "// Not implemented for this ordering, please change to global ordering." |
| | 5230 | ); |
| | 5231 | } |
| | 5232 | i=simplify(i,2); |
| | 5233 | if ((i[1]==0)||(i[1]==1)) |
| | 5234 | { |
| | 5235 | list L=list(ideal(i[1]),ideal(i[1])); |
| | 5236 | return(list(L)); |
| | 5237 | } |
| | 5238 | if(minpoly!=0) |
| | 5239 | { |
| | 5240 | return(algeDeco(i,1)); |
| | 5241 | } |
| | 5242 | if (size(#)==1) |
| | 5243 | { return(prim_dec(i,#[1])); } |
| | 5244 | else |
| | 5245 | { return(prim_dec(i,1)); } |
| | 5246 | } |
| | 5247 | example |
| | 5248 | { "EXAMPLE:"; echo = 2; |
| | 5249 | ring r = 0,(x,y,z),lp; |
| | 5250 | poly p = z2+1; |
| | 5251 | poly q = z3+2; |
| | 5252 | ideal i = p*q^2,y-z2; |
| | 5253 | list pr = primdecSY(i); |
| | 5254 | pr; |
| | 5255 | } |
| | 5256 | /////////////////////////////////////////////////////////////////////////////// |
| | 5257 | proc minAssGTZ(ideal i,list #) |
| | 5258 | "USAGE: minAssGTZ(I[, l]); I ideal, l list (optional) |
| | 5259 | @* Optional parameters in list l (can be entered in any order): |
| | 5260 | @* 0, \"facstd\" -> uses facstd to first decompose the ideal (default) |
| | 5261 | @* 1, \"noFacstd\" -> does not use facstd |
| | 5262 | @* \"GTZ\" -> the original algorithm by Gianni, Trager and Zacharias is used |
| | 5263 | @* \"SL\" -> GTZ algorithm with modificiations by Laplagne is used (default) |
| | 5264 | |
| | 5265 | RETURN: a list, the minimal associated prime ideals of I. |
| | 5266 | NOTE: Designed for characteristic 0, works also in char k > 0 based |
| | 5267 | on an algorithm of Yokoyama |
| | 5268 | EXAMPLE: example minAssGTZ; shows an example |
| | 5269 | " |
| | 5270 | { |
| | 5271 | int j; |
| | 5272 | string algorithm; |
| | 5273 | string facstdOption; |
| | 5274 | int useFac; |
| | 5275 | |
| | 5276 | // Set input parameters |
| | 5277 | algorithm = "SL"; // Default: SL algorithm |
| | 5278 | facstdOption = "facstd"; |
| | 5279 | if(size(#) > 0) |
| | 5280 | { |
| | 5281 | int valid; |
| | 5282 | for(j = 1; j <= size(#); j++) |
| | 5283 | { |
| | 5284 | valid = 0; |
| | 5285 | if((typeof(#[j]) == "int") or (typeof(#[j]) == "number")) |
| | 5286 | { |
| | 5287 | if (#[j] == 1) {facstdOption = "noFacstd"; valid = 1;} // If #[j] == 1, facstd is not used. |
| | 5288 | if (#[j] == 0) {facstdOption = "facstd"; valid = 1;} // If #[j] == 0, facstd is used. |
| | 5289 | } |
| | 5290 | if(typeof(#[j]) == "string") |
| | 5291 | { |
| | 5292 | if((#[j] == "GTZ") || (#[j] == "SL")) |
| | 5293 | { |
| | 5294 | algorithm = #[j]; |
| | 5295 | valid = 1; |
| | 5296 | } |
| | 5297 | if((#[j] == "noFacstd") || (#[j] == "facstd")) |
| | 5298 | { |
| | 5299 | facstdOption = #[j]; |
| | 5300 | valid = 1; |
| | 5301 | } |
| | 5302 | } |
| | 5303 | if(valid == 0) |
| | 5304 | { |
| | 5305 | dbprint(1, "Warning! The following input parameter was not recognized:", #[j]); |
| | 5306 | } |
| | 5307 | } |
| | 5308 | } |
| | 5309 | |
| | 5310 | if(attrib(basering,"global")!=1) |
| | 5311 | { |
| | 5312 | ERROR( |
| | 5313 | "// Not implemented for this ordering, please change to global ordering." |
| | 5314 | ); |
| | 5315 | } |
| | 5316 | if(minpoly!=0) |
| | 5317 | { |
| | 5318 | return(algeDeco(i,2)); |
| | 5319 | } |
| | 5320 | |
| | 5321 | list result = minAssPrimes(i, facstdOption, algorithm); |
| | 5322 | return(result); |
| | 5323 | } |
| | 5324 | example |
| | 5325 | { "EXAMPLE:"; echo = 2; |
| | 5326 | ring r = 0,(x,y,z),dp; |
| | 5327 | poly p = z2+1; |
| | 5328 | poly q = z3+2; |
| | 5329 | ideal i = p*q^2,y-z2; |
| | 5330 | list pr = minAssGTZ(i); |
| | 5331 | pr; |
| | 5332 | } |
| | 5333 | |
| | 5334 | /////////////////////////////////////////////////////////////////////////////// |
| | 5335 | proc minAssChar(ideal i, list #) |
| | 5336 | "USAGE: minAssChar(I[,c]); i ideal, c int (optional). |
| | 5337 | RETURN: list, the minimal associated prime ideals of i. |
| | 5338 | NOTE: If c=0, the given ordering of the variables is used. @* |
| | 5339 | Otherwise, the system tries to find an optimal ordering, |
| | 5340 | which in some cases may considerably speed up the algorithm. @* |
| | 5341 | Due to a bug in the factorization, the result may be not completely |
| | 5342 | decomposed in small characteristic. |
| | 5343 | EXAMPLE: example minAssChar; shows an example |
| | 5344 | " |
| | 5345 | { |
| | 5346 | if(attrib(basering,"global")!=1) |
| | 5347 | { |
| | 5348 | ERROR( |
| | 5349 | "// Not implemented for this ordering, please change to global ordering." |
| | 5350 | ); |
| | 5351 | } |
| | 5352 | if (size(#)==1) |
| | 5353 | { return(min_ass_prim_charsets(i,#[1])); } |
| | 5354 | else |
| | 5355 | { return(min_ass_prim_charsets(i,1)); } |
| | 5356 | } |
| | 5357 | example |
| | 5358 | { "EXAMPLE:"; echo = 2; |
| | 5359 | ring r = 0,(x,y,z),dp; |
| | 5360 | poly p = z2+1; |
| | 5361 | poly q = z3+2; |
| | 5362 | ideal i = p*q^2,y-z2; |
| | 5363 | list pr = minAssChar(i); |
| | 5364 | pr; |
| | 5365 | } |
| | 5366 | /////////////////////////////////////////////////////////////////////////////// |
| | 5367 | proc equiRadical(ideal i) |
| | 5368 | "USAGE: equiRadical(I); I ideal |
| | 5369 | RETURN: ideal, intersection of associated primes of I of maximal dimension. |
| | 5370 | NOTE: A combination of the algorithms of Krick/Logar (with modifications by Laplagne) and Kemper is used. |
| | 5371 | Works also in positive characteristic (Kempers algorithm). |
| | 5372 | EXAMPLE: example equiRadical; shows an example |
| | 5373 | " |
| | 5374 | { |
| | 5375 | if(attrib(basering,"global")!=1) |
| | 5376 | { |
| | 5377 | ERROR( |
| | 5378 | "// Not implemented for this ordering, please change to global ordering." |
| | 5379 | ); |
| | 5380 | } |
| | 5381 | return(radical(i, 1)); |
| | 5382 | } |
| | 5383 | example |
| | 5384 | { "EXAMPLE:"; echo = 2; |
| | 5385 | ring r = 0,(x,y,z),dp; |
| | 5386 | poly p = z2+1; |
| | 5387 | poly q = z3+2; |
| | 5388 | ideal i = p*q^2,y-z2; |
| | 5389 | ideal pr= equiRadical(i); |
| | 5390 | pr; |
| | 5391 | } |
| | 5392 | |
| | 5393 | /////////////////////////////////////////////////////////////////////////////// |
| | 5394 | proc radical(ideal i, list #) |
| | 5395 | "USAGE: radical(I[, l]); I ideal, l list (optional) |
| | 5396 | @* Optional parameters in list l (can be entered in any order): |
| | 5397 | @* 0, \"fullRad\" -> full radical is computed (default) |
| | 5398 | @* 1, \"equiRad\" -> equiRadical is computed |
| | 5399 | @* \"KL\" -> Krick/Logar algorithm is used |
| | 5400 | @* \"SL\" -> modifications by Laplagne are used (default) |
| | 5401 | @* \"facstd\" -> uses facstd to first decompose the ideal (default for non homogeneous ideals) |
| | 5402 | @* \"noFacstd\" -> does not use facstd (default for homogeneous ideals) |
| | 5403 | RETURN: ideal, the radical of I (or the equiradical if required in the input parameters) |
| | 5404 | NOTE: A combination of the algorithms of Krick/Logar (with modifications by Laplagne) and Kemper is used. |
| | 5405 | Works also in positive characteristic (Kempers algorithm). |
| | 5406 | EXAMPLE: example radical; shows an example |
| | 5407 | " |
| | 5408 | { |
| | 5409 | dbprint(printlevel - voice, "Radical, version 2006.05.08"); |
| | 5410 | if(attrib(basering,"global")!=1) |
| | 5411 | { |
| | 5412 | ERROR( |
| | 5413 | "// Not implemented for this ordering, please change to global ordering." |
| | 5414 | ); |
| | 5415 | } |
| | 5416 | if(size(i) == 0){return(ideal(0));} |
| | 5417 | int j; |
| | 5418 | def P0 = basering; |
| | 5419 | list Pl=ringlist(P0); |
| | 5420 | intvec dp_w; |
| | 5421 | for(j=nvars(P0);j>0;j--) {dp_w[j]=1;} |
| | 5422 | Pl[3]=list(list("dp",dp_w),list("C",0)); |
| | 5423 | def @P=ring(Pl); |
| | 5424 | setring @P; |
| | 5425 | ideal i=imap(P0,i); |
| | 5426 | |
| | 5427 | int il; |
| | 5428 | string algorithm; |
| | 5429 | int useFac; |
| | 5430 | |
| | 5431 | // Set input parameters |
| | 5432 | algorithm = "SL"; // Default: SL algorithm |
| | 5433 | il = 0; // Default: Full radical (not only equiRadical) |
| | 5434 | if (homog(i) == 1) |
| | 5435 | { // Default: facStd is used, except if the ideal is homogeneous. |
| | 5436 | useFac = 0; |
| | 5437 | } |
| | 5438 | else |
| | 5439 | { |
| | 5440 | useFac = 1; |
| | 5441 | } |
| | 5442 | if(size(#) > 0) |
| | 5443 | { |
| | 5444 | int valid; |
| | 5445 | for(j = 1; j <= size(#); j++) |
| | 5446 | { |
| | 5447 | valid = 0; |
| | 5448 | if((typeof(#[j]) == "int") or (typeof(#[j]) == "number")) |
| | 5449 | { |
| | 5450 | il = #[j]; // If il == 1, equiRadical is computed |
| | 5451 | valid = 1; |
| | 5452 | } |
| | 5453 | if(typeof(#[j]) == "string") |
| | 5454 | { |
| | 5455 | if(#[j] == "KL") |
| | 5456 | { |
| | 5457 | algorithm = "KL"; |
| | 5458 | valid = 1; |
| | 5459 | } |
| | 5460 | if(#[j] == "SL") |
| | 5461 | { |
| | 5462 | algorithm = "SL"; |
| | 5463 | valid = 1; |
| | 5464 | } |
| | 5465 | if(#[j] == "noFacstd") |
| | 5466 | { |
| | 5467 | useFac = 0; |
| | 5468 | valid = 1; |
| | 5469 | } |
| | 5470 | if(#[j] == "facstd") |
| | 5471 | { |
| | 5472 | useFac = 1; |
| | 5473 | valid = 1; |
| | 5474 | } |
| | 5475 | if(#[j] == "equiRad") |
| | 5476 | { |
| | 5477 | il = 1; |
| | 5478 | valid = 1; |
| | 5479 | } |
| | 5480 | if(#[j] == "fullRad") |
| | 5481 | { |
| | 5482 | il = 0; |
| | 5483 | valid = 1; |
| | 5484 | } |
| | 5485 | } |
| | 5486 | if(valid == 0) |
| | 5487 | { |
| | 5488 | dbprint(1, "Warning! The following input parameter was not recognized:", #[j]); |
| | 5489 | } |
| | 5490 | } |
| | 5491 | } |
| | 5492 | |
| | 5493 | ideal rad = 1; |
| | 5494 | intvec op = option(get); |
| | 5495 | list qr = simplifyIdeal(i); |
| | 5496 | map phi = @P, qr[2]; |
| | 5497 | |
| | 5498 | option(redSB); |
| | 5499 | i = groebner(qr[1]); |
| | 5500 | option(set, op); |
| | 5501 | int di = dim(i); |
| | 5502 | |
| | 5503 | if(di == 0) |
| | 5504 | { |
| | 5505 | i = zeroRad(i, qr[1]); |
| | 5506 | i=interred(phi(i)); |
| | 5507 | setring(P0); |
| | 5508 | i=imap(@P,i); |
| | 5509 | return(i); |
| | 5510 | } |
| | 5511 | |
| | 5512 | option(redSB); |
| | 5513 | list pr; |
| | 5514 | if(useFac == 1) |
| | 5515 | { |
| | 5516 | pr = facstd(i); |
| | 5517 | } |
| | 5518 | else |
| | 5519 | { |
| | 5520 | pr = i; |
| | 5521 | } |
| | 5522 | option(set, op); |
| | 5523 | int s = size(pr); |
| | 5524 | if(useFac == 1) |
| | 5525 | { |
| | 5526 | dbprint(printlevel - voice, "Number of components returned by facstd: ", s); |
| | 5527 | } |
| | 5528 | for(j = 1; j <= s; j++) |
| | 5529 | { |
| | 5530 | attrib(pr[s + 1 - j], "isSB", 1); |
| | 5531 | if((size(reduce(rad, pr[s + 1 - j], 1)) != 0) && ((dim(pr[s + 1 - j]) == di) || !il)) |
| | 5532 | { |
| | 5533 | // SL Debug messages |
| | 5534 | dbprint(printlevel-voice, "We shall compute the radical of ", pr[s + 1 - j]); |
| | 5535 | dbprint(printlevel-voice, "The dimension is: ", dim(pr[s+1-j])); |
| | 5536 | |
| | 5537 | if(algorithm == "KL") |
| | 5538 | { |
| | 5539 | rad = intersect(rad, radicalKL(pr[s + 1 - j], rad, il)); |
| | 5540 | } |
| | 5541 | if(algorithm == "SL") |
| | 5542 | { |
| | 5543 | rad = intersect(rad, radicalSL(pr[s + 1 - j], il)); |
| | 5544 | } |
| | 5545 | } |
| | 5546 | else |
| | 5547 | { |
| | 5548 | // SL Debug |
| | 5549 | dbprint(printlevel-voice, "The radical of this component is not needed."); |
| | 5550 | dbprint(printlevel-voice, "size(reduce(rad, pr[s + 1 - j], 1))", |
| | 5551 | size(reduce(rad, pr[s + 1 - j], 1))); |
| | 5552 | dbprint(printlevel-voice, "dim(pr[s + 1 - j])", dim(pr[s + 1 - j])); |
| | 5553 | dbprint(printlevel-voice, "il", il); |
| | 5554 | } |
| | 5555 | } |
| | 5556 | rad=interred(phi(rad)); |
| | 5557 | setring(P0); |
| | 5558 | i=imap(@P,rad); |
| | 5559 | return(i); |
| | 5560 | } |
| | 5561 | example |
| | 5562 | { "EXAMPLE:"; echo = 2; |
| | 5563 | ring r = 0,(x,y,z),dp; |
| | 5564 | poly p = z2+1; |
| | 5565 | poly q = z3+2; |
| | 5566 | ideal i = p*q^2,y-z2; |
| | 5567 | ideal pr = radical(i); |
| | 5568 | pr; |
| | 5569 | } |
| | 5570 | |
| | 5571 | /////////////////////////////////////////////////////////////////////////////// |
| | 5572 | // |
| | 5573 | // Computes the radical of I using KL algorithm. |
| | 5574 | // The only difference with the the previous implementation of KL algorithm is |
| | 5575 | // that now it uses block dp instead of lp ordering for the reduction to the |
| | 5576 | // zerodimensional case. |
| | 5577 | // The reduction step has been moved to the new routine radicalReduction, so that it can be |
| | 5578 | // used also by radicalSL procedure. |
| | 5579 | // |
| | 5580 | static proc radicalKL(ideal I, ideal ser, list #) |
| | 5581 | { |
| | 5582 | // ideal I The ideal for which the radical is computed |
| | 5583 | // ideal ser Used to reduce components already obtained |
| | 5584 | // list # If #[1] = 1, equiradical is computed. |
| | 5585 | |
| | 5586 | // I needs to be a Groebner basis. |
| | 5587 | if (attrib(I, "isSB") != 1) |
| | 5588 | { |
| | 5589 | I = groebner(I); |
| | 5590 | } |
| | 5591 | |
| | 5592 | ideal rad; // The radical |
| | 5593 | int allIndep = 1; // All max independent sets are used |
| | 5594 | |
| | 5595 | list result = radicalReduction(I, ser, allIndep, #); |
| | 5596 | int done = result[3]; |
| | 5597 | rad = result[1]; |
| | 5598 | if (done == 0) |
| | 5599 | { |
| | 5600 | rad = intersect(rad, radicalKL(result[2], ideal(1), #)); |
| | 5601 | } |
| | 5602 | return(rad); |
| | 5603 | } |
| | 5604 | |
| | 5605 | |
| | 5606 | /////////////////////////////////////////////////////////////////////////////// |
| | 5607 | // |
| | 5608 | // Computes the radical of I via Laplagne algorithm, using zerodimensional radical in |
| | 5609 | // the zero dimensional case. |
| | 5610 | // For the reduction to the zerodimensional case, it uses the procedure |
| | 5611 | // radical, with some modifications to avoid the recursion. |
| | 5612 | // |
| | 5613 | static proc radicalSL(ideal I, list #) |
| | 5614 | // Input = I, ideal |
| | 5615 | // #, list. If #[1] = 1, then computes only the equiradical. |
| | 5616 | // Output = (P, primaryDec) where P = rad(I) and primaryDec is the list of the radicals |
| | 5617 | // obtained in intermediate steps. |
| | 5618 | { |
| | 5619 | ideal rad = 1; |
| | 5620 | ideal equiRad = 1; |
| | 5621 | list primes; |
| | 5622 | int k; // Counter |
| | 5623 | int il; // If il = 1, only the equiradical is required. |
| | 5624 | int iDim; // The dimension of I |
| | 5625 | int stop = 0; // Checks if the radical has been obtained |
| | 5626 | |
| | 5627 | if (attrib(I, "isSB") != 1) |
| | 5628 | { |
| | 5629 | I = groebner(I); |
| | 5630 | } |
| | 5631 | iDim = dim(I); |
| | 5632 | |
| | 5633 | // Checks if only equiradical is required |
| | 5634 | if (size(#) > 0) |
| | 5635 | { |
| | 5636 | il = #[1]; |
| | 5637 | } |
| | 5638 | |
| | 5639 | while(stop == 0) |
| | 5640 | { |
| | 5641 | dbprint (printlevel-voice, "// We call radLoopR to find new prime ideals."); |
| | 5642 | primes = radicalSLIteration(I, rad); // A list of primes or intersections of primes, not included in P |
| | 5643 | dbprint (printlevel - voice, "// Output of Iteration Step:"); |
| | 5644 | dbprint (printlevel - voice, primes); |
| | 5645 | if (size(primes) > 0) |
| | 5646 | { |
| | 5647 | dbprint (printlevel - voice, "// We intersect P with the ideal just obtained."); |
| | 5648 | for(k = 1; k <= size(primes); k++) |
| | 5649 | { |
| | 5650 | rad = intersect(rad, primes[k]); |
| | 5651 | if (il == 1) |
| | 5652 | { |
| | 5653 | if (attrib(primes[k], "isSB") != 1) |
| | 5654 | { |
| | 5655 | primes[k] = groebner(primes[k]); |
| | 5656 | } |
| | 5657 | if (iDim == dim(primes[k])) |
| | 5658 | { |
| | 5659 | equiRad = intersect(equiRad, primes[k]); |
| | 5660 | } |
| | 5661 | } |
| | 5662 | } |
| | 5663 | } |
| | 5664 | else |
| | 5665 | { |
| | 5666 | stop = 1; |
| | 5667 | } |
| | 5668 | } |
| | 5669 | if (il == 0) |
| | 5670 | { |
| | 5671 | return(rad); |
| | 5672 | } |
| | 5673 | else |
| | 5674 | { |
| | 5675 | return(equiRad); |
| | 5676 | } |
| | 5677 | } |
| | 5678 | |
| | 5679 | ////////////////////////////////////////////////////////////////////////// |
| | 5680 | // Based on radicalKL. |
| | 5681 | // It contains all of old version of proc radicalKL except the recursion call. |
| | 5682 | // |
| | 5683 | // Output: |
| | 5684 | // #1 -> output ideal, the part of the radical that has been computed |
| | 5685 | // #2 -> complementary ideal, the part of the ideal I whose radical remains to be computed |
| | 5686 | // = (I, h) in KL algorithm |
| | 5687 | // This is not used in the new algorithm. It is part of KL algorithm |
| | 5688 | // #3 -> done, 1: output = radical, there is no need to continue |
| | 5689 | // 0: radical = output \cap \sqrt{complementary ideal} |
| | 5690 | // This is not used in the new algorithm. It is part of KL algorithm |
| | 5691 | |
| | 5692 | static proc radicalReduction(ideal I, ideal ser, int allIndep, list #) |
| | 5693 | { |
| | 5694 | // allMaximal 1 -> Indicates that the reduction to the zerodim case |
| | 5695 | // must be done for all indep set of the leading terms ideal |
| | 5696 | // 0 -> Otherwise |
| | 5697 | // ideal ser Only for radicalKL. (Same as in radicalKL) |
| | 5698 | // list # Only for radicalKL (If #[1] = 1, only equiradical is required. |
| | 5699 | // It is used to set the value of done.) |
| | 5700 | |
| | 5701 | attrib(I, "isSB", 1); // I needs to be a reduced standard basis |
| | 5702 | list indep, fett; |
| | 5703 | intvec @w, @hilb, op; |
| | 5704 | int @wr, @n, @m, lauf, di; |
| | 5705 | ideal fac, @h, collectrad, lsau; |
| | 5706 | poly @q; |
| | 5707 | string @va, quotring; |
| | 5708 | |
| | 5709 | def @P = basering; |
| | 5710 | int jdim = dim(I); // Computes the dimension of I |
| | 5711 | int homo = homog(I); // Finds out if I is homogeneous |
| | 5712 | ideal rad = ideal(1); // The unit ideal |
| | 5713 | ideal te = ser; |
| | 5714 | if(size(#) > 0) |
| | 5715 | { |
| | 5716 | @wr = #[1]; |
| | 5717 | } |
| | 5718 | if(homo == 1) |
| | 5719 | { |
| | 5720 | for(@n = 1; @n <= nvars(basering); @n++) |
| | 5721 | { |
| | 5722 | @w[@n] = ord(var(@n)); |
| | 5723 | } |
| | 5724 | @hilb = hilb(I, 1, @w); |
| | 5725 | } |
| | 5726 | |
| | 5727 | // SL 2006.04.11 1 Debug messages |
| | 5728 | dbprint(printlevel-voice, "//Computes the radical of the ideal:", I); |
| | 5729 | // SL 2006.04.11 2 Debug messages |
| | 5730 | |
| | 5731 | //--------------------------------------------------------------------------- |
| | 5732 | //j is the ring |
| | 5733 | //--------------------------------------------------------------------------- |
| | 5734 | |
| | 5735 | if (jdim==-1) |
| | 5736 | { |
| | 5737 | return(ideal(1), ideal(1), 1); |
| | 5738 | } |
| | 5739 | |
| | 5740 | //--------------------------------------------------------------------------- |
| | 5741 | //the zero-dimensional case |
| | 5742 | //--------------------------------------------------------------------------- |
| | 5743 | |
| | 5744 | if (jdim==0) |
| | 5745 | { |
| | 5746 | return(zeroRad(I), ideal(1), 1); |
| | 5747 | } |
| | 5748 | |
| | 5749 | //------------------------------------------------------------------------- |
| | 5750 | //search for a maximal independent set indep,i.e. |
| | 5751 | //look for subring such that the intersection with the ideal is zero |
| | 5752 | //j intersected with K[var(indep[3]+1),...,var(nvar)] is zero, |
| | 5753 | //indep[1] is the new varstring, indep[2] the string for the block-ordering |
| | 5754 | //------------------------------------------------------------------------- |
| | 5755 | |
| | 5756 | // SL 2006-04-24 1 If allIndep = 0, then it only computes one maximal |
| | 5757 | // independent set. |
| | 5758 | // This looks better for the new algorithm but not for KL |
| | 5759 | // algorithm |
| | 5760 | list parameters = allIndep; |
| | 5761 | indep = newMaxIndependSetDp(I, parameters); |
| | 5762 | // SL 2006-04-24 2 |
| | 5763 | |
| | 5764 | for(@m = 1; @m <= size(indep); @m++) |
| | 5765 | { |
| | 5766 | if((indep[@m][1] == varstr(basering)) && (@m == 1)) |
| | 5767 | //this is the good case, nothing to do, just to have the same notations |
| | 5768 | //change the ring |
| | 5769 | { |
| | 5770 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
| | 5771 | +ordstr(basering)+");"); |
| | 5772 | ideal @j = fetch(@P, I); |
| | 5773 | attrib(@j, "isSB", 1); |
| | 5774 | } |
| | 5775 | else |
| | 5776 | { |
| | 5777 | @va = string(maxideal(1)); |
| | 5778 | |
| | 5779 | execute("ring gnir1 = (" + charstr(basering) + "), (" + indep[@m][1] + "),(" |
| | 5780 | + indep[@m][2] + ");"); |
| | 5781 | execute("map phi = @P," + @va + ";"); |
| | 5782 | if(homo == 1) |
| | 5783 | { |
| | 5784 | ideal @j = std(phi(I), @hilb, @w); |
| | 5785 | } |
| | 5786 | else |
| | 5787 | { |
| | 5788 | ideal @j = groebner(phi(I)); |
| | 5789 | } |
| | 5790 | } |
| | 5791 | if((deg(@j[1]) == 0) || (dim(@j) < jdim)) |
| | 5792 | { |
| | 5793 | setring @P; |
| | 5794 | break; |
| | 5795 | } |
| | 5796 | for (lauf = 1; lauf <= size(@j); lauf++) |
| | 5797 | { |
| | 5798 | fett[lauf] = size(@j[lauf]); |
| | 5799 | } |
| | 5800 | //------------------------------------------------------------------------ |
| | 5801 | // We have now the following situation: |
| | 5802 | // j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
| | 5803 | // to this quotientring, j is there still a standardbasis, the |
| | 5804 | // leading coefficients of the polynomials there (polynomials in |
| | 5805 | // K[var(nnp+1),..,var(nva)]) are collected in the list h, |
| | 5806 | // we need their LCM, gh, because of the following: |
| | 5807 | // let (j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..rest..] |
| | 5808 | // intersected with K[var(1),...,var(nva)] is (j:gh^n) |
| | 5809 | // on the other hand j = ((j, gh^n) intersected with (j : gh^n)) |
| | 5810 | |
| | 5811 | //------------------------------------------------------------------------ |
| | 5812 | // The arrangement for the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
| | 5813 | // and the map phi:K[var(1),...,var(nva)] -----> |
| | 5814 | // K(var(nnpr+1),..,var(nva))[..the rest..] |
| | 5815 | //------------------------------------------------------------------------ |
| | 5816 | quotring = prepareQuotientRingDp(nvars(basering) - indep[@m][3]); |
| | 5817 | //------------------------------------------------------------------------ |
| | 5818 | // We pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
| | 5819 | //------------------------------------------------------------------------ |
| | 5820 | |
| | 5821 | execute(quotring); |
| | 5822 | |
| | 5823 | // @j considered in the quotientring |
| | 5824 | ideal @j = imap(gnir1, @j); |
| | 5825 | |
| | 5826 | kill gnir1; |
| | 5827 | |
| | 5828 | // j is a standardbasis in the quotientring but usually not minimal |
| | 5829 | // here it becomes minimal |
| | 5830 | |
| | 5831 | @j = clearSB(@j, fett); |
| | 5832 | |
| | 5833 | // We need later LCM(h[1],...) = gh for saturation |
| | 5834 | ideal @h; |
| | 5835 | if(deg(@j[1]) > 0) |
| | 5836 | { |
| | 5837 | for(@n = 1; @n <= size(@j); @n++) |
| | 5838 | { |
| | 5839 | @h[@n] = leadcoef(@j[@n]); |
| | 5840 | } |
| | 5841 | op = option(get); |
| | 5842 | option(redSB); |
| | 5843 | @j = interred(@j); //to obtain a reduced standardbasis |
| | 5844 | attrib(@j, "isSB", 1); |
| | 5845 | option(set, op); |
| | 5846 | |
| | 5847 | // SL 1 Debug messages |
| | 5848 | dbprint(printlevel - voice, "zero_rad", basering, @j, dim(groebner(@j))); |
| | 5849 | ideal zero_rad = zeroRad(@j); |
| | 5850 | dbprint(printlevel - voice, "zero_rad passed"); |
| | 5851 | // SL 2 |
| | 5852 | } |
| | 5853 | else |
| | 5854 | { |
| | 5855 | ideal zero_rad = ideal(1); |
| | 5856 | } |
| | 5857 | |
| | 5858 | // We need the intersection of the ideals in the list quprimary with the |
| | 5859 | // polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
| | 5860 | // but fi polynomials, then the intersection of q with the polynomialring |
| | 5861 | // is the saturation of the ideal generated by f1,...,fr with respect to |
| | 5862 | // h which is the lcm of the leading coefficients of the fi considered in |
| | 5863 | // the quotientring: this is coded in saturn |
| | 5864 | |
| | 5865 | zero_rad = std(zero_rad); |
| | 5866 | |
| | 5867 | ideal hpl; |
| | 5868 | |
| | 5869 | for(@n = 1; @n <= size(zero_rad); @n++) |
| | 5870 | { |
| | 5871 | hpl = hpl, leadcoef(zero_rad[@n]); |
| | 5872 | } |
| | 5873 | |
| | 5874 | //------------------------------------------------------------------------ |
| | 5875 | // We leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
| | 5876 | // back to the polynomialring |
| | 5877 | //------------------------------------------------------------------------ |
| | 5878 | setring @P; |
| | 5879 | |
| | 5880 | collectrad = imap(quring, zero_rad); |
| | 5881 | lsau = simplify(imap(quring, hpl), 2); |
| | 5882 | @h = imap(quring, @h); |
| | 5883 | |
| | 5884 | kill quring; |
| | 5885 | |
| | 5886 | // Here the intersection with the polynomialring |
| | 5887 | // mentioned above is really computed |
| | 5888 | |
| | 5889 | collectrad = sat2(collectrad, lsau)[1]; |
| | 5890 | if(deg(@h[1])>=0) |
| | 5891 | { |
| | 5892 | fac = ideal(0); |
| | 5893 | for(lauf = 1; lauf <= ncols(@h); lauf++) |
| | 5894 | { |
| | 5895 | if(deg(@h[lauf]) > 0) |
| | 5896 | { |
| | 5897 | fac = fac + factorize(@h[lauf], 1); |
| | 5898 | } |
| | 5899 | } |
| | 5900 | fac = simplify(fac, 6); |
| | 5901 | @q = 1; |
| | 5902 | for(lauf = 1; lauf <= size(fac); lauf++) |
| | 5903 | { |
| | 5904 | @q = @q * fac[lauf]; |
| | 5905 | } |
| | 5906 | op = option(get); |
| | 5907 | option(returnSB); |
| | 5908 | option(redSB); |
| | 5909 | I = quotient(I + ideal(@q), rad); |
| | 5910 | attrib(I, "isSB", 1); |
| | 5911 | option(set, op); |
| | 5912 | } |
| | 5913 | if((deg(rad[1]) > 0) && (deg(collectrad[1]) > 0)) |
| | 5914 | { |
| | 5915 | rad = intersect(rad, collectrad); |
| | 5916 | te = intersect(te, collectrad); |
| | 5917 | te = simplify(reduce(te, I, 1), 2); |
| | 5918 | } |
| | 5919 | else |
| | 5920 | { |
| | 5921 | if(deg(collectrad[1]) > 0) |
| | 5922 | { |
| | 5923 | rad = collectrad; |
| | 5924 | te = intersect(te, collectrad); |
| | 5925 | te = simplify(reduce(te, I, 1), 2); |
| | 5926 | } |
| | 5927 | } |
| | 5928 | |
| | 5929 | if((dim(I) < jdim)||(size(te) == 0)) |
| | 5930 | { |
| | 5931 | break; |
| | 5932 | } |
| | 5933 | if(homo==1) |
| | 5934 | { |
| | 5935 | @hilb = hilb(I, 1, @w); |
| | 5936 | } |
| | 5937 | } |
| | 5938 | |
| | 5939 | // SL 2006.04.11 1 Debug messages |
| | 5940 | dbprint (printlevel-voice, "// Part of the Radical already computed:", rad); |
| | 5941 | dbprint (printlevel-voice, "// Dimension:", dim(groebner(rad))); |
| | 5942 | // SL 2006.04.11 2 Debug messages |
| | 5943 | |
| | 5944 | // SL 2006.04.21 1 New variable "done". |
| | 5945 | // It tells if the radical is already computed or |
| | 5946 | // if it still has to be computed the radical of the new ideal I |
| | 5947 | int done; |
| | 5948 | if(((@wr == 1) && (dim(I)<jdim)) || (deg(I[1])==0) || (size(te) == 0)) |
| | 5949 | { |
| | 5950 | done = 1; |
| | 5951 | } |
| | 5952 | else |
| | 5953 | { |
| | 5954 | done = 0; |
| | 5955 | } |
| | 5956 | // SL 2006.04.21 2 |
| | 5957 | |
| | 5958 | // SL 2006.04.21 1 See details of the output at the beggining of this proc. |
| | 5959 | list result = rad, I, done; |
| | 5960 | return(result); |
| | 5961 | // SL 2006.04.21 2 |
| | 5962 | } |
| | 5963 | |
| | 5964 | |
| | 5965 | |
| | 5966 | /////////////////////////////////////////////////////////////////////////////// |
| | 5967 | // Given an ideal I and an ideal P (intersection of some minimal prime ideals |
| | 5968 | // associated to I), it calculates the intersection of new minimal prime ideals |
| | 5969 | // associated to I which where not used to calculate P. |
| | 5970 | // This version uses ZD Radical in the zerodimensional case. |
| | 5971 | static proc radicalSLIteration (ideal I, ideal P); |
| | 5972 | // Input: I, ideal. The ideal from which new prime components will be obtained. |
| | 5973 | // P, ideal. Intersection of some prime ideals of I. |
| | 5974 | // Output: ideal. Intersection of some primes of I different from the ones in P. |
| | 5975 | { |
| | 5976 | int k = 1; // Counter |
| | 5977 | int good = 0; // Checks if an element of P is in rad(I) |
| | 5978 | |
| | 5979 | dbprint (printlevel-voice, "// We search for an element in P - sqrt(I)."); |
| | 5980 | while ((k <= size(P)) and (good == 0)) |
| | 5981 | { |
| | 5982 | dbprint (printlevel-voice, "// We try with:", P[k]); |
| | 5983 | good = 1 - rad_con(P[k], I); |
| | 5984 | k++; |
| | 5985 | } |
| | 5986 | k--; |
| | 5987 | if (good == 0) |
| | 5988 | { |
| | 5989 | dbprint (printlevel-voice, "// No element was found, P = sqrt(I)."); |
| | 5990 | list emptyList = list(); |
| | 5991 | return (emptyList); |
| | 5992 | } |
| | 5993 | dbprint(printlevel - voice, "// That one was good!"); |
| | 5994 | dbprint(printlevel - voice, "// We saturate I with respect to this element."); |
| | 5995 | if (P[k] != 1) |
| | 5996 | { |
| | 5997 | ideal J = sat(I, P[k])[1]; |
| | 5998 | } |
| | 5999 | else |
| | 6000 | { |
| | 6001 | dbprint(printlevel - voice, "// The polynomial is 1, the saturation in not actually computed."); |
| | 6002 | ideal J = I; |
| | 6003 | } |
| | 6004 | |
| | 6005 | // We now call proc radicalNew; |
| | 6006 | dbprint(printlevel - voice, "// We do the reduction to the zerodimensional case, via radical."); |
| | 6007 | dbprint(printlevel - voice, "// The ideal is ", J); |
| | 6008 | dbprint(printlevel - voice, "// The dimension is ", dim(groebner(J))); |
| | 6009 | |
| | 6010 | int allMaximal = 0; // Compute the zerodim reduction for only one indep set. |
| | 6011 | ideal re = 1; // No reduction is need, there are not redundant components. |
| | 6012 | list emptyList = list(); // Look for primes of any dimension, not only of max dimension. |
| | 6013 | list result = radicalReduction(J, re, allMaximal, emptyList); |
| | 6014 | |
| | 6015 | return(result[1]); |
| | 6016 | } |
| | 6017 | |
| | 6018 | /////////////////////////////////////////////////////////////////////////////////// |
| | 6019 | // Based on maxIndependSet |
| | 6020 | // Added list # as parameter |
| | 6021 | // If the first element of # is 0, the output is only 1 max indep set. |
| | 6022 | // If no list is specified or #[1] = 1, the output is all the max indep set of the |
| | 6023 | // leading terms ideal. This is the original output of maxIndependSet |
| | 6024 | |
| | 6025 | // The ordering given in the output has been changed to block dp instead of lp. |
| | 6026 | |
| | 6027 | proc newMaxIndependSetDp(ideal j, list #) |
| | 6028 | "USAGE: newMaxIndependentSetDp(I); I ideal (returns all maximal independent sets of the corresponding leading terms ideal) |
| | 6029 | newMaxIndependentSetDp(I, 0); I ideal (returns only one maximal independent set) |
| | 6030 | RETURN: list = #1. new varstring with the maximal independent set at the end, |
| | 6031 | #2. ordstring with the corresponding dp block ordering, |
| | 6032 | #3. the number of independent variables |
| | 6033 | NOTE: |
| | 6034 | EXAMPLE: example newMaxIndependentSetDp; shows an example |
| | 6035 | " |
| | 6036 | { |
| | 6037 | int n, k, di; |
| | 6038 | list resu, hilf; |
| | 6039 | string var1, var2; |
| | 6040 | list v = indepSet(j, 0); |
| | 6041 | |
| | 6042 | // SL 2006.04.21 1 Lines modified to use only one independent Set |
| | 6043 | int allMaximal; |
| | 6044 | if (size(#) > 0) |
| | 6045 | { |
| | 6046 | allMaximal = #[1]; |
| | 6047 | } |
| | 6048 | else |
| | 6049 | { |
| | 6050 | allMaximal = 1; |
| | 6051 | } |
| | 6052 | |
| | 6053 | int nMax; |
| | 6054 | if (allMaximal == 1) |
| | 6055 | { |
| | 6056 | nMax = size(v); |
| | 6057 | } |
| | 6058 | else |
| | 6059 | { |
| | 6060 | nMax = 1; |
| | 6061 | } |
| | 6062 | |
| | 6063 | for(n = 1; n <= nMax; n++) |
| | 6064 | // SL 2006.04.21 2 |
| | 6065 | { |
| | 6066 | di = 0; |
| | 6067 | var1 = ""; |
| | 6068 | var2 = ""; |
| | 6069 | for(k = 1; k <= size(v[n]); k++) |
| | 6070 | { |
| | 6071 | if(v[n][k] != 0) |
| | 6072 | { |
| | 6073 | di++; |
| | 6074 | var2 = var2 + "var(" + string(k) + "), "; |
| | 6075 | } |
| | 6076 | else |
| | 6077 | { |
| | 6078 | var1 = var1 + "var(" + string(k) + "), "; |
| | 6079 | } |
| | 6080 | } |
| | 6081 | if(di > 0) |
| | 6082 | { |
| | 6083 | var1 = var1 + var2; |
| | 6084 | var1 = var1[1..size(var1) - 2]; // The "- 2" removes the trailer comma |
| | 6085 | hilf[1] = var1; |
| | 6086 | // SL 2006.21.04 1 The order is now block dp instead of lp |
| | 6087 | hilf[2] = "dp(" + string(nvars(basering) - di) + "), dp(" + string(di) + ")"; |
| | 6088 | // SL 2006.21.04 2 |
| | 6089 | hilf[3] = di; |
| | 6090 | resu[n] = hilf; |
| | 6091 | } |
| | 6092 | else |
| | 6093 | { |
| | 6094 | resu[n] = varstr(basering), ordstr(basering), 0; |
| | 6095 | } |
| | 6096 | } |
| | 6097 | return(resu); |
| | 6098 | } |
| | 6099 | example |
| | 6100 | { "EXAMPLE:"; echo = 2; |
| | 6101 | ring s1 = (0, x, y), (a, b, c, d, e, f, g), lp; |
| | 6102 | ideal i = ea - fbg, fa + be, ec - fdg, fc + de; |
| | 6103 | i = std(i); |
| | 6104 | list l = newMaxIndependSetDp(i); |
| | 6105 | l; |
| | 6106 | i = i, g; |
| | 6107 | l = newMaxIndependSetDp(i); |
| | 6108 | l; |
| | 6109 | |
| | 6110 | ring s = 0, (x, y, z), lp; |
| | 6111 | ideal i = z, yx; |
| | 6112 | list l = newMaxIndependSetDp(i); |
| | 6113 | l; |
| | 6114 | } |
| | 6115 | |
| | 6116 | |
| | 6117 | /////////////////////////////////////////////////////////////////////////////// |
| | 6118 | // based on prepareQuotientring |
| | 6119 | // The order returned is now (C, dp) instead of (C, lp) |
| | 6120 | |
| | 6121 | static proc prepareQuotientRingDp (int nnp) |
| | 6122 | "USAGE: prepareQuotientRingDp(nnp); nnp int |
| | 6123 | RETURN: string = to define Kvar(nnp+1),...,var(nvars)[..rest ] |
| | 6124 | NOTE: |
| | 6125 | EXAMPLE: example prepareQuotientRingDp; shows an example |
| | 6126 | " |
| | 6127 | { |
| | 6128 | ideal @ih,@jh; |
| | 6129 | int npar=npars(basering); |
| | 6130 | int @n; |
| | 6131 | |
| | 6132 | string quotring= "ring quring = ("+charstr(basering); |
| | 6133 | for(@n = nnp + 1; @n <= nvars(basering); @n++) |
| | 6134 | { |
| | 6135 | quotring = quotring + ", var(" + string(@n) + ")"; |
| | 6136 | @ih = @ih + var(@n); |
| | 6137 | } |
| | 6138 | |
| | 6139 | quotring = quotring+"),(var(1)"; |
| | 6140 | @jh = @jh + var(1); |
| | 6141 | for(@n = 2; @n <= nnp; @n++) |
| | 6142 | { |
| | 6143 | quotring = quotring + ", var(" + string(@n) + ")"; |
| | 6144 | @jh = @jh + var(@n); |
| | 6145 | } |
| | 6146 | // SL 2006-04-21 1 The order returned is now (C, dp) instead of (C, lp) |
| | 6147 | quotring = quotring + "), (C, dp);"; |
| | 6148 | // SL 2006-04-21 2 |
| | 6149 | |
| | 6150 | return(quotring); |
| | 6151 | } |
| | 6152 | example |
| | 6153 | { "EXAMPLE:"; echo = 2; |
| | 6154 | ring s1=(0,x),(a,b,c,d,e,f,g),lp; |
| | 6155 | def @Q=basering; |
| | 6156 | list l= prepareQuotientRingDp(3); |
| | 6157 | l; |
| | 6158 | execute(l[1]); |
| | 6159 | execute(l[2]); |
| | 6160 | basering; |
| | 6161 | phi; |
| | 6162 | setring @Q; |
| | 6163 | |
| | 6164 | } |
| | 6165 | |
| | 6166 | /////////////////////////////////////////////////////////////////////////////// |
| | 6167 | proc prepareAss(ideal i) |
| | 6168 | "USAGE: prepareAss(I); I ideal |
| | 6169 | RETURN: list, the radicals of the maximal dimensional components of I. |
| | 6170 | NOTE: Uses algorithm of Eisenbud/Huneke/Vasconcelos. |
| | 6171 | EXAMPLE: example prepareAss; shows an example |
| | 6172 | " |
| | 6173 | { |
| | 6174 | if(attrib(basering,"global")!=1) |
| | 6175 | { |
| | 6176 | ERROR( |
| | 6177 | "// Not implemented for this ordering, please change to global ordering." |
| | 6178 | ); |
| | 6179 | } |
| | 6180 | ideal j=std(i); |
| | 6181 | int cod=nvars(basering)-dim(j); |
| | 6182 | int e; |
| | 6183 | list er; |
| | 6184 | ideal ann; |
| | 6185 | if(homog(i)==1) |
| | 6186 | { |
| | 6187 | list re=sres(j,0); //the resolution |
| | 6188 | re=minres(re); //minimized resolution |
| | 6189 | } |
| | 6190 | else |
| | 6191 | { |
| | 6192 | list re=mres(i,0); |
| | 6193 | } |
| | 6194 | for(e=cod;e<=nvars(basering);e++) |
| | 6195 | { |
| | 6196 | ann=AnnExt_R(e,re); |
| | 6197 | |
| | 6198 | if(nvars(basering)-dim(std(ann))==e) |
| | 6199 | { |
| | 6200 | er[size(er)+1]=equiRadical(ann); |
| | 6201 | } |
| | 6202 | } |
| | 6203 | return(er); |
| | 6204 | } |
| | 6205 | example |
| | 6206 | { "EXAMPLE:"; echo = 2; |
| | 6207 | ring r = 0,(x,y,z),dp; |
| | 6208 | poly p = z2+1; |
| | 6209 | poly q = z3+2; |
| | 6210 | ideal i = p*q^2,y-z2; |
| | 6211 | list pr = prepareAss(i); |
| | 6212 | pr; |
| | 6213 | } |
| | 6214 | /////////////////////////////////////////////////////////////////////////////// |
| | 6215 | proc equidimMaxEHV(ideal i) |
| | 6216 | "USAGE: equidimMaxEHV(I); I ideal |
| | 6217 | RETURN: ideal, the equidimensional component (of maximal dimension) of I. |
| | 6218 | NOTE: Uses algorithm of Eisenbud, Huneke and Vasconcelos. |
| | 6219 | EXAMPLE: example equidimMaxEHV; shows an example |
| | 6220 | " |
| | 6221 | { |
| | 6222 | if(attrib(basering,"global")!=1) |
| | 6223 | { |
| | 6224 | ERROR( |
| | 6225 | "// Not implemented for this ordering, please change to global ordering." |
| | 6226 | ); |
| | 6227 | } |
| | 6228 | ideal j=groebner(i); |
| | 6229 | int cod=nvars(basering)-dim(j); |
| | 6230 | int e; |
| | 6231 | ideal ann; |
| | 6232 | if(homog(i)==1) |
| | 6233 | { |
| | 6234 | list re=sres(j,0); //the resolution |
| | 6235 | re=minres(re); //minimized resolution |
| | 6236 | } |
| | 6237 | else |
| | 6238 | { |
| | 6239 | list re=mres(i,0); |
| | 6240 | } |
| | 6241 | ann=AnnExt_R(cod,re); |
| | 6242 | return(ann); |
| | 6243 | } |
| | 6244 | example |
| | 6245 | { "EXAMPLE:"; echo = 2; |
| | 6246 | ring r = 0,(x,y,z),dp; |
| | 6247 | ideal i=intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
| | 6248 | equidimMaxEHV(i); |
| | 6249 | } |
| | 6250 | |
| | 6251 | proc testPrimary(list pr, ideal k) |
| | 6252 | "USAGE: testPrimary(pr,k); pr a list, k an ideal. |
| | 6253 | ASSUME: pr is the result of primdecGTZ(k) or primdecSY(k). |
| | 6254 | RETURN: int, 1 if the intersection of the ideals in pr is k, 0 if not |
| | 6255 | EXAMPLE: example testPrimary; shows an example |
| | 6256 | " |
| | 6257 | { |
| | 6258 | int i; |
| | 6259 | pr=reconvList(pr); |
| | 6260 | ideal j=pr[1]; |
| | 6261 | for (i=2;i<=size(pr)/2;i++) |
| | 6262 | { |
| | 6263 | j=intersect(j,pr[2*i-1]); |
| | 6264 | } |
| | 6265 | return(idealsEqual(j,k)); |
| | 6266 | } |
| | 6267 | example |
| | 6268 | { "EXAMPLE:"; echo = 2; |
| | 6269 | ring r = 32003,(x,y,z),dp; |
| | 6270 | poly p = z2+1; |
| | 6271 | poly q = z4+2; |
| | 6272 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
| | 6273 | list pr = primdecGTZ(i); |
| | 6274 | testPrimary(pr,i); |
| | 6275 | } |
| | 6276 | |
| | 6277 | /////////////////////////////////////////////////////////////////////////////// |
| | 6278 | proc zerodec(ideal I) |
| | 6279 | "USAGE: zerodec(I); I ideal |
| | 6280 | ASSUME: I is zero-dimensional, the characteristic of the ground field is 0 |
| | 6281 | RETURN: list of primary ideals, the zero-dimensional decomposition of I |
| | 6282 | NOTE: The algorithm (of Monico), works well only for a small total number |
| | 6283 | of solutions (@code{vdim(std(I))} should be < 100) and without |
| | 6284 | parameters. In practice, it works also in large characteristic p>0 |
| | 6285 | but may fail for small p. |
| | 6286 | @* If printlevel > 0 (default = 0) additional information is displayed. |
| | 6287 | EXAMPLE: example zerodec; shows an example |
| | 6288 | " |
| | 6289 | { |
| | 6290 | if(attrib(basering,"global")!=1) |
| | 6291 | { |
| | 6292 | ERROR( |
| | 6293 | "// Not implemented for this ordering, please change to global ordering." |
| | 6294 | ); |
| | 6295 | } |
| | 6296 | def R=basering; |
| | 6297 | poly q; |
| | 6298 | int j,time; |
| | 6299 | matrix m; |
| | 6300 | list re; |
| | 6301 | poly va=var(1); |
| | 6302 | ideal J=groebner(I); |
| | 6303 | ideal ba=kbase(J); |
| | 6304 | int d=vdim(J); |
| | 6305 | dbprint(printlevel-voice+2,"// multiplicity of ideal : "+ string(d)); |
| | 6306 | //------ compute matrix of multiplication on R/I with generic element p ----- |
| | 6307 | int e=nvars(basering); |
| | 6308 | poly p=randomLast(100)[e]+random(-50,50); //the generic element |
| | 6309 | matrix n[d][d]; |
| | 6310 | time = timer; |
| | 6311 | for(j=2;j<=e;j++) |
| | 6312 | { |
| | 6313 | va=va*var(j); |
| | 6314 | } |
| | 6315 | for(j=1;j<=d;j++) |
| | 6316 | { |
| | 6317 | q=reduce(p*ba[j],J); |
| | 6318 | m=coeffs(q,ba,va); |
| | 6319 | n[j,1..d]=m[1..d,1]; |
| | 6320 | } |
| | 6321 | dbprint(printlevel-voice+2, |
| | 6322 | "// time for computing multiplication matrix (with generic element) : "+ |
| | 6323 | string(timer-time)); |
| | 6324 | //---------------- compute characteristic polynomial of matrix -------------- |
| | 6325 | execute("ring P1=("+charstr(R)+"),T,dp;"); |
| | 6326 | matrix n=imap(R,n); |
| | 6327 | time = timer; |
| | 6328 | poly charpol=det(n-T*freemodule(d)); |
| | 6329 | dbprint(printlevel-voice+2,"// time for computing char poly: "+ |
| | 6330 | string(timer-time)); |
| | 6331 | //------------------- factorize characteristic polynomial ------------------- |
| | 6332 | //check first if constant term of charpoly is != 0 (which is true for |
| | 6333 | //sufficiently generic element) |
| | 6334 | if(charpol[size(charpol)]!=0) |
| | 6335 | { |
| | 6336 | time = timer; |
| | 6337 | list fac=factor(charpol); |
| | 6338 | testFactor(fac,charpol); |
| | 6339 | dbprint(printlevel-voice+2,"// time for factorizing char poly: "+ |
| | 6340 | string(timer-time)); |
| | 6341 | int f=size(fac[1]); |
| | 6342 | //--------------------------- the irreducible case -------------------------- |
| | 6343 | if(f==1) |
| | 6344 | { |
| | 6345 | setring R; |
| | 6346 | re=I; |
| | 6347 | return(re); |
| | 6348 | } |
| | 6349 | //---------------------------- the reducible case --------------------------- |
| | 6350 | //if f_i are the irreducible factors of charpoly, mult=ri, then <I,g_i^ri> |
| | 6351 | //are the primary components where g_i = f_i(p). However, substituting p in |
| | 6352 | //f_i may result in a huge object although the final result may be small. |
| | 6353 | //Hence it is better to simultaneously reduce with I. For this we need a new |
| | 6354 | //ring. |
| | 6355 | execute("ring P=("+charstr(R)+"),(T,"+varstr(R)+"),(dp(1),dp);"); |
| | 6356 | list rfac=imap(P1,fac); |
| | 6357 | intvec ov=option(get);; |
| | 6358 | option(redSB); |
| | 6359 | list re1; |
| | 6360 | ideal new = T-imap(R,p),imap(R,J); |
| | 6361 | attrib(new, "isSB",1); //we know that new is a standard basis |
| | 6362 | for(j=1;j<=f;j++) |
| | 6363 | { |
| | 6364 | re1[j]=reduce(rfac[1][j]^rfac[2][j],new); |
| | 6365 | } |
| | 6366 | setring R; |
| | 6367 | re = imap(P,re1); |
| | 6368 | for(j=1;j<=f;j++) |
| | 6369 | { |
| | 6370 | J=I,re[j]; |
| | 6371 | re[j]=interred(J); |
| | 6372 | } |
| | 6373 | option(set,ov); |
| | 6374 | return(re); |
| | 6375 | } |
| | 6376 | else |
| | 6377 | //------------------- choice of generic element failed ------------------- |
| | 6378 | { |
| | 6379 | dbprint(printlevel-voice+2,"// try new generic element!"); |
| | 6380 | setring R; |
| | 6381 | return(zerodec(I)); |
| | 6382 | } |
| | 6383 | } |
| | 6384 | example |
| | 6385 | { "EXAMPLE:"; echo = 2; |
| | 6386 | ring r = 0,(x,y),dp; |
| | 6387 | ideal i = x2-2,y2-2; |
| | 6388 | list pr = zerodec(i); |
| | 6389 | pr; |
| | 6390 | } |
| | 6391 | /////////////////////////////////////////////////////////////////////////////// |
| | 6392 | static proc newDecompStep(ideal i, list #) |
| | 6393 | "USAGE: newDecompStep(i); i ideal (for primary decomposition) |
| | 6394 | newDecompStep(i,1); (for the associated primes of dimension of i) |
| | 6395 | newDecompStep(i,2); (for the minimal associated primes) |
| | 6396 | newDecompStep(i,3); (for the absolute primary decomposition (not tested!)) |
| | 6397 | "oneIndep"; (for using only one max indep set) |
| | 6398 | "intersect"; (returns alse the intersection of the components founded) |
| | 6399 | |
| | 6400 | RETURN: list = list of primary ideals and their associated primes |
| | 6401 | (at even positions in the list) |
| | 6402 | (resp. a list of the minimal associated primes) |
| | 6403 | NOTE: Algorithm of Gianni/Trager/Zacharias |
| | 6404 | EXAMPLE: example newDecompStep; shows an example |
| | 6405 | " |
| | 6406 | { |
| | 6407 | intvec op,@vv; |
| | 6408 | def @P = basering; |
| | 6409 | list primary,indep,ltras; |
| | 6410 | intvec @vh,isat,@w; |
| | 6411 | int @wr,@k,@n,@m,@n1,@n2,@n3,homo,seri,keepdi,abspri,ab,nn; |
| | 6412 | ideal peek=i; |
| | 6413 | ideal ser,tras; |
| | 6414 | list data; |
| | 6415 | list result; |
| | 6416 | intvec @hilb; |
| | 6417 | int isS=(attrib(i,"isSB")==1); |
| | 6418 | |
| | 6419 | // Debug |
| | 6420 | dbprint(printlevel - voice, "newDecompStep, v2.0"); |
| | 6421 | |
| | 6422 | string indepOption = "allIndep"; |
| | 6423 | string intersectOption = "noIntersect"; |
| | 6424 | |
| | 6425 | if(size(#)>0) |
| | 6426 | { |
| | 6427 | int count = 1; |
| | 6428 | if(typeof(#[count]) == "string") |
| | 6429 | { |
| | 6430 | if ((#[count] == "oneIndep") or (#[count] == "allIndep")) |
| | 6431 | { |
| | 6432 | indepOption = #[count]; |
| | 6433 | count++; |
| | 6434 | } |
| | 6435 | } |
| | 6436 | if(typeof(#[count]) == "string") |
| | 6437 | { |
| | 6438 | if ((#[count] == "intersect") or (#[count] == "noIntersect")) |
| | 6439 | { |
| | 6440 | intersectOption = #[count]; |
| | 6441 | count++; |
| | 6442 | } |
| | 6443 | } |
| | 6444 | if((typeof(#[count]) == "int") or (typeof(#[count]) == "number")) |
| | 6445 | { |
| | 6446 | if ((#[count]==1)||(#[count]==2)||(#[count]==3)) |
| | 6447 | { |
| | 6448 | @wr=#[count]; |
| | 6449 | if(@wr==3){abspri = 1; @wr = 0;} |
| | 6450 | count++; |
| | 6451 | } |
| | 6452 | } |
| | 6453 | if(size(#)>count) |
| | 6454 | { |
| | 6455 | seri=1; |
| | 6456 | peek=#[count + 1]; |
| | 6457 | ser=#[count + 2]; |
| | 6458 | } |
| | 6459 | } |
| | 6460 | if(abspri) |
| | 6461 | { |
| | 6462 | list absprimary,abskeep,absprimarytmp,abskeeptmp; |
| | 6463 | } |
| | 6464 | homo=homog(i); |
| | 6465 | if(homo==1) |
| | 6466 | { |
| | 6467 | if(attrib(i,"isSB")!=1) |
| | 6468 | { |
| | 6469 | //ltras=mstd(i); |
| | 6470 | tras=groebner(i); |
| | 6471 | ltras=tras,tras; |
| | 6472 | attrib(ltras[1],"isSB",1); |
| | 6473 | } |
| | 6474 | else |
| | 6475 | { |
| | 6476 | ltras=i,i; |
| | 6477 | attrib(ltras[1],"isSB",1); |
| | 6478 | } |
| | 6479 | tras = ltras[1]; |
| | 6480 | attrib(tras,"isSB",1); |
| | 6481 | if(dim(tras)==0) |
| | 6482 | { |
| | 6483 | primary[1]=ltras[2]; |
| | 6484 | primary[2]=maxideal(1); |
| | 6485 | if(@wr>0) |
| | 6486 | { |
| | 6487 | list l; |
| | 6488 | l[2]=maxideal(1); |
| | 6489 | l[1]=maxideal(1); |
| | 6490 | if (intersectOption == "intersect") |
| | 6491 | { |
| | 6492 | return(list(l, maxideal(1))); |
| | 6493 | } |
| | 6494 | else |
| | 6495 | { |
| | 6496 | return(l); |
| | 6497 | } |
| | 6498 | } |
| | 6499 | if (intersectOption == "intersect") |
| | 6500 | { |
| | 6501 | return(list(primary, primary[1])); |
| | 6502 | } |
| | 6503 | else |
| | 6504 | { |
| | 6505 | return(primary); |
| | 6506 | } |
| | 6507 | } |
| | 6508 | for(@n=1;@n<=nvars(basering);@n++) |
| | 6509 | { |
| | 6510 | @w[@n]=ord(var(@n)); |
| | 6511 | } |
| | 6512 | @hilb=hilb(tras,1,@w); |
| | 6513 | intvec keephilb=@hilb; |
| | 6514 | } |
| | 6515 | |
| | 6516 | //---------------------------------------------------------------- |
| | 6517 | //i is the zero-ideal |
| | 6518 | //---------------------------------------------------------------- |
| | 6519 | |
| | 6520 | if(size(i)==0) |
| | 6521 | { |
| | 6522 | primary=i,i; |
| | 6523 | if (intersectOption == "intersect") |
| | 6524 | { |
| | 6525 | return(list(primary, i)); |
| | 6526 | } |
| | 6527 | else |
| | 6528 | { |
| | 6529 | return(primary); |
| | 6530 | } |
| | 6531 | } |
| | 6532 | |
| | 6533 | //---------------------------------------------------------------- |
| | 6534 | //pass to the lexicographical ordering and compute a standardbasis |
| | 6535 | //---------------------------------------------------------------- |
| | 6536 | |
| | 6537 | int lp=islp(); |
| | 6538 | |
| | 6539 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
| | 6540 | op=option(get); |
| | 6541 | option(redSB); |
| | 6542 | |
| | 6543 | ideal ser=fetch(@P,ser); |
| | 6544 | if(homo==1) |
| | 6545 | { |
| | 6546 | if(!lp) |
| | 6547 | { |
| | 6548 | ideal @j=std(fetch(@P,i),@hilb,@w); |
| | 6549 | } |
| | 6550 | else |
| | 6551 | { |
| | 6552 | ideal @j=fetch(@P,tras); |
| | 6553 | attrib(@j,"isSB",1); |
| | 6554 | } |
| | 6555 | } |
| | 6556 | else |
| | 6557 | { |
| | 6558 | if(lp&&isS) |
| | 6559 | { |
| | 6560 | ideal @j=fetch(@P,i); |
| | 6561 | attrib(@j,"isSB",1); |
| | 6562 | } |
| | 6563 | else |
| | 6564 | { |
| | 6565 | ideal @j=groebner(fetch(@P,i)); |
| | 6566 | } |
| | 6567 | } |
| | 6568 | option(set,op); |
| | 6569 | if(seri==1) |
| | 6570 | { |
| | 6571 | ideal peek=fetch(@P,peek); |
| | 6572 | attrib(peek,"isSB",1); |
| | 6573 | } |
| | 6574 | else |
| | 6575 | { |
| | 6576 | ideal peek=@j; |
| | 6577 | } |
| | 6578 | if((size(ser)==0)&&(!abspri)) |
| | 6579 | { |
| | 6580 | ideal fried; |
| | 6581 | @n=size(@j); |
| | 6582 | for(@k=1;@k<=@n;@k++) |
| | 6583 | { |
| | 6584 | if(deg(lead(@j[@k]))==1) |
| | 6585 | { |
| | 6586 | fried[size(fried)+1]=@j[@k]; |
| | 6587 | @j[@k]=0; |
| | 6588 | } |
| | 6589 | } |
| | 6590 | if(size(fried)==nvars(basering)) |
| | 6591 | { |
| | 6592 | setring @P; |
| | 6593 | primary[1]=i; |
| | 6594 | primary[2]=i; |
| | 6595 | if (intersectOption == "intersect") |
| | 6596 | { |
| | 6597 | return(list(primary, i)); |
| | 6598 | } |
| | 6599 | else |
| | 6600 | { |
| | 6601 | return(primary); |
| | 6602 | } |
| | 6603 | } |
| | 6604 | if(size(fried)>0) |
| | 6605 | { |
| | 6606 | string newva; |
| | 6607 | string newma; |
| | 6608 | for(@k=1;@k<=nvars(basering);@k++) |
| | 6609 | { |
| | 6610 | @n1=0; |
| | 6611 | for(@n=1;@n<=size(fried);@n++) |
| | 6612 | { |
| | 6613 | if(leadmonom(fried[@n])==var(@k)) |
| | 6614 | { |
| | 6615 | @n1=1; |
| | 6616 | break; |
| | 6617 | } |
| | 6618 | } |
| | 6619 | if(@n1==0) |
| | 6620 | { |
| | 6621 | newva=newva+string(var(@k))+","; |
| | 6622 | newma=newma+string(var(@k))+","; |
| | 6623 | } |
| | 6624 | else |
| | 6625 | { |
| | 6626 | newma=newma+string(0)+","; |
| | 6627 | } |
| | 6628 | } |
| | 6629 | newva[size(newva)]=")"; |
| | 6630 | newma[size(newma)]=";"; |
| | 6631 | execute("ring @deirf=("+charstr(gnir)+"),("+newva+",lp;"); |
| | 6632 | execute("map @kappa=gnir,"+newma); |
| | 6633 | ideal @j= @kappa(@j); |
| | 6634 | @j=simplify(@j, 2); |
| | 6635 | attrib(@j,"isSB",1); |
| | 6636 | result = newDecompStep(@j, indepOption, intersectOption, @wr); |
| | 6637 | if (intersectOption == "intersect") |
| | 6638 | { |
| | 6639 | list pr = result[1]; |
| | 6640 | ideal intersection = result[2]; |
| | 6641 | } |
| | 6642 | else |
| | 6643 | { |
| | 6644 | list pr = result; |
| | 6645 | } |
| | 6646 | |
| | 6647 | setring gnir; |
| | 6648 | list pr=imap(@deirf,pr); |
| | 6649 | for(@k=1;@k<=size(pr);@k++) |
| | 6650 | { |
| | 6651 | @j=pr[@k]+fried; |
| | 6652 | pr[@k]=@j; |
| | 6653 | } |
| | 6654 | if (intersectOption == "intersect") |
| | 6655 | { |
| | 6656 | ideal intersection = imap(@deirf, intersection); |
| | 6657 | @j = intersection + fried; |
| | 6658 | intersection = @j; |
| | 6659 | } |
| | 6660 | setring @P; |
| | 6661 | if (intersectOption == "intersect") |
| | 6662 | { |
| | 6663 | return(list(imap(gnir,pr), imap(gnir,intersection))); |
| | 6664 | } |
| | 6665 | else |
| | 6666 | { |
| | 6667 | return(imap(gnir,pr)); |
| | 6668 | } |
| | 6669 | } |
| | 6670 | } |
| | 6671 | //---------------------------------------------------------------- |
| | 6672 | //j is the ring |
| | 6673 | //---------------------------------------------------------------- |
| | 6674 | |
| | 6675 | if (dim(@j)==-1) |
| | 6676 | { |
| | 6677 | setring @P; |
| | 6678 | primary=ideal(1),ideal(1); |
| | 6679 | if (intersectOption == "intersect") |
| | 6680 | { |
| | 6681 | return(list(primary, ideal(1))); |
| | 6682 | } |
| | 6683 | else |
| | 6684 | { |
| | 6685 | return(primary); |
| | 6686 | } |
| | 6687 | } |
| | 6688 | |
| | 6689 | //---------------------------------------------------------------- |
| | 6690 | // the case of one variable |
| | 6691 | //---------------------------------------------------------------- |
| | 6692 | |
| | 6693 | if(nvars(basering)==1) |
| | 6694 | { |
| | 6695 | list fac=factor(@j[1]); |
| | 6696 | list gprimary; |
| | 6697 | poly generator; |
| | 6698 | ideal gIntersection; |
| | 6699 | for(@k=1;@k<=size(fac[1]);@k++) |
| | 6700 | { |
| | 6701 | if(@wr==0) |
| | 6702 | { |
| | 6703 | gprimary[2*@k-1]=ideal(fac[1][@k]^fac[2][@k]); |
| | 6704 | gprimary[2*@k]=ideal(fac[1][@k]); |
| | 6705 | } |
| | 6706 | else |
| | 6707 | { |
| | 6708 | gprimary[2*@k-1]=ideal(fac[1][@k]); |
| | 6709 | gprimary[2*@k]=ideal(fac[1][@k]); |
| | 6710 | } |
| | 6711 | if (intersectOption == "intersect") |
| | 6712 | { |
| | 6713 | generator = generator * fac[1][@k]; |
| | 6714 | } |
| | 6715 | } |
| | 6716 | if (intersectOption == "intersect") |
| | 6717 | { |
| | 6718 | gIntersection = generator; |
| | 6719 | } |
| | 6720 | setring @P; |
| | 6721 | primary=fetch(gnir,gprimary); |
| | 6722 | if (intersectOption == "intersect") |
| | 6723 | { |
| | 6724 | ideal intersection = fetch(gnir,gIntersection); |
| | 6725 | } |
| | 6726 | |
| | 6727 | //HIER |
| | 6728 | if(abspri) |
| | 6729 | { |
| | 6730 | list resu,tempo; |
| | 6731 | string absotto; |
| | 6732 | for(ab=1;ab<=size(primary)/2;ab++) |
| | 6733 | { |
| | 6734 | absotto= absFactorize(primary[2*ab][1],77); |
| | 6735 | tempo=primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
| | 6736 | resu[ab]=tempo; |
| | 6737 | } |
| | 6738 | primary=resu; |
| | 6739 | intersection = 1; |
| | 6740 | for(ab=1;ab<=size(primary);ab++) |
| | 6741 | { |
| | 6742 | intersection = intersect(intersection, primary[ab][2]); |
| | 6743 | } |
| | 6744 | } |
| | 6745 | if (intersectOption == "intersect") |
| | 6746 | { |
| | 6747 | return(list(primary, intersection)); |
| | 6748 | } |
| | 6749 | else |
| | 6750 | { |
| | 6751 | return(primary); |
| | 6752 | } |
| | 6753 | } |
| | 6754 | |
| | 6755 | //------------------------------------------------------------------ |
| | 6756 | //the zero-dimensional case |
| | 6757 | //------------------------------------------------------------------ |
| | 6758 | if (dim(@j)==0) |
| | 6759 | { |
| | 6760 | op=option(get); |
| | 6761 | option(redSB); |
| | 6762 | list gprimary= newZero_decomp(@j,ser,@wr); |
| | 6763 | |
| | 6764 | setring @P; |
| | 6765 | primary=fetch(gnir,gprimary); |
| | 6766 | |
| | 6767 | if(size(ser)>0) |
| | 6768 | { |
| | 6769 | primary=cleanPrimary(primary); |
| | 6770 | } |
| | 6771 | //HIER |
| | 6772 | if(abspri) |
| | 6773 | { |
| | 6774 | list resu,tempo; |
| | 6775 | string absotto; |
| | 6776 | for(ab=1;ab<=size(primary)/2;ab++) |
| | 6777 | { |
| | 6778 | absotto= absFactorize(primary[2*ab][1],77); |
| | 6779 | tempo=primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
| | 6780 | resu[ab]=tempo; |
| | 6781 | } |
| | 6782 | primary=resu; |
| | 6783 | } |
| | 6784 | if (intersectOption == "intersect") |
| | 6785 | { |
| | 6786 | return(list(primary, fetch(gnir,@j))); |
| | 6787 | } |
| | 6788 | else |
| | 6789 | { |
| | 6790 | return(primary); |
| | 6791 | } |
| | 6792 | } |
| | 6793 | |
| | 6794 | poly @gs,@gh,@p; |
| | 6795 | string @va,quotring; |
| | 6796 | list quprimary,htprimary,collectprimary,lsau,lnew,allindep,restindep; |
| | 6797 | ideal @h; |
| | 6798 | int jdim=dim(@j); |
| | 6799 | list fett; |
| | 6800 | int lauf,di,newtest; |
| | 6801 | //------------------------------------------------------------------ |
| | 6802 | //search for a maximal independent set indep,i.e. |
| | 6803 | //look for subring such that the intersection with the ideal is zero |
| | 6804 | //j intersected with K[var(indep[3]+1),...,var(nvar] is zero, |
| | 6805 | //indep[1] is the new varstring and indep[2] the string for block-ordering |
| | 6806 | //------------------------------------------------------------------ |
| | 6807 | if(@wr!=1) |
| | 6808 | { |
| | 6809 | allindep = newMaxIndependSetLp(@j, indepOption); |
| | 6810 | for(@m=1;@m<=size(allindep);@m++) |
| | 6811 | { |
| | 6812 | if(allindep[@m][3]==jdim) |
| | 6813 | { |
| | 6814 | di++; |
| | 6815 | indep[di]=allindep[@m]; |
| | 6816 | } |
| | 6817 | else |
| | 6818 | { |
| | 6819 | lauf++; |
| | 6820 | restindep[lauf]=allindep[@m]; |
| | 6821 | } |
| | 6822 | } |
| | 6823 | } |
| | 6824 | else |
| | 6825 | { |
| | 6826 | indep = newMaxIndependSetLp(@j, indepOption); |
| | 6827 | } |
| | 6828 | |
| | 6829 | ideal jkeep=@j; |
| | 6830 | if(ordstr(@P)[1]=="w") |
| | 6831 | { |
| | 6832 | execute("ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),("+ordstr(@P)+");"); |
| | 6833 | } |
| | 6834 | else |
| | 6835 | { |
| | 6836 | execute( "ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),(C,dp);"); |
| | 6837 | } |
| | 6838 | |
| | 6839 | if(homo==1) |
| | 6840 | { |
| | 6841 | if((ordstr(@P)[3]=="d")||(ordstr(@P)[1]=="d")||(ordstr(@P)[1]=="w") |
| | 6842 | ||(ordstr(@P)[3]=="w")) |
| | 6843 | { |
| | 6844 | ideal jwork=imap(@P,tras); |
| | 6845 | attrib(jwork,"isSB",1); |
| | 6846 | } |
| | 6847 | else |
| | 6848 | { |
| | 6849 | ideal jwork=std(imap(gnir,@j),@hilb,@w); |
| | 6850 | } |
| | 6851 | } |
| | 6852 | else |
| | 6853 | { |
| | 6854 | ideal jwork=groebner(imap(gnir,@j)); |
| | 6855 | } |
| | 6856 | list hquprimary; |
| | 6857 | poly @p,@q; |
| | 6858 | ideal @h,fac,ser; |
| | 6859 | //Aenderung================ |
| | 6860 | ideal @Ptest=1; |
| | 6861 | //========================= |
| | 6862 | di=dim(jwork); |
| | 6863 | keepdi=di; |
| | 6864 | |
| | 6865 | ser = 1; |
| | 6866 | |
| | 6867 | setring gnir; |
| | 6868 | for(@m=1; @m<=size(indep); @m++) |
| | 6869 | { |
| | 6870 | data[1] = indep[@m]; |
| | 6871 | result = newReduction(@j, ser, @hilb, @w, jdim, abspri, @wr, data); |
| | 6872 | quprimary = quprimary + result[1]; |
| | 6873 | if(abspri) |
| | 6874 | { |
| | 6875 | absprimary = absprimary + result[2]; |
| | 6876 | abskeep = abskeep + result[3]; |
| | 6877 | } |
| | 6878 | @h = result[5]; |
| | 6879 | ser = result[4]; |
| | 6880 | if(size(@h)>0) |
| | 6881 | { |
| | 6882 | //--------------------------------------------------------------- |
| | 6883 | //we change to @Phelp to have the ordering dp for saturation |
| | 6884 | //--------------------------------------------------------------- |
| | 6885 | |
| | 6886 | setring @Phelp; |
| | 6887 | @h=imap(gnir,@h); |
| | 6888 | //Aenderung================================== |
| | 6889 | if(defined(@LL)){kill @LL;} |
| | 6890 | list @LL=minSat(jwork,@h); |
| | 6891 | @Ptest=intersect(@Ptest,@LL[1]); |
| | 6892 | ser = intersect(ser, @LL[1]); |
| | 6893 | //=========================================== |
| | 6894 | |
| | 6895 | if(@wr!=1) |
| | 6896 | { |
| | 6897 | //Aenderung================================== |
| | 6898 | @q=@LL[2]; |
| | 6899 | //=========================================== |
| | 6900 | //@q=minSat(jwork,@h)[2]; |
| | 6901 | } |
| | 6902 | else |
| | 6903 | { |
| | 6904 | fac=ideal(0); |
| | 6905 | for(lauf=1;lauf<=ncols(@h);lauf++) |
| | 6906 | { |
| | 6907 | if(deg(@h[lauf])>0) |
| | 6908 | { |
| | 6909 | fac=fac+factorize(@h[lauf],1); |
| | 6910 | } |
| | 6911 | } |
| | 6912 | fac=simplify(fac,6); |
| | 6913 | @q=1; |
| | 6914 | for(lauf=1;lauf<=size(fac);lauf++) |
| | 6915 | { |
| | 6916 | @q=@q*fac[lauf]; |
| | 6917 | } |
| | 6918 | } |
| | 6919 | jwork = std(jwork,@q); |
| | 6920 | keepdi = dim(jwork); |
| | 6921 | if(keepdi < di) |
| | 6922 | { |
| | 6923 | setring gnir; |
| | 6924 | @j = imap(@Phelp, jwork); |
| | 6925 | ser = imap(@Phelp, ser); |
| | 6926 | break; |
| | 6927 | } |
| | 6928 | if(homo == 1) |
| | 6929 | { |
| | 6930 | @hilb = hilb(jwork, 1, @w); |
| | 6931 | } |
| | 6932 | |
| | 6933 | setring gnir; |
| | 6934 | ser = imap(@Phelp, ser); |
| | 6935 | @j = imap(@Phelp, jwork); |
| | 6936 | } |
| | 6937 | } |
| | 6938 | |
| | 6939 | if((size(quprimary)==0)&&(@wr==1)) |
| | 6940 | { |
| | 6941 | @j=ideal(1); |
| | 6942 | quprimary[1]=ideal(1); |
| | 6943 | quprimary[2]=ideal(1); |
| | 6944 | } |
| | 6945 | if((size(quprimary)==0)) |
| | 6946 | { |
| | 6947 | keepdi = di - 1; |
| | 6948 | quprimary[1]=ideal(1); |
| | 6949 | quprimary[2]=ideal(1); |
| | 6950 | } |
| | 6951 | //--------------------------------------------------------------- |
| | 6952 | //notice that j=sat(j,gh) intersected with (j,gh^n) |
| | 6953 | //we finished with sat(j,gh) and have to start with (j,gh^n) |
| | 6954 | //--------------------------------------------------------------- |
| | 6955 | if((deg(@j[1])!=0)&&(@wr!=1)) |
| | 6956 | { |
| | 6957 | if(size(quprimary)>0) |
| | 6958 | { |
| | 6959 | setring @Phelp; |
| | 6960 | ser=imap(gnir,ser); |
| | 6961 | |
| | 6962 | hquprimary=imap(gnir,quprimary); |
| | 6963 | if(@wr==0) |
| | 6964 | { |
| | 6965 | //Aenderung==================================================== |
| | 6966 | //HIER STATT DURCHSCHNITT SATURIEREN! |
| | 6967 | ideal htest=@Ptest; |
| | 6968 | /* |
| | 6969 | ideal htest=hquprimary[1]; |
| | 6970 | for (@n1=2;@n1<=size(hquprimary)/2;@n1++) |
| | 6971 | { |
| | 6972 | htest=intersect(htest,hquprimary[2*@n1-1]); |
| | 6973 | } |
| | 6974 | */ |
| | 6975 | //============================================================= |
| | 6976 | } |
| | 6977 | else |
| | 6978 | { |
| | 6979 | ideal htest=hquprimary[2]; |
| | 6980 | |
| | 6981 | for (@n1=2;@n1<=size(hquprimary)/2;@n1++) |
| | 6982 | { |
| | 6983 | htest=intersect(htest,hquprimary[2*@n1]); |
| | 6984 | } |
| | 6985 | } |
| | 6986 | |
| | 6987 | if(size(ser)>0) |
| | 6988 | { |
| | 6989 | ser=intersect(htest,ser); |
| | 6990 | } |
| | 6991 | else |
| | 6992 | { |
| | 6993 | ser=htest; |
| | 6994 | } |
| | 6995 | setring gnir; |
| | 6996 | ser=imap(@Phelp,ser); |
| | 6997 | } |
| | 6998 | if(size(reduce(ser,peek,1))!=0) |
| | 6999 | { |
| | 7000 | for(@m=1;@m<=size(restindep);@m++) |
| | 7001 | { |
| | 7002 | // if(restindep[@m][3]>=keepdi) |
| | 7003 | // { |
| | 7004 | isat=0; |
| | 7005 | @n2=0; |
| | 7006 | |
| | 7007 | if(restindep[@m][1]==varstr(basering)) |
| | 7008 | //the good case, nothing to do, just to have the same notations |
| | 7009 | //change the ring |
| | 7010 | { |
| | 7011 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
| | 7012 | varstr(basering)+"),("+ordstr(basering)+");"); |
| | 7013 | ideal @j=fetch(gnir,jkeep); |
| | 7014 | attrib(@j,"isSB",1); |
| | 7015 | } |
| | 7016 | else |
| | 7017 | { |
| | 7018 | @va=string(maxideal(1)); |
| | 7019 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
| | 7020 | restindep[@m][1]+"),(" +restindep[@m][2]+");"); |
| | 7021 | execute("map phi=gnir,"+@va+";"); |
| | 7022 | op=option(get); |
| | 7023 | option(redSB); |
| | 7024 | if(homo==1) |
| | 7025 | { |
| | 7026 | ideal @j=std(phi(jkeep),keephilb,@w); |
| | 7027 | } |
| | 7028 | else |
| | 7029 | { |
| | 7030 | ideal @j=groebner(phi(jkeep)); |
| | 7031 | } |
| | 7032 | ideal ser=phi(ser); |
| | 7033 | option(set,op); |
| | 7034 | } |
| | 7035 | |
| | 7036 | for (lauf=1;lauf<=size(@j);lauf++) |
| | 7037 | { |
| | 7038 | fett[lauf]=size(@j[lauf]); |
| | 7039 | } |
| | 7040 | //------------------------------------------------------------------ |
| | 7041 | //we have now the following situation: |
| | 7042 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may |
| | 7043 | //pass to this quotientring, j is their still a standardbasis, the |
| | 7044 | //leading coefficients of the polynomials there (polynomials in |
| | 7045 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
| | 7046 | //we need their ggt, gh, because of the following: |
| | 7047 | //let (j:gh^n)=(j:gh^infinity) then |
| | 7048 | //j*K(var(nnp+1),..,var(nva))[..the rest..] |
| | 7049 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
| | 7050 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
| | 7051 | |
| | 7052 | //------------------------------------------------------------------ |
| | 7053 | |
| | 7054 | //the arrangement for the quotientring |
| | 7055 | // K(var(nnp+1),..,var(nva))[..the rest..] |
| | 7056 | //and the map phi:K[var(1),...,var(nva)] ----> |
| | 7057 | //--->K(var(nnpr+1),..,var(nva))[..the rest..] |
| | 7058 | //------------------------------------------------------------------ |
| | 7059 | |
| | 7060 | quotring=prepareQuotientring(nvars(basering)-restindep[@m][3]); |
| | 7061 | |
| | 7062 | //------------------------------------------------------------------ |
| | 7063 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
| | 7064 | //------------------------------------------------------------------ |
| | 7065 | |
| | 7066 | execute(quotring); |
| | 7067 | |
| | 7068 | // @j considered in the quotientring |
| | 7069 | ideal @j=imap(gnir1,@j); |
| | 7070 | ideal ser=imap(gnir1,ser); |
| | 7071 | |
| | 7072 | kill gnir1; |
| | 7073 | |
| | 7074 | //j is a standardbasis in the quotientring but usually not minimal |
| | 7075 | //here it becomes minimal |
| | 7076 | @j=clearSB(@j,fett); |
| | 7077 | attrib(@j,"isSB",1); |
| | 7078 | |
| | 7079 | //we need later ggt(h[1],...)=gh for saturation |
| | 7080 | ideal @h; |
| | 7081 | |
| | 7082 | for(@n=1;@n<=size(@j);@n++) |
| | 7083 | { |
| | 7084 | @h[@n]=leadcoef(@j[@n]); |
| | 7085 | } |
| | 7086 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..rest..] |
| | 7087 | |
| | 7088 | op=option(get); |
| | 7089 | option(redSB); |
| | 7090 | list uprimary= newZero_decomp(@j,ser,@wr); |
| | 7091 | //HIER |
| | 7092 | if(abspri) |
| | 7093 | { |
| | 7094 | ideal II; |
| | 7095 | ideal jmap; |
| | 7096 | map sigma; |
| | 7097 | nn=nvars(basering); |
| | 7098 | map invsigma=basering,maxideal(1); |
| | 7099 | for(ab=1;ab<=size(uprimary)/2;ab++) |
| | 7100 | { |
| | 7101 | II=uprimary[2*ab]; |
| | 7102 | attrib(II,"isSB",1); |
| | 7103 | if(deg(II[1])!=vdim(II)) |
| | 7104 | { |
| | 7105 | jmap=randomLast(50); |
| | 7106 | sigma=basering,jmap; |
| | 7107 | jmap[nn]=2*var(nn)-jmap[nn]; |
| | 7108 | invsigma=basering,jmap; |
| | 7109 | II=groebner(sigma(II)); |
| | 7110 | } |
| | 7111 | absprimarytmp[ab]= absFactorize(II[1],77); |
| | 7112 | II=var(nn); |
| | 7113 | abskeeptmp[ab]=string(invsigma(II)); |
| | 7114 | invsigma=basering,maxideal(1); |
| | 7115 | } |
| | 7116 | } |
| | 7117 | option(set,op); |
| | 7118 | |
| | 7119 | //we need the intersection of the ideals in the list quprimary with |
| | 7120 | //the polynomialring, i.e. let q=(f1,...,fr) in the quotientring |
| | 7121 | //such an ideal but fi polynomials, then the intersection of q with |
| | 7122 | //the polynomialring is the saturation of the ideal generated by |
| | 7123 | //f1,...,fr with respect toh which is the lcm of the leading |
| | 7124 | //coefficients of the fi considered in the quotientring: |
| | 7125 | //this is coded in saturn |
| | 7126 | |
| | 7127 | list saturn; |
| | 7128 | ideal hpl; |
| | 7129 | |
| | 7130 | for(@n=1;@n<=size(uprimary);@n++) |
| | 7131 | { |
| | 7132 | hpl=0; |
| | 7133 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
| | 7134 | { |
| | 7135 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
| | 7136 | } |
| | 7137 | saturn[@n]=hpl; |
| | 7138 | } |
| | 7139 | //------------------------------------------------------------------ |
| | 7140 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
| | 7141 | //back to the polynomialring |
| | 7142 | //------------------------------------------------------------------ |
| | 7143 | setring gnir; |
| | 7144 | collectprimary=imap(quring,uprimary); |
| | 7145 | lsau=imap(quring,saturn); |
| | 7146 | @h=imap(quring,@h); |
| | 7147 | |
| | 7148 | kill quring; |
| | 7149 | |
| | 7150 | |
| | 7151 | @n2=size(quprimary); |
| | 7152 | //================NEU========================================= |
| | 7153 | if(deg(quprimary[1][1])<=0){ @n2=0; } |
| | 7154 | //============================================================ |
| | 7155 | |
| | 7156 | @n3=@n2; |
| | 7157 | |
| | 7158 | for(@n1=1;@n1<=size(collectprimary)/2;@n1++) |
| | 7159 | { |
| | 7160 | if(deg(collectprimary[2*@n1][1])>0) |
| | 7161 | { |
| | 7162 | @n2++; |
| | 7163 | quprimary[@n2]=collectprimary[2*@n1-1]; |
| | 7164 | lnew[@n2]=lsau[2*@n1-1]; |
| | 7165 | @n2++; |
| | 7166 | lnew[@n2]=lsau[2*@n1]; |
| | 7167 | quprimary[@n2]=collectprimary[2*@n1]; |
| | 7168 | if(abspri) |
| | 7169 | { |
| | 7170 | absprimary[@n2/2]=absprimarytmp[@n1]; |
| | 7171 | abskeep[@n2/2]=abskeeptmp[@n1]; |
| | 7172 | } |
| | 7173 | } |
| | 7174 | } |
| | 7175 | |
| | 7176 | |
| | 7177 | //here the intersection with the polynomialring |
| | 7178 | //mentioned above is really computed |
| | 7179 | |
| | 7180 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
| | 7181 | { |
| | 7182 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
| | 7183 | { |
| | 7184 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
| | 7185 | quprimary[2*@n]=quprimary[2*@n-1]; |
| | 7186 | } |
| | 7187 | else |
| | 7188 | { |
| | 7189 | if(@wr==0) |
| | 7190 | { |
| | 7191 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
| | 7192 | } |
| | 7193 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
| | 7194 | } |
| | 7195 | } |
| | 7196 | if(@n2>=@n3+2) |
| | 7197 | { |
| | 7198 | setring @Phelp; |
| | 7199 | ser=imap(gnir,ser); |
| | 7200 | hquprimary=imap(gnir,quprimary); |
| | 7201 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
| | 7202 | { |
| | 7203 | if(@wr==0) |
| | 7204 | { |
| | 7205 | ser=intersect(ser,hquprimary[2*@n-1]); |
| | 7206 | } |
| | 7207 | else |
| | 7208 | { |
| | 7209 | ser=intersect(ser,hquprimary[2*@n]); |
| | 7210 | } |
| | 7211 | } |
| | 7212 | setring gnir; |
| | 7213 | ser=imap(@Phelp,ser); |
| | 7214 | } |
| | 7215 | |
| | 7216 | // } |
| | 7217 | } |
| | 7218 | //HIER |
| | 7219 | if(abspri) |
| | 7220 | { |
| | 7221 | list resu,tempo; |
| | 7222 | for(ab=1;ab<=size(quprimary)/2;ab++) |
| | 7223 | { |
| | 7224 | if (deg(quprimary[2*ab][1])!=0) |
| | 7225 | { |
| | 7226 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
| | 7227 | absprimary[ab],abskeep[ab]; |
| | 7228 | resu[ab]=tempo; |
| | 7229 | } |
| | 7230 | } |
| | 7231 | quprimary=resu; |
| | 7232 | @wr=3; |
| | 7233 | } |
| | 7234 | if(size(reduce(ser,peek,1))!=0) |
| | 7235 | { |
| | 7236 | if(@wr>0) |
| | 7237 | { |
| | 7238 | // The following line was dropped to avoid the recursion step: |
| | 7239 | //htprimary=newDecompStep(@j,@wr,peek,ser); |
| | 7240 | htprimary = list(); |
| | 7241 | } |
| | 7242 | else |
| | 7243 | { |
| | 7244 | // The following line was dropped to avoid the recursion step: |
| | 7245 | //htprimary=newDecompStep(@j,peek,ser); |
| | 7246 | htprimary = list(); |
| | 7247 | } |
| | 7248 | // here we collect now both results primary(sat(j,gh)) |
| | 7249 | // and primary(j,gh^n) |
| | 7250 | @n=size(quprimary); |
| | 7251 | if (deg(quprimary[1][1])<=0) { @n=0; } |
| | 7252 | for (@k=1;@k<=size(htprimary);@k++) |
| | 7253 | { |
| | 7254 | quprimary[@n+@k]=htprimary[@k]; |
| | 7255 | } |
| | 7256 | } |
| | 7257 | } |
| | 7258 | } |
| | 7259 | else |
| | 7260 | { |
| | 7261 | if(abspri) |
| | 7262 | { |
| | 7263 | list resu,tempo; |
| | 7264 | for(ab=1;ab<=size(quprimary)/2;ab++) |
| | 7265 | { |
| | 7266 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
| | 7267 | absprimary[ab],abskeep[ab]; |
| | 7268 | resu[ab]=tempo; |
| | 7269 | } |
| | 7270 | quprimary=resu; |
| | 7271 | } |
| | 7272 | } |
| | 7273 | //--------------------------------------------------------------------------- |
| | 7274 | //back to the ring we started with |
| | 7275 | //the final result: primary |
| | 7276 | //--------------------------------------------------------------------------- |
| | 7277 | |
| | 7278 | setring @P; |
| | 7279 | primary=imap(gnir,quprimary); |
| | 7280 | |
| | 7281 | if (intersectOption == "intersect") |
| | 7282 | { |
| | 7283 | return(list(primary, imap(gnir, ser))); |
| | 7284 | } |
| | 7285 | else |
| | 7286 | { |
| | 7287 | return(primary); |
| | 7288 | } |
| | 7289 | } |
| | 7290 | example |
| | 7291 | { "EXAMPLE:"; echo = 2; |
| | 7292 | ring r = 32003,(x,y,z),lp; |
| | 7293 | poly p = z2+1; |
| | 7294 | poly q = z4+2; |
| | 7295 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
| | 7296 | list pr= newDecompStep(i); |
| | 7297 | pr; |
| | 7298 | testPrimary( pr, i); |
| | 7299 | } |
| | 7300 | |
| | 7301 | // This was part of proc decomp. |
| | 7302 | // In proc newDecompStep, used for the computation of the minimal associated primes, |
| | 7303 | // this part was separated as a soubrutine to make the code more clear. |
| | 7304 | // Also, since the reduction is performed twice in proc newDecompStep, it should use both times this routine. |
| | 7305 | // This is not yet implemented, since the reduction is not exactly the same and some changes should be made. |
| | 7306 | static proc newReduction(ideal @j, ideal ser, intvec @hilb, intvec @w, int jdim, int abspri, int @wr, list data) |
| | 7307 | { |
| | 7308 | string @va; |
| | 7309 | string quotring; |
| | 7310 | intvec op; |
| | 7311 | intvec @vv; |
| | 7312 | def gnir = basering; |
| | 7313 | ideal isat=0; |
| | 7314 | int @n; |
| | 7315 | int @n1 = 0; |
| | 7316 | int @n2 = 0; |
| | 7317 | int @n3 = 0; |
| | 7318 | int homo = homog(@j); |
| | 7319 | int lauf; |
| | 7320 | int @k; |
| | 7321 | list fett; |
| | 7322 | int keepdi; |
| | 7323 | list collectprimary; |
| | 7324 | list lsau; |
| | 7325 | list lnew; |
| | 7326 | ideal @h; |
| | 7327 | |
| | 7328 | list indepInfo = data[1]; |
| | 7329 | list quprimary = list(); |
| | 7330 | |
| | 7331 | //if(abspri) |
| | 7332 | //{ |
| | 7333 | int ab; |
| | 7334 | list absprimarytmp,abskeeptmp; |
| | 7335 | list absprimary, abskeep; |
| | 7336 | //} |
| | 7337 | // Debug |
| | 7338 | dbprint(printlevel - voice, "newReduction, v2.0"); |
| | 7339 | |
| | 7340 | if((indepInfo[1]==varstr(basering))) // &&(@m==1) |
| | 7341 | //this is the good case, nothing to do, just to have the same notations |
| | 7342 | //change the ring |
| | 7343 | { |
| | 7344 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
| | 7345 | +ordstr(basering)+");"); |
| | 7346 | ideal @j = fetch(gnir, @j); |
| | 7347 | attrib(@j,"isSB",1); |
| | 7348 | ideal ser = fetch(gnir, ser); |
| | 7349 | } |
| | 7350 | else |
| | 7351 | { |
| | 7352 | @va=string(maxideal(1)); |
| | 7353 | //Aenderung============== |
| | 7354 | //if(@m==1) |
| | 7355 | //{ |
| | 7356 | // @j=fetch(@P,i); |
| | 7357 | //} |
| | 7358 | //======================= |
| | 7359 | execute("ring gnir1 = ("+charstr(basering)+"),("+indepInfo[1]+"),(" |
| | 7360 | +indepInfo[2]+");"); |
| | 7361 | execute("map phi=gnir,"+@va+";"); |
| | 7362 | op=option(get); |
| | 7363 | option(redSB); |
| | 7364 | if(homo==1) |
| | 7365 | { |
| | 7366 | ideal @j=std(phi(@j),@hilb,@w); |
| | 7367 | } |
| | 7368 | else |
| | 7369 | { |
| | 7370 | ideal @j=groebner(phi(@j)); |
| | 7371 | } |
| | 7372 | ideal ser=phi(ser); |
| | 7373 | |
| | 7374 | option(set,op); |
| | 7375 | } |
| | 7376 | if((deg(@j[1])==0)||(dim(@j)<jdim)) |
| | 7377 | { |
| | 7378 | setring gnir; |
| | 7379 | break; |
| | 7380 | } |
| | 7381 | for (lauf=1;lauf<=size(@j);lauf++) |
| | 7382 | { |
| | 7383 | fett[lauf]=size(@j[lauf]); |
| | 7384 | } |
| | 7385 | //------------------------------------------------------------------------ |
| | 7386 | //we have now the following situation: |
| | 7387 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
| | 7388 | //to this quotientring, j is their still a standardbasis, the |
| | 7389 | //leading coefficients of the polynomials there (polynomials in |
| | 7390 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
| | 7391 | //we need their ggt, gh, because of the following: let |
| | 7392 | //(j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..the rest..] |
| | 7393 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
| | 7394 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
| | 7395 | |
| | 7396 | //------------------------------------------------------------------------ |
| | 7397 | |
| | 7398 | //arrangement for quotientring K(var(nnp+1),..,var(nva))[..the rest..] and |
| | 7399 | //map phi:K[var(1),...,var(nva)] --->K(var(nnpr+1),..,var(nva))[..rest..] |
| | 7400 | //------------------------------------------------------------------------ |
| | 7401 | |
| | 7402 | quotring=prepareQuotientring(nvars(basering)-indepInfo[3]); |
| | 7403 | |
| | 7404 | //--------------------------------------------------------------------- |
| | 7405 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
| | 7406 | //--------------------------------------------------------------------- |
| | 7407 | |
| | 7408 | ideal @jj=lead(@j); //!! vorn vereinbaren |
| | 7409 | execute(quotring); |
| | 7410 | |
| | 7411 | ideal @jj=imap(gnir1,@jj); |
| | 7412 | @vv=clearSBNeu(@jj,fett); //!! vorn vereinbaren |
| | 7413 | setring gnir1; |
| | 7414 | @k=size(@j); |
| | 7415 | for (lauf=1;lauf<=@k;lauf++) |
| | 7416 | { |
| | 7417 | if(@vv[lauf]==1) |
| | 7418 | { |
| | 7419 | @j[lauf]=0; |
| | 7420 | } |
| | 7421 | } |
| | 7422 | @j=simplify(@j,2); |
| | 7423 | setring quring; |
| | 7424 | // @j considered in the quotientring |
| | 7425 | ideal @j=imap(gnir1,@j); |
| | 7426 | |
| | 7427 | ideal ser=imap(gnir1,ser); |
| | 7428 | |
| | 7429 | kill gnir1; |
| | 7430 | |
| | 7431 | //j is a standardbasis in the quotientring but usually not minimal |
| | 7432 | //here it becomes minimal |
| | 7433 | |
| | 7434 | attrib(@j,"isSB",1); |
| | 7435 | |
| | 7436 | //we need later ggt(h[1],...)=gh for saturation |
| | 7437 | ideal @h; |
| | 7438 | if(deg(@j[1])>0) |
| | 7439 | { |
| | 7440 | for(@n=1;@n<=size(@j);@n++) |
| | 7441 | { |
| | 7442 | @h[@n]=leadcoef(@j[@n]); |
| | 7443 | } |
| | 7444 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..the rest..] |
| | 7445 | op=option(get); |
| | 7446 | option(redSB); |
| | 7447 | |
| | 7448 | int zeroMinAss = @wr; |
| | 7449 | if (@wr == 2) {zeroMinAss = 1;} |
| | 7450 | list uprimary= newZero_decomp(@j, ser, zeroMinAss); |
| | 7451 | |
| | 7452 | //HIER |
| | 7453 | if(abspri) |
| | 7454 | { |
| | 7455 | ideal II; |
| | 7456 | ideal jmap; |
| | 7457 | map sigma; |
| | 7458 | nn=nvars(basering); |
| | 7459 | map invsigma=basering,maxideal(1); |
| | 7460 | for(ab=1;ab<=size(uprimary)/2;ab++) |
| | 7461 | { |
| | 7462 | II=uprimary[2*ab]; |
| | 7463 | attrib(II,"isSB",1); |
| | 7464 | if(deg(II[1])!=vdim(II)) |
| | 7465 | { |
| | 7466 | jmap=randomLast(50); |
| | 7467 | sigma=basering,jmap; |
| | 7468 | jmap[nn]=2*var(nn)-jmap[nn]; |
| | 7469 | invsigma=basering,jmap; |
| | 7470 | II=groebner(sigma(II)); |
| | 7471 | } |
| | 7472 | absprimarytmp[ab]= absFactorize(II[1],77); |
| | 7473 | II=var(nn); |
| | 7474 | abskeeptmp[ab]=string(invsigma(II)); |
| | 7475 | invsigma=basering,maxideal(1); |
| | 7476 | } |
| | 7477 | } |
| | 7478 | option(set,op); |
| | 7479 | } |
| | 7480 | else |
| | 7481 | { |
| | 7482 | list uprimary; |
| | 7483 | uprimary[1]=ideal(1); |
| | 7484 | uprimary[2]=ideal(1); |
| | 7485 | } |
| | 7486 | //we need the intersection of the ideals in the list quprimary with the |
| | 7487 | //polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
| | 7488 | //but fi polynomials, then the intersection of q with the polynomialring |
| | 7489 | //is the saturation of the ideal generated by f1,...,fr with respect to |
| | 7490 | //h which is the lcm of the leading coefficients of the fi considered in |
| | 7491 | //in the quotientring: this is coded in saturn |
| | 7492 | |
| | 7493 | list saturn; |
| | 7494 | ideal hpl; |
| | 7495 | |
| | 7496 | for(@n=1;@n<=size(uprimary);@n++) |
| | 7497 | { |
| | 7498 | uprimary[@n]=interred(uprimary[@n]); // temporary fix |
| | 7499 | hpl=0; |
| | 7500 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
| | 7501 | { |
| | 7502 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
| | 7503 | } |
| | 7504 | saturn[@n]=hpl; |
| | 7505 | } |
| | 7506 | |
| | 7507 | //-------------------------------------------------------------------- |
| | 7508 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
| | 7509 | //back to the polynomialring |
| | 7510 | //--------------------------------------------------------------------- |
| | 7511 | setring gnir; |
| | 7512 | |
| | 7513 | collectprimary=imap(quring,uprimary); |
| | 7514 | lsau=imap(quring,saturn); |
| | 7515 | @h=imap(quring,@h); |
| | 7516 | |
| | 7517 | kill quring; |
| | 7518 | |
| | 7519 | @n2=size(quprimary); |
| | 7520 | @n3=@n2; |
| | 7521 | |
| | 7522 | for(@n1=1;@n1<=size(collectprimary)/2;@n1++) |
| | 7523 | { |
| | 7524 | if(deg(collectprimary[2*@n1][1])>0) |
| | 7525 | { |
| | 7526 | @n2++; |
| | 7527 | quprimary[@n2]=collectprimary[2*@n1-1]; |
| | 7528 | lnew[@n2]=lsau[2*@n1-1]; |
| | 7529 | @n2++; |
| | 7530 | lnew[@n2]=lsau[2*@n1]; |
| | 7531 | quprimary[@n2]=collectprimary[2*@n1]; |
| | 7532 | if(abspri) |
| | 7533 | { |
| | 7534 | absprimary[@n2/2]=absprimarytmp[@n1]; |
| | 7535 | abskeep[@n2/2]=abskeeptmp[@n1]; |
| | 7536 | } |
| | 7537 | } |
| | 7538 | } |
| | 7539 | |
| | 7540 | //here the intersection with the polynomialring |
| | 7541 | //mentioned above is really computed |
| | 7542 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
| | 7543 | { |
| | 7544 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
| | 7545 | { |
| | 7546 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
| | 7547 | quprimary[2*@n]=quprimary[2*@n-1]; |
| | 7548 | } |
| | 7549 | else |
| | 7550 | { |
| | 7551 | if(@wr==0) |
| | 7552 | { |
| | 7553 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
| | 7554 | } |
| | 7555 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
| | 7556 | } |
| | 7557 | } |
| | 7558 | |
| | 7559 | return(quprimary, absprimary, abskeep, ser, @h); |
| | 7560 | } |
| | 7561 | |
| | 7562 | |
| | 7563 | //////////////////////////////////////////////////////////////////////////// |
| | 7564 | |
| | 7565 | |
| | 7566 | |
| | 7567 | |
| | 7568 | /////////////////////////////////////////////////////////////////////////////// |
| | 7569 | // Based on minAssGTZ |
| | 7570 | |
| | 7571 | proc minAss(ideal i,list #) |
| | 7572 | "USAGE: minAss(I[, l]); i ideal, l list (optional) of parameters, same as minAssGTZ |
| | 7573 | RETURN: a list, the minimal associated prime ideals of I. |
| | 7574 | NOTE: Designed for characteristic 0, works also in char k > 0 based |
| | 7575 | on an algorithm of Yokoyama |
| | 7576 | EXAMPLE: example minAss; shows an example |
| | 7577 | " |
| | 7578 | { |
| | 7579 | return(minAssGTZ(i,#)); |
| | 7580 | } |
| | 7581 | example |
| | 7582 | { "EXAMPLE:"; echo = 2; |
| | 7583 | ring r = 0, (x, y, z), dp; |
| | 7584 | poly p = z2 + 1; |
| | 7585 | poly q = z3 + 2; |
| | 7586 | ideal i = p * q^2, y - z2; |
| | 7587 | list pr = minAss(i); |
| | 7588 | pr; |
| | 7589 | } |
| | 7590 | |
| | 7591 | |
| | 7592 | /////////////////////////////////////////////////////////////////////////////// |
| | 7593 | // |
| | 7594 | // Computes the minimal associated primes of I via Laplagne algorithm, |
| | 7595 | // using primary decomposition in the zero dimensional case. |
| | 7596 | // For reduction to the zerodimensional case, it uses the procedure |
| | 7597 | // decomp, with some modifications to avoid the recursion. |
| | 7598 | // |
| | 7599 | |
| | 7600 | static proc minAssSL(ideal I) |
| | 7601 | // Input = I, ideal |
| | 7602 | // Output = primaryDec where primaryDec is the list of the minimal |
| | 7603 | // associated primes and the primary components corresponding to these primes. |
| | 7604 | { |
| | 7605 | ideal P = 1; |
| | 7606 | list pd = list(); |
| | 7607 | int k; |
| | 7608 | int stop = 0; |
| | 7609 | list primaryDec = list(); |
| | 7610 | |
| | 7611 | while (stop == 0) |
| | 7612 | { |
| | 7613 | // Debug |
| | 7614 | dbprint(printlevel - voice, "// We call minAssSLIteration to find new prime ideals!"); |
| | 7615 | pd = minAssSLIteration(I, P); |
| | 7616 | // Debug |
| | 7617 | dbprint(printlevel - voice, "// Output of minAssSLIteration:"); |
| | 7618 | dbprint(printlevel - voice, pd); |
| | 7619 | if (size(pd[1]) > 0) |
| | 7620 | { |
| | 7621 | primaryDec = primaryDec + pd[1]; |
| | 7622 | // Debug |
| | 7623 | dbprint(printlevel - voice, "// We intersect the prime ideals obtained."); |
| | 7624 | P = intersect(P, pd[2]); |
| | 7625 | // Debug |
| | 7626 | dbprint(printlevel - voice, "// Intersection finished."); |
| | 7627 | } |
| | 7628 | else |
| | 7629 | { |
| | 7630 | stop = 1; |
| | 7631 | } |
| | 7632 | } |
| | 7633 | |
| | 7634 | // Returns only the primary components, not the radical. |
| | 7635 | return(primaryDec); |
| | 7636 | } |
| | 7637 | |
| | 7638 | /////////////////////////////////////////////////////////////////////////////// |
| | 7639 | // Given an ideal I and an ideal P (intersection of some minimal prime ideals |
| | 7640 | // associated to I), it calculates new minimal prime ideals associated to I |
| | 7641 | // which were not used to calculate P. |
| | 7642 | // This version uses Primary Decomposition in the zerodimensional case. |
| | 7643 | static proc minAssSLIteration(ideal I, ideal P); |
| | 7644 | { |
| | 7645 | int k = 1; |
| | 7646 | int good = 0; |
| | 7647 | list primaryDec = list(); |
| | 7648 | // Debug |
| | 7649 | dbprint (printlevel-voice, "// We search for an element in P - sqrt(I)."); |
| | 7650 | while ((k <= size(P)) and (good == 0)) |
| | 7651 | { |
| | 7652 | good = 1 - rad_con(P[k], I); |
| | 7653 | k++; |
| | 7654 | } |
| | 7655 | k--; |
| | 7656 | if (good == 0) |
| | 7657 | { |
| | 7658 | // Debug |
| | 7659 | dbprint (printlevel - voice, "// No element was found, P = sqrt(I)."); |
| | 7660 | return (list(primaryDec, ideal(0))); |
| | 7661 | } |
| | 7662 | // Debug |
| | 7663 | dbprint (printlevel - voice, "// We found h = ", P[k]); |
| | 7664 | dbprint (printlevel - voice, "// We calculate the saturation of I with respect to the element just founded."); |
| | 7665 | ideal J = sat(I, P[k])[1]; |
| | 7666 | |
| | 7667 | // Uses decomp from primdec, modified to avoid the recursion. |
| | 7668 | // Debug |
| | 7669 | dbprint(printlevel - voice, "// We do the reduction to the zerodimensional case, via decomp."); |
| | 7670 | |
| | 7671 | primaryDec = newDecompStep(J, "oneIndep", "intersect", 2); |
| | 7672 | // Debug |
| | 7673 | dbprint(printlevel - voice, "// Proc decomp has found", size(primaryDec) / 2, "new primary components."); |
| | 7674 | |
| | 7675 | return(primaryDec); |
| | 7676 | } |
| | 7677 | |
| | 7678 | |
| | 7679 | |
| | 7680 | /////////////////////////////////////////////////////////////////////////////////// |
| | 7681 | // Based on maxIndependSet |
| | 7682 | // Added list # as parameter |
| | 7683 | // If the first element of # is 0, the output is only 1 max indep set. |
| | 7684 | // If no list is specified or #[1] = 1, the output is all the max indep set of the |
| | 7685 | // leading terms ideal. This is the original output of maxIndependSet |
| | 7686 | |
| | 7687 | proc newMaxIndependSetLp(ideal j, list #) |
| | 7688 | "USAGE: newMaxIndependentSetLp(i); i ideal (returns all maximal independent sets of the corresponding leading terms ideal) |
| | 7689 | newMaxIndependentSetLp(i, 0); i ideal (returns only one maximal independent set) |
| | 7690 | RETURN: list = #1. new varstring with the maximal independent set at the end, |
| | 7691 | #2. ordstring with the lp ordering, |
| | 7692 | #3. the number of independent variables |
| | 7693 | NOTE: |
| | 7694 | EXAMPLE: example newMaxIndependentSetLp; shows an example |
| | 7695 | " |
| | 7696 | { |
| | 7697 | int n, k, di; |
| | 7698 | list resu, hilf; |
| | 7699 | string var1, var2; |
| | 7700 | list v = indepSet(j, 0); |
| | 7701 | |
| | 7702 | // SL 2006.04.21 1 Lines modified to use only one independent Set |
| | 7703 | string indepOption; |
| | 7704 | if (size(#) > 0) |
| | 7705 | { |
| | 7706 | indepOption = #[1]; |
| | 7707 | } |
| | 7708 | else |
| | 7709 | { |
| | 7710 | indepOption = "allIndep"; |
| | 7711 | } |
| | 7712 | |
| | 7713 | int nMax; |
| | 7714 | if (indepOption == "allIndep") |
| | 7715 | { |
| | 7716 | nMax = size(v); |
| | 7717 | } |
| | 7718 | else |
| | 7719 | { |
| | 7720 | nMax = 1; |
| | 7721 | } |
| | 7722 | |
| | 7723 | for(n = 1; n <= nMax; n++) |
| | 7724 | // SL 2006.04.21 2 |
| | 7725 | { |
| | 7726 | di = 0; |
| | 7727 | var1 = ""; |
| | 7728 | var2 = ""; |
| | 7729 | for(k = 1; k <= size(v[n]); k++) |
| | 7730 | { |
| | 7731 | if(v[n][k] != 0) |
| | 7732 | { |
| | 7733 | di++; |
| | 7734 | var2 = var2 + "var(" + string(k) + "), "; |
| | 7735 | } |
| | 7736 | else |
| | 7737 | { |
| | 7738 | var1 = var1 + "var(" + string(k) + "), "; |
| | 7739 | } |
| | 7740 | } |
| | 7741 | if(di > 0) |
| | 7742 | { |
| | 7743 | var1 = var1 + var2; |
| | 7744 | var1 = var1[1..size(var1) - 2]; // The "- 2" removes the trailer comma |
| | 7745 | hilf[1] = var1; |
| | 7746 | // SL 2006.21.04 1 The order is now block dp instead of lp |
| | 7747 | //hilf[2] = "dp(" + string(nvars(basering) - di) + "), dp(" + string(di) + ")"; |
| | 7748 | // SL 2006.21.04 2 |
| | 7749 | // For decomp, lp ordering is needed. Nothing is changed. |
| | 7750 | hilf[2] = "lp"; |
| | 7751 | hilf[3] = di; |
| | 7752 | resu[n] = hilf; |
| | 7753 | } |
| | 7754 | else |
| | 7755 | { |
| | 7756 | resu[n] = varstr(basering), ordstr(basering), 0; |
| | 7757 | } |
| | 7758 | } |
| | 7759 | return(resu); |
| | 7760 | } |
| | 7761 | example |
| | 7762 | { "EXAMPLE:"; echo = 2; |
| | 7763 | ring s1 = (0, x, y), (a, b, c, d, e, f, g), lp; |
| | 7764 | ideal i = ea - fbg, fa + be, ec - fdg, fc + de; |
| | 7765 | i = std(i); |
| | 7766 | list l = newMaxIndependSetLp(i); |
| | 7767 | l; |
| | 7768 | i = i, g; |
| | 7769 | l = newMaxIndependSetLp(i); |
| | 7770 | l; |
| | 7771 | |
| | 7772 | ring s = 0, (x, y, z), lp; |
| | 7773 | ideal i = z, yx; |
| | 7774 | list l = newMaxIndependSetLp(i); |
| | 7775 | l; |
| | 7776 | } |
| | 7777 | |
| | 7778 | |
| | 7779 | /////////////////////////////////////////////////////////////////////////////// |
| | 7780 | |
| | 7781 | proc newZero_decomp (ideal j, ideal ser, int @wr, list #) |
| | 7782 | "USAGE: newZero_decomp(j,ser,@wr); j,ser ideals, @wr=0 or 1 |
| | 7783 | (@wr=0 for primary decomposition, @wr=1 for computation of associated |
| | 7784 | primes) |
| | 7785 | if #[1] = "nest", then #[2] indicates the nest level (number of recursive calls) |
| | 7786 | When the nest level is high it indicates that the computation is difficult, |
| | 7787 | and different methods are applied. |
| | 7788 | RETURN: list = list of primary ideals and their radicals (at even positions |
| | 7789 | in the list) if the input is zero-dimensional and a standardbases |
| | 7790 | with respect to lex-ordering |
| | 7791 | If ser!=(0) and ser is contained in j or if j is not zero-dimen- |
| | 7792 | sional then ideal(1),ideal(1) is returned |
| | 7793 | NOTE: Algorithm of Gianni/Trager/Zacharias |
| | 7794 | EXAMPLE: example newZero_decomp; shows an example |
| | 7795 | " |
| | 7796 | { |
| | 7797 | def @P = basering; |
| | 7798 | int uytrewq; |
| | 7799 | int nva = nvars(basering); |
| | 7800 | int @k,@s,@n,@k1,zz; |
| | 7801 | list primary,lres0,lres1,act,@lh,@wh; |
| | 7802 | map phi,psi,phi1,psi1; |
| | 7803 | ideal jmap,jmap1,jmap2,helpprim,@qh,@qht,ser1; |
| | 7804 | intvec @vh,@hilb; |
| | 7805 | string @ri; |
| | 7806 | poly @f; |
| | 7807 | |
| | 7808 | // Debug |
| | 7809 | dbprint(printlevel - voice, "proc newZero_decomp"); |
| | 7810 | |
| | 7811 | if (dim(j)>0) |
| | 7812 | { |
| | 7813 | primary[1]=ideal(1); |
| | 7814 | primary[2]=ideal(1); |
| | 7815 | return(primary); |
| | 7816 | } |
| | 7817 | j=interred(j); |
| | 7818 | |
| | 7819 | attrib(j,"isSB",1); |
| | 7820 | |
| | 7821 | int nestLevel = 0; |
| | 7822 | if (size(#) > 0) |
| | 7823 | { |
| | 7824 | if (typeof(#[1]) == "string") |
| | 7825 | { |
| | 7826 | if (#[1] == "nest") |
| | 7827 | { |
| | 7828 | nestLevel = #[2]; |
| | 7829 | } |
| | 7830 | # = list(); |
| | 7831 | } |
| | 7832 | } |
| | 7833 | |
| | 7834 | if(vdim(j)==deg(j[1])) |
| | 7835 | { |
| | 7836 | act=factor(j[1]); |
| | 7837 | for(@k=1;@k<=size(act[1]);@k++) |
| | 7838 | { |
| | 7839 | @qh=j; |
| | 7840 | if(@wr==0) |
| | 7841 | { |
| | 7842 | @qh[1]=act[1][@k]^act[2][@k]; |
| | 7843 | } |
| | 7844 | else |
| | 7845 | { |
| | 7846 | @qh[1]=act[1][@k]; |
| | 7847 | } |
| | 7848 | primary[2*@k-1]=interred(@qh); |
| | 7849 | @qh=j; |
| | 7850 | @qh[1]=act[1][@k]; |
| | 7851 | primary[2*@k]=interred(@qh); |
| | 7852 | attrib( primary[2*@k-1],"isSB",1); |
| | 7853 | |
| | 7854 | if((size(ser)>0)&&(size(reduce(ser,primary[2*@k-1],1))==0)) |
| | 7855 | { |
| | 7856 | primary[2*@k-1]=ideal(1); |
| | 7857 | primary[2*@k]=ideal(1); |
| | 7858 | } |
| | 7859 | } |
| | 7860 | return(primary); |
| | 7861 | } |
| | 7862 | |
| | 7863 | if(homog(j)==1) |
| | 7864 | { |
| | 7865 | primary[1]=j; |
| | 7866 | if((size(ser)>0)&&(size(reduce(ser,j,1))==0)) |
| | 7867 | { |
| | 7868 | primary[1]=ideal(1); |
| | 7869 | primary[2]=ideal(1); |
| | 7870 | return(primary); |
| | 7871 | } |
| | 7872 | if(dim(j)==-1) |
| | 7873 | { |
| | 7874 | primary[1]=ideal(1); |
| | 7875 | primary[2]=ideal(1); |
| | 7876 | } |
| | 7877 | else |
| | 7878 | { |
| | 7879 | primary[2]=maxideal(1); |
| | 7880 | } |
| | 7881 | return(primary); |
| | 7882 | } |
| | 7883 | |
| | 7884 | //the first element in the standardbase is factorized |
| | 7885 | if(deg(j[1])>0) |
| | 7886 | { |
| | 7887 | act=factor(j[1]); |
| | 7888 | testFactor(act,j[1]); |
| | 7889 | } |
| | 7890 | else |
| | 7891 | { |
| | 7892 | primary[1]=ideal(1); |
| | 7893 | primary[2]=ideal(1); |
| | 7894 | return(primary); |
| | 7895 | } |
| | 7896 | |
| | 7897 | //with the factors new ideals (hopefully the primary decomposition) |
| | 7898 | //are created |
| | 7899 | if(size(act[1])>1) |
| | 7900 | { |
| | 7901 | if(size(#)>1) |
| | 7902 | { |
| | 7903 | primary[1]=ideal(1); |
| | 7904 | primary[2]=ideal(1); |
| | 7905 | primary[3]=ideal(1); |
| | 7906 | primary[4]=ideal(1); |
| | 7907 | return(primary); |
| | 7908 | } |
| | 7909 | for(@k=1;@k<=size(act[1]);@k++) |
| | 7910 | { |
| | 7911 | if(@wr==0) |
| | 7912 | { |
| | 7913 | primary[2*@k-1]=std(j,act[1][@k]^act[2][@k]); |
| | 7914 | } |
| | 7915 | else |
| | 7916 | { |
| | 7917 | primary[2*@k-1]=std(j,act[1][@k]); |
| | 7918 | } |
| | 7919 | if((act[2][@k]==1)&&(vdim(primary[2*@k-1])==deg(act[1][@k]))) |
| | 7920 | { |
| | 7921 | primary[2*@k] = primary[2*@k-1]; |
| | 7922 | } |
| | 7923 | else |
| | 7924 | { |
| | 7925 | primary[2*@k] = primaryTest(primary[2*@k-1],act[1][@k]); |
| | 7926 | } |
| | 7927 | } |
| | 7928 | } |
| | 7929 | else |
| | 7930 | { |
| | 7931 | primary[1]=j; |
| | 7932 | if((size(#)>0)&&(act[2][1]>1)) |
| | 7933 | { |
| | 7934 | act[2]=1; |
| | 7935 | primary[1]=std(primary[1],act[1][1]); |
| | 7936 | } |
| | 7937 | if(@wr!=0) |
| | 7938 | { |
| | 7939 | primary[1]=std(j,act[1][1]); |
| | 7940 | } |
| | 7941 | if((act[2][1]==1)&&(vdim(primary[1])==deg(act[1][1]))) |
| | 7942 | { |
| | 7943 | primary[2]=primary[1]; |
| | 7944 | } |
| | 7945 | else |
| | 7946 | { |
| | 7947 | primary[2]=primaryTest(primary[1],act[1][1]); |
| | 7948 | } |
| | 7949 | } |
| | 7950 | |
| | 7951 | if(size(#)==0) |
| | 7952 | { |
| | 7953 | primary=splitPrimary(primary,ser,@wr,act); |
| | 7954 | } |
| | 7955 | |
| | 7956 | if((voice>=6)&&(char(basering)<=181)) |
| | 7957 | { |
| | 7958 | primary=splitCharp(primary); |
| | 7959 | } |
| | 7960 | |
| | 7961 | if((@wr==2)&&(npars(basering)>0)&&(voice>=6)&&(char(basering)>0)) |
| | 7962 | { |
| | 7963 | //the prime decomposition of Yokoyama in characteristic p |
| | 7964 | &nbs |
