Changeset 08d847 in git
 Timestamp:
 May 12, 2008, 4:55:16 PM (15 years ago)
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 (u'jengelhdatetime', 'ceac47cbc86fe4a15902392bdbb9bd2ae0ea02c6')(u'spielwiese', 'a800fe4b3e9d37a38c5a10cc0ae9dfa0c15a4ee6')
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 69fb8d5542e0a6b5c38f18c4c9462982a9f00e03
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 f7a73ed1209d4fc117e52cdbefbe3d5ce10759e2
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Singular/LIB/freegb.lib
rf7a73ed r08d847 1 1 ////////////////////////////////////////////////////////////////////////////// 2 version="$Id: freegb.lib,v 1. 8 20080313 19:26:16 levandov Exp $";2 version="$Id: freegb.lib,v 1.9 20080512 14:55:16 levandov Exp $"; 3 3 category="Noncommutative"; 4 4 info=" 5 LIBRARY: ratgb.lib Twosided Noncommutative Groebner bases in Free Algebras5 LIBRARY: freegb.lib Twosided Noncommutative Groebner bases in Free Algebras 6 6 AUTHOR: Viktor Levandovskyy, levandov@math.rwthaachen.de 7 7 8 8 PROCEDURES: 9 9 freegbasis(list L, int n); compute twosided Groebner basis of ideal, encoded via L, up to degree n 10 lst2str(list L); convert a list (of modules) into polynomials in free algebra 11 mod2str(module M); convert a module into a polynomial in free algebra 10 11 freegbRing(int d); creates a ring with d blocks of shifted original variables 12 13 CONVERSION ROUTINES: 14 15 lp2lstr(ideal K, def save): converts letterplace ideal to a list of modules 16 lst2str(list L[,int n]); convert a list (of modules) into polynomials in free algebra 17 mod2str(module M[,int n]); convert a module into a polynomial in free algebra 18 vct2str(module M[,int n]); convert a vector into a word in free algebra 12 19 " 13 20 … … 19 26 // V[1+i] = the corresponding symbol 20 27 28 LIB "discretize.lib"; // for replace 21 29 LIB "qhmoduli.lib"; // for Max 22 30 … … 95 103 } 96 104 97 proc lst2str(list L )98 "USAGE: lst2str(L ); L a list of modules105 proc lst2str(list L, list #) 106 "USAGE: lst2str(L[,n]); L a list of modules, n an optional integer 99 107 RETURN: list (of strings) 100 108 PURPOSE: convert a list (of modules) into polynomials in free algebra 101 109 EXAMPLE: example lst2str; shows examples 110 NOTE: if an optional integer is not 0, stars signs are used in multiplication 102 111 " 103 112 { 104 113 // returns a list of strings 105 114 // being sentences in words built from L 115 // if #[1] = 1, use * between generators 116 int useStar = 0; 117 if ( size(#)>0 ) 118 { 119 if (#[1]) 120 { 121 useStar = 1; 122 } 123 } 106 124 int i; 107 125 int s = size(L); … … 111 129 if ((typeof(L[i]) == "module")  (typeof(L[i]) == "matrix") ) 112 130 { 113 N[i] = mod2str(L[i] );131 N[i] = mod2str(L[i],useStar); 114 132 } 115 133 else … … 129 147 list L; L[1] = M; L[2] = N; 130 148 lst2str(L); 131 } 132 133 134 proc mod2str(module M) 135 "USAGE: mod2str(M); M a module 149 lst2str(L[1],1); 150 } 151 152 153 proc mod2str(module M, list #) 154 "USAGE: mod2str(M[,n]); M a module, n an optional integer 136 155 RETURN: string 137 PURPOSE: convert a module sinto a polynomial in free algebra156 PURPOSE: convert a module into a polynomial in free algebra 138 157 EXAMPLE: example mod2str; shows examples 158 NOTE: if an optional integer is not 0, stars signs are used in multiplication 139 159 " 140 160 { 141 161 // returns a string 142 162 // a sentence in words built from M 163 // if #[1] = 1, use * between generators 164 int useStar = 0; 165 if ( size(#)>0 ) 166 { 167 if (#[1]) 168 { 169 useStar = 1; 170 } 171 } 143 172 int i; 144 173 int s = ncols(M); … … 147 176 for(i=1; i<=s; i++) 148 177 { 149 mp = vct2str(M[i] );178 mp = vct2str(M[i],useStar); 150 179 if (mp[1] == "") 151 180 { … … 169 198 module M = [1,x,y,x,y],[2,y,x,y,x],[6,x,y,y,x,y]; 170 199 mod2str(M); 171 } 172 173 proc vct2str(vector v) 174 { 200 mod2str(M,1); 201 } 202 203 proc vct2str(vector v, list #) 204 "USAGE: vct2str(v[,n]); v a vector, n an optional integer 205 RETURN: string 206 PURPOSE: convert a vector into a word in free algebra 207 EXAMPLE: example vct2str; shows examples 208 NOTE: if an optional integer is not 0, stars signs are used in multiplication 209 " 210 { 211 // if #[1] = 1, use * between generators 212 int useStar = 0; 213 if ( size(#)>0 ) 214 { 215 if (#[1]) 216 { 217 useStar = 1; 218 } 219 } 175 220 int ppl = printlevelvoice+2; 176 221 // for a word, encoded by v … … 192 237 if (p==1) 193 238 { 194 vs = vs + string(v[i+1]); 239 if (useStar && (size(vs) >0)) { vs = vs + "*"; } 240 vs = vs + string(v[i+1]); 195 241 } 196 242 else //power … … 212 258 for(j=1;j<=p;j++) 213 259 { 214 vs = vs + vp; 260 if (useStar && (size(vs) >0)) { vs = vs + "*"; } 261 vs = vs + vp; 215 262 } 216 263 } … … 229 276 } 230 277 } 278 if (useStar && (size(scf) >0) && (scf!="") ) { scf = scf + "*"; } 231 279 vs = scf + vs; 232 280 return(vs); … … 234 282 example 235 283 { 284 "EXAMPLE:"; echo = 2; 236 285 ring r = (0,a),(x,y3,z(1)),dp; 237 286 vector v = [7,x,y3^4,x2,z(1)^3]; 238 287 vct2str(v); 288 vct2str(v,1); 239 289 vector w = [7a^5+6a,x,y3,y3,x,z(1),z(1)]; 240 290 vct2str(w); 291 vct2str(w,1); 241 292 } 242 293 … … 267 318 example 268 319 { 320 "EXAMPLE:"; echo = 2; 269 321 ring r = 0,(x,y),dp; 270 322 poly f = xy+1; … … 282 334 proc id2words(ideal I, int d) 283 335 { 336 // NOT FINISHED 284 337 // input: ideal I of polys in letterplace notation 285 338 // in the ring with d real vars … … 318 371 { 319 372 "EXAMPLE:"; echo = 2; 320 ring r = 0,(x(1),y(1),z(1) ),dp;373 ring r = 0,(x(1),y(1),z(1),x(2),y(2),z(2)),dp; 321 374 ideal I = x(1)*y(2) z(1)*x(2); 322 375 id2words(I,3); 323 376 } 324 325 326 377 327 378 proc mono2word(poly p, int d) … … 1384 1435 // t*h  h*t, t*f  f*t, t*e  e*t 1385 1436 } 1437 1438 proc adem(int i, int j) 1439 { 1440 // produces Adem relations for i<2j in char 0 1441 // assume: 0<i<2j 1442 // requires presence of vars up to i+j 1443 if ( (i<0)  (i >= 2*j) ) 1444 { 1445 ERROR("arguments out of range"); return(0); 1446 } 1447 ring @r = 0,(s(i+j..0)),lp; 1448 poly p,q; 1449 number n; 1450 int ii = i div 2; int k; 1451 // k=0 => s(0)=1 1452 n = binomial(j1,i); 1453 q = n*s(i+j)*s(0); 1454 printf("k=0, term=%s",q); 1455 p = p + q; 1456 for (k=1; k<= ii; k++) 1457 { 1458 n = binomial(jk1,i2*k); 1459 q = n*s(i+jk)*s(k);; 1460 printf("k=%s, term=%s",k,q); 1461 p = p + q; 1462 } 1463 poly AdemRel = p; 1464 export AdemRel; 1465 return(@r); 1466 } 1467 example 1468 { 1469 "EXAMPLE:"; echo = 2; 1470 def A = adem(2,5); 1471 setring A; 1472 AdemRel; 1473 } 1474 1475 /* 1476 1,1: 0 1477 1,2: s(3)*s(0) == s(3) > def for s(3):=s(1)s(2) 1478 2,1: adm 1479 2,2: s(3)*s(1) == s(1)s(2)s(1) 1480 1,3: 0 ( since 2*s(4)*s(0) = 0 mod 2) 1481 3,1: adm 1482 2,3: s(5)*s(0)+s(4)*s(1) == s(5)+s(4)*s(1) 1483 3,2: 0 1484 3,3: s(5)*s(1) 1485 1,4: 3*s(5)*s(0) == s(5) > def for s(5):=s(1)*s(4) 1486 4,1: adm 1487 2,4: 3*s(6)*s(0)+s(5)*s(1) == s(6) + s(5)*s(1) == s(6) + s(1)*s(4)*s(1) 1488 4,2: adm 1489 4,3: s(5)*s(2) 1490 3,4: s(7)*s(0)+2*s(6)*s(1) == s(7) > def for s(7):=s(3)*s(4) 1491 4,4: s(7)*s(1)+s(6)*s(2) 1492 */ 1493 1494 /* s1,s2: 1495 s1*s1 =0, s2*s2 = s1*s2*s1 1496 */ 1497 1498 /* 1499 try char 0: 1500 s1,s2: 1501 s1*s1 =0, s2*s2 = s1*s2*s1, s(1)*s(3)== s(1)*s(1)*s(3) == 0 = 2*s(4) >def for s(4) 1502 hence 2==0! only in char 2 1503 */ 1504 1505 proc adem2mod(int n) 1506 { 1507 // Adem rels modulo 2 1508 } 1509 1510 proc stringpoly2lplace(string s) 1511 { 1512 // decomposes sentence into terms 1513 s = replace(s,newline,""); // get rid of newlines 1514 s = replace(s," ",""); // get rid of empties 1515 //arith symbols: +, 1516 // decompose into words with coeffs 1517 list LS; 1518 int i,j,ie,je,k,cnt; 1519 // s[1]="" situation 1520 if (s[1]=="") 1521 { 1522 LS = stringpoly2lplace(string(s[2..size(s)])); 1523 LS[1] = string(""+string(LS[1])); 1524 return(LS); 1525 } 1526 i = find(s,"",2); 1527 // i==1 might happen if the 1st symbol coeff is negative 1528 j = find(s,"+"); 1529 list LL; 1530 if (i==j) 1531 { 1532 "return a monomial"; 1533 // that is both are 0 > s is a monomial 1534 LS[1] = s; 1535 return(LS); 1536 } 1537 if (i==0) 1538 { 1539 "i==0 situation"; 1540 // no minuses at all => pluses only 1541 cnt++; 1542 LS[cnt] = string(s[1..j1]); 1543 s = s[j+1..size(s)]; 1544 while (s!= "") 1545 { 1546 j = find(s,"+"); 1547 cnt++; 1548 if (j==0) 1549 { 1550 LS[cnt] = string(s); 1551 s = ""; 1552 } 1553 else 1554 { 1555 LS[cnt] = string(s[1..j1]); 1556 s = s[j+1..size(s)]; 1557 } 1558 } 1559 return(LS); 1560 } 1561 if (j==0) 1562 { 1563 "j==0 situation"; 1564 // no pluses at all except the lead coef => the rest are minuses only 1565 cnt++; 1566 LS[cnt] = string(s[1..i1]); 1567 s = s[i..size(s)]; 1568 while (s!= "") 1569 { 1570 i = find(s,"",2); 1571 cnt++; 1572 if (i==0) 1573 { 1574 LS[cnt] = string(s); 1575 s = ""; 1576 } 1577 else 1578 { 1579 LS[cnt] = string(s[1..i1]); 1580 s = s[i..size(s)]; 1581 } 1582 } 1583 return(LS); 1584 } 1585 // now i, j are nonzero 1586 if (i>j) 1587 { 1588 "i>j situation"; 1589 // + comes first, at place j 1590 cnt++; 1591 // "cnt:"; cnt; "j:"; j; 1592 LS[cnt] = string(s[1..j1]); 1593 s = s[j+1..size(s)]; 1594 LL = stringpoly2lplace(s); 1595 LS = LS + LL; 1596 kill LL; 1597 return(LS); 1598 } 1599 else 1600 { 1601 "j>i situation"; 1602 //  might come first, at place i 1603 if (i>1) 1604 { 1605 cnt++; 1606 LS[cnt] = string(s[1..i1]); 1607 s = s[i..size(s)]; 1608 } 1609 else 1610 { 1611 // i==1> minus at leadcoef 1612 ie = find(s,"",i+1); 1613 je = find(s,"+",i+1); 1614 if (je == ie) 1615 { 1616 "ie=je situation"; 1617 //monomial 1618 cnt++; 1619 LS[cnt] = s; 1620 return(LS); 1621 } 1622 if (je < ie) 1623 { 1624 "je<ie situation"; 1625 // + comes first 1626 cnt++; 1627 LS[cnt] = s[1..je1]; 1628 s = s[je+1..size(s)]; 1629 } 1630 else 1631 { 1632 // ie < je 1633 "ie<je situation"; 1634 cnt++; 1635 LS[cnt] = s[1..ie1]; 1636 s = s[ie..size(s)]; 1637 } 1638 } 1639 "going into recursion with "+s; 1640 LL = stringpoly2lplace(s); 1641 LS = LS + LL; 1642 return(LS); 1643 } 1644 } 1645 example 1646 { 1647 "EXAMPLE:"; echo = 2; 1648 string s = "x*y+y*z+z*t"; // + only 1649 stringpoly2lplace(s); 1650 string s2 = "x*y  y*zz*t*w*w"; // +1,  only 1651 stringpoly2lplace(s2); 1652 string s3 = "x*y + y  2*x +7*w*w*w"; 1653 stringpoly2lplace(s3); 1654 } 1655 1656 proc addplaces(list L) 1657 { 1658 // adds places to the list of strings 1659 // according to their order in the list 1660 int s = size(L); 1661 int i; 1662 for (i=1; i<=s; i++) 1663 { 1664 if (typeof(L[i]) == "string") 1665 { 1666 L[i] = L[i] + "(" + string(i) + ")"; 1667 } 1668 else 1669 { 1670 ERROR("entry of type string expected"); 1671 return(0); 1672 } 1673 } 1674 return(L); 1675 } 1676 example 1677 { 1678 "EXAMPLE:"; echo = 2; 1679 string a = "f1"; string b = "f2"; 1680 list L = a,b,a; 1681 addplaces(L); 1682 } 1683 1684 proc sent2lplace(string s) 1685 { 1686 list L = stringpoly2lplace(s); 1687 int i; int ss = size(L); 1688 for(i=1; i<=ss; i++) 1689 { 1690 L[i] = str2lplace(L[i]); 1691 } 1692 return(L); 1693 } 1694 example 1695 { 1696 "EXAMPLE:"; echo = 2; 1697 ring r = 0,(f2,f1),dp; 1698 string s = "f2*f1*f1  2*f1*f2*f1+ f1*f1*f2"; 1699 sent2lplace(s); 1700 } 1701 1702 proc testnumber(string s) 1703 { 1704 string t; 1705 if (s[1]=="") 1706 { 1707 // two situations: either there's a negative number 1708 t = s[2..size(s)]; 1709 if (testnumber(t)) 1710 { 1711 //a negative number 1712 } 1713 else 1714 { 1715 // a variable times 1 1716 } 1717 // or just a "" for 1 1718 } 1719 t = "ring @r=("; 1720 t = t + charstr(basering)+"),"; 1721 t = t + string(var(1))+",dp;"; 1722 // write(":w tstnum.tst",t); 1723 t = t+ "number @@Nn = " + s + ";"+"$"; 1724 write(":w tstnum.tst",t); 1725 string runsing = system("Singular"); 1726 int k; 1727 t = runsing+ " teq <tstnum.tst >tstnum.out"; 1728 k = system("sh",t); 1729 if (k!=0) 1730 { 1731 ERROR("Problems running Singular"); 1732 } 1733 int i = system("sh", "grep error tstnum.out > /dev/NULL"); 1734 if (i!=0) 1735 { 1736 // no error: s is a number 1737 i = 1; 1738 } 1739 k = system("sh","rm tstnum.tst tstnum.out > /dev/NULL"); 1740 return(i); 1741 } 1742 example 1743 { 1744 "EXAMPLE:"; echo = 2; 1745 ring r = (0,a),x,dp; 1746 string s = "a^2+7*a2"; 1747 testnumber(s); 1748 s = "b+a"; 1749 testnumber(s); 1750 } 1751 1752 proc str2lplace(string s) 1753 { 1754 // converts a word (monomial) with coeff into letterplace 1755 // string: coef*var1^exp1*var2^exp2*...varN^expN 1756 s = strpower2rep(s); // expand powers 1757 if (size(s)==0) { return(0); } 1758 int i,j,k,insC; 1759 string a,b,c,d,t; 1760 // 1. get coeff 1761 i = find(s,"*"); 1762 if (i==0) { return(s); } 1763 list VN; 1764 c = s[1..i1]; // incl. the case like (a^2+1) 1765 int tn = testnumber(c); 1766 if (tn == 0) 1767 { 1768 // failed test 1769 if (c[1]=="") 1770 { 1771 // two situations: either there's a negative number 1772 t = c[2..size(c)]; 1773 if (testnumber(t)) 1774 { 1775 //a negative number 1776 // nop here 1777 } 1778 else 1779 { 1780 // a variable times 1 1781 c = "1"; 1782 j++; VN[j] = t; //string(c[2..size(c)]); 1783 insC = 1; 1784 } 1785 } 1786 else 1787 { 1788 // just a variable with coeff 1 1789 j++; VN[j] = string(c); 1790 c = "1"; 1791 insC = 1; 1792 } 1793 } 1794 // get vars 1795 t = s; 1796 // t = s[i+1..size(s)]; 1797 k = size(t); //j = 0; 1798 while (k>0) 1799 { 1800 t = t[i+1..size(t)]; //next part 1801 i = find(t,"*"); // next * 1802 if (i==0) 1803 { 1804 // last monomial 1805 j++; 1806 VN[j] = t; 1807 k = size(t); 1808 break; 1809 } 1810 b = t[1..i1]; 1811 // print(b); 1812 j++; 1813 VN[j] = b; 1814 k = size(t); 1815 } 1816 VN = addplaces(VN); 1817 VN[size(VN)+1] = string(c); 1818 return(VN); 1819 } 1820 example 1821 { 1822 "EXAMPLE:"; echo = 2; 1823 ring r = (0,a),(f2,f1),dp; 1824 str2lplace("2*f2^2*f1^2*f2"); 1825 str2lplace("f1*f2"); 1826 str2lplace("(a^2+7a)*f1*f2"); 1827 } 1828 1829 proc strpower2rep(string s) 1830 { 1831 // makes x*x*x*x out of x^4 ., rep statys for repetitions 1832 // looks for "" problem 1833 // exception: "" as coeff 1834 string ex,t; 1835 int i,j,k; 1836 1837 i = find(s,"^"); // first ^ 1838 if (i==0) { return(s); } // no ^ signs 1839 1840 if (s[1] == "") 1841 { 1842 // either coef or 1 1843 // got the coeff: 1844 i = find(s,"*"); 1845 if (i==0) 1846 { 1847 // no *'s => coef == 1 or s == 23 1848 i = size(s)+1; 1849 } 1850 t = string(s[2..i1]); // without "" 1851 if ( testnumber(t) ) 1852 { 1853 // a good number 1854 t = strpower2rep(string(s[2..size(s)])); 1855 t = ""+t; 1856 return(t); 1857 } 1858 else 1859 { 1860 // a variable 1861 t = strpower2rep(string(s[2..size(s)])); 1862 t = "1*"+ t; 1863 return(t); 1864 } 1865 } 1866 // the case when leadcoef is a number in () 1867 if (s[1] == "(") 1868 { 1869 i = find(s,")",2); // must be nonzero 1870 t = s[2..i1]; 1871 if ( testnumber(t) ) 1872 { 1873 // a good number 1874 } 1875 else {"strpower2rep: bad number as coef";} 1876 ex = string(s[i+2..size(s)]); // 2 because of * 1877 ex = strpower2rep(ex); 1878 t = "("+t+")*"+ex; 1879 return(t); 1880 } 1881 1882 i = find(s,"^"); // first ^ 1883 j = find(s,"*",i+1); // next * == end of ^ 1884 if (j==0) 1885 { 1886 ex = s[i+1..size(s)]; 1887 } 1888 else 1889 { 1890 ex = s[i+1..j1]; 1891 } 1892 execute("int @exp = " + ex + ";"); //@exp = exponent 1893 // got varname 1894 for (k=i1; k>0; k) 1895 { 1896 if (s[k] == "*") break; 1897 } 1898 string varn = s[k+1..i1]; 1899 // "varn:"; varn; 1900 string pref; 1901 if (k>0) 1902 { 1903 pref = s[1..k]; // with * on the kth place 1904 } 1905 // "pref:"; pref; 1906 string suf; 1907 if ( (j>0) && (j+1 <= size(s)) ) 1908 { 1909 suf = s[j+1..size(s)]; // without * on the 1st place 1910 } 1911 // "suf:"; suf; 1912 string toins; 1913 for (k=1; k<=@exp; k++) 1914 { 1915 toins = toins + varn+"*"; 1916 } 1917 // "toins: "; toins; 1918 if (size(suf) == 0) 1919 { 1920 toins = toins[1..size(toins)1]; // get rid of trailing * 1921 } 1922 else 1923 { 1924 suf = strpower2rep(suf); 1925 } 1926 ex = pref + toins + suf; 1927 return(ex); 1928 // return(strpower2rep(ex)); 1929 } 1930 example 1931 { 1932 "EXAMPLE:"; echo = 2; 1933 ring r = (0,a),(x,y,z,t),dp; 1934 strpower2rep("x^4"); 1935 strpower2rep("2*x^4*y^3*z*t^2"); 1936 strpower2rep("a^2*x^4"); 1937 } 1938 1939 proc Liebr(poly a, poly b) 1940 { 1941 // alias ppLiebr 1942 poly q; 1943 while (b!=0) 1944 { 1945 q = q + pmLiebr(a,lead(b)); 1946 b = b  lead(b); 1947 } 1948 return(q); 1949 } 1950 example 1951 { 1952 "EXAMPLE:"; echo = 2; 1953 ring r = 0,(x(1),x(2),x(3),y(1),y(2),y(3)),dp; 1954 poly a = x(1)*y(2); poly b = y(1); 1955 int uptodeg=3; int lV=2; 1956 export uptodeg; export lV; 1957 Liebr(a,b); 1958 } 1959 1960 proc pmLiebr(poly a, poly b) 1961 { 1962 // a poly, b mono 1963 poly s; 1964 while (a!=0) 1965 { 1966 s = s + mmLiebr(lead(a),lead(b)); 1967 a = a  lead(a); 1968 } 1969 return(s); 1970 } 1971 1972 //proc pshift(poly a, int i, int uptodeg, int lV) 1973 proc pshift(poly a, int i) 1974 { 1975 // shifts a monomial a by i 1976 // calls pLPshift(p,sh,uptodeg,lVblock); 1977 return(system("stest",a,i,uptodeg,lV)); 1978 } 1979 1980 proc mmLiebr(poly a, poly b) 1981 { 1982 // a,b, monomials 1983 a = lead(a); 1984 b = lead(b); 1985 int sa = deg(a); 1986 int sb = deg(b); 1987 poly v = a*pshift(b,sa)  b*pshift(a,sb); 1988 return(v); 1989 } 1990 1991 static proc test_shift() 1992 { 1993 LIB "freegb.lib"; 1994 ring r = 0,(a,b),dp; 1995 int d =5; 1996 def R = freegbRing(d); 1997 setring R; 1998 int uptodeg = d; export uptodeg; 1999 int lV = 2; export lV; 2000 poly p = mmLiebr(a(1),b(1)); 2001 poly p = Liebr(a(1),b(1)); 2002 } 2003 2004 proc Serre(intmat A, int zu) 2005 { 2006 // zu = 1 > with commutators [f_i,f_j]; zu == 0 without them 2007 // suppose that A is cartan matrix 2008 // then Serre's relations are 2009 // (ad f_j)^{1A_{ij}} ( f_i) 2010 int ppl = printlevelvoice+2; 2011 int n = ncols(A); // hence n variables 2012 int i,j,k,l; 2013 poly p,q; 2014 ideal I; 2015 for (i=1; i<=n; i++) 2016 { 2017 for (j=1; j<=n; j++) 2018 { 2019 l = 1  A[i,j]; 2020 // printf("i:%s, j: %s, l: %s",i,j,l); 2021 dbprint(ppl,"i, j, l: ",i,j,l); 2022 // if ((i!=j) && (l >0)) 2023 // if ( (i!=j) && ( ((zu ==0) && (l >=2))  ((zu ==1) && (l >=1)) ) ) 2024 if ((i!=j) && (l >0)) 2025 { 2026 q = Liebr(var(j),var(i)); 2027 // printf("first bracket: %s",q); 2028 dbprint(ppl,"first bracket: ",q); 2029 // if (l >=2) 2030 // { 2031 for (k=1; k<=l1; k++) 2032 { 2033 q = Liebr(var(j),q); 2034 // printf("further bracket: %s",q); 2035 dbprint(ppl,"further bracket:",q); 2036 } 2037 // } 2038 } 2039 if (q!=0) { I = I,q; q=0;} 2040 } 2041 } 2042 I = simplify(I,2); 2043 return(I); 2044 } 2045 example 2046 { 2047 "EXAMPLE:"; echo = 2; 2048 intmat A[2][2] = 2, 1, 1, 2; // sl_3 == A_2 2049 ring r = 0,(f1,f2),dp; 2050 int uptodeg = 3; int lV = 2; 2051 export uptodeg; export lV; 2052 def R = freegbRing(uptodeg); 2053 setring R; 2054 ideal I = Serre(A,1); 2055 I; 2056 Serre(A,0); 2057 } 2058 2059 proc lp2lstr(ideal K, def save) 2060 "USAGE: lp2lstr(K,save); K an ideal, save a ring 2061 RETURN: nothing (exports object LN into save) 2062 PURPOSE: converts letterplace ideal to list of modules 2063 EXAMPLE: example lp2lstr; shows examples 2064 " 2065 { 2066 def @R = basering; 2067 string err; 2068 int s = nvars(save); 2069 int i,j,k; 2070 // K contains vars x(1),...z(1) = images of originals 2071 // 5. go back to orig vars, produce strings/modules 2072 int sk = size(K); 2073 int sp, sx, a, b; 2074 intvec x; 2075 poly p,q; 2076 poly pn; 2077 // vars in 'save' 2078 setring save; 2079 module N; 2080 list LN; 2081 vector V; 2082 poly pn; 2083 // test and skip exponents >=2 2084 setring @R; 2085 for(i=1; i<=sk; i++) 2086 { 2087 p = K[i]; 2088 while (p!=0) 2089 { 2090 q = lead(p); 2091 // "processing q:";q; 2092 x = leadexp(q); 2093 sx = size(x); 2094 for(k=1; k<=sx; k++) 2095 { 2096 if ( x[k] >= 2 ) 2097 { 2098 err = "skip: the value x[k] is " + string(x[k]); 2099 dbprint(ppl,err); 2100 // return(0); 2101 K[i] = 0; 2102 p = 0; 2103 q = 0; 2104 break; 2105 } 2106 } 2107 p = p  q; 2108 } 2109 } 2110 K = simplify(K,2); 2111 sk = size(K); 2112 for(i=1; i<=sk; i++) 2113 { 2114 // setring save; 2115 // V = 0; 2116 setring @R; 2117 p = K[i]; 2118 while (p!=0) 2119 { 2120 q = lead(p); 2121 err = "processing q:" + string(q); 2122 dbprint(ppl,err); 2123 x = leadexp(q); 2124 sx = size(x); 2125 pn = leadcoef(q); 2126 setring save; 2127 pn = imap(@R,pn); 2128 V = V + leadcoef(pn)*gen(1); 2129 for(k=1; k<=sx; k++) 2130 { 2131 if (x[k] ==1) 2132 { 2133 a = k / s; // block number=a+1, a!=0 2134 b = k % s; // remainder 2135 // printf("a: %s, b: %s",a,b); 2136 if (b == 0) 2137 { 2138 // that is it's the last var in the block 2139 b = s; 2140 a = a1; 2141 } 2142 V = V + var(b)*gen(a+2); 2143 } 2144 } 2145 err = "V: " + string(V); 2146 dbprint(ppl,err); 2147 // printf("V: %s", string(V)); 2148 N = N,V; 2149 V = 0; 2150 setring @R; 2151 p = p  q; 2152 pn = 0; 2153 } 2154 setring save; 2155 LN[i] = simplify(N,2); 2156 N = 0; 2157 } 2158 setring save; 2159 export LN; 2160 // return(LN); 2161 } 2162 example 2163 { 2164 "EXAMPLE:"; echo = 2; 2165 intmat A[2][2] = 2, 1, 1, 2; // sl_3 == A_2 2166 ring r = 0,(f1,f2),dp; 2167 int uptodeg = 3; int lV = 2; 2168 export uptodeg; export lV; 2169 def R = freegbRing(uptodeg); 2170 setring R; 2171 ideal I = Serre(A); 2172 lp2lstr(I,r); 2173 setring r; 2174 lst2str(LN,1); 2175 kill uptodeg; kill lV; 2176 } 2177 2178 proc strList2poly(list L) 2179 { 2180 // list L comes from sent2lplace (which takes a poly on the input) 2181 // each entry of L is a sublist with the coef on the last place 2182 int s = size(L); int t; 2183 int i,j; 2184 list M; 2185 poly p,q; 2186 string Q; 2187 for(i=1; i<=s; i++) 2188 { 2189 M = L[i]; 2190 t = size(M); 2191 // q = M[t]; // a constant 2192 Q = string(M[t]); 2193 for(j=1; j<t; j++) 2194 { 2195 // q = q*M[j]; 2196 Q = Q+"*"+string(M[j]); 2197 } 2198 execute("q="+Q+";"); 2199 // q; 2200 p = p + q; 2201 } 2202 kill Q; 2203 return(p); 2204 } 2205 example 2206 { 2207 "EXAMPLE:"; echo = 2; 2208 ring r =0,(x,y,z,t),Dp; 2209 def A = freegbRing(4); 2210 setring A; 2211 string t = "2*y*z*y*z + y*t*z*z  z*x*x*y + 2*z*y*z*y"; 2212 list L = sent2lplace(t); 2213 L; 2214 poly p = strList2poly(L); 2215 p; 2216 } 2217 2218 proc file2lplace(string fname) 2219 { 2220 string s = read(fname); 2221 // assume: file is a commasep list of polys 2222 // the vars are declared before 2223 // ends with ";" 2224 string t; int i; 2225 ideal I; 2226 list tst; 2227 while (s!="") 2228 { 2229 i = find(s,","); 2230 "i"; i; 2231 if (i==0) 2232 { 2233 i = find(s,";"); 2234 if (i==0) 2235 { 2236 // no ; ?? 2237 "no colon or semicolon found anymore"; 2238 return(I); 2239 } 2240 // no "," but ";" on the ith place 2241 t = s[1..i1]; 2242 s = ""; 2243 "processing: "; t; 2244 tst = sent2lplace(t); 2245 tst; 2246 I = I, strList2poly(tst); 2247 return(I); 2248 } 2249 // here i !=0 2250 t = s[1..i1]; 2251 s = s[i+1..size(s)]; 2252 "processing: "; t; 2253 tst = sent2lplace(t); 2254 tst; 2255 I = I, strList2poly(tst); 2256 } 2257 return(I); 2258 } 2259 example 2260 { 2261 "EXAMPLE:"; echo = 2; 2262 ring r =0,(x,y,z,t),dp; 2263 def A = freegbRing(4); 2264 setring A; 2265 string fn = "myfile"; 2266 string s1 = "z*y*y*y  3*y*z*x*y + 3*y*y*z*y  y*x*y*z,"; 2267 string s2 = "2*y*x*y*z + y*y*z*z  z*z*y*y + 2*z*y*z*y,"; 2268 string s3 = "z*y*x*t  2*y*z*y*t + y*y*z*t  t*z*y*y + 2*t*y*z*y  t*x*y*z;"; 2269 write(":w "+fn,s1); write(":a "+fn,s2); write(":a "+fn,s3); 2270 read(fn); 2271 ideal I = file2lplace(fn); 2272 I; 2273 } 2274 2275 static proc get_ls3nilp() 2276 { 2277 //first app of file2lplace 2278 ring r =0,(x,y,z,t),dp; 2279 int d = 10; 2280 def A = freegbRing(d); 2281 setring A; 2282 ideal I = file2lplace("./ls3nilp.bg"); 2283 // and now test the correctness: go back from lplace to strings 2284 lp2lstr(I,r); 2285 setring r; 2286 lst2str(LN,1); // agree! 2287 }
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