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 |