2048:
129:, realized an implementation of this method and reported experimental data in his 1987 pioneer article titled "A zero structure theorem for polynomial equations solving". To put this work into context, let us recall what was the common idea of an algebraic set decomposition at the time this article was written.
727:
The abundant literature on the subject can be explained by the many equivalent definitions of a regular chain. Actually, the original formulation of
Kalkbrener is quite different from that of Yang and Zhang. A bridge between these two notions, the point of view of Kalkbrener and that of Yang and
716:
In
Kalkbrener's vision, regular chains are used to represent the generic zeros of the irreducible components of an algebraic variety. In the original work of Yang and Zhang, they are used to decide whether a hypersurface intersects a quasi-variety (given by a regular chain).
1805:
1318:
1589:
950:
632:
Nowadays, decomposing an algebraic variety into irreducible components is not essential to process most application problems, since weaker notions of decompositions, less costly to compute, are sufficient.
2204:
2043:{\displaystyle {\begin{cases}z=0\\y=1\\x=0\end{cases}}\cup {\begin{cases}z=0\\y=0\\x=1\end{cases}}\cup {\begin{cases}z=1\\y=0\\x=0\end{cases}}\cup {\begin{cases}z^{2}+2z-1=0\\y=z\\x=z\end{cases}}}
1050:
1125:
379:
687:
610:
568:
528:
1456:
1217:
479:
In order to lead to a computer program, this algorithm specification should prescribe how irreducible components are represented. Such an encoding is introduced by
622:
described a method for solving polynomial systems based on polynomial factorization over field extensions and computation of characteristic sets of prime ideals.
1424:
2128:
M. Kalkbrener: A Generalized
Euclidean Algorithm for Computing Triangular Representations of Algebraic Varieties. J. Symb. Comput. 15(2): 143–167 (1993)
125:
is the first factorization-free algorithm, which was proposed for decomposing an algebraic variety into equidimensional components. Moreover, the Author,
713:, introduced independently by Michael Kalkbrener in 1991 in his PhD Thesis and, by Lu Yang and Jingzhong Zhang led to important algorithmic discoveries.
721:
have, in fact, several interesting properties and are the key notion in many algorithms for decomposing systems of algebraic or differential equations.
1464:
867:
2106:
A. M. Steel
Conquering inseparability: Primary decomposition and multivariate factorization over algebraic function fields of positive characteristic
629:
Method was introduced, polynomial factorization was an active research area and certain fundamental questions on this subject were solved recently
470:
and can be regarded as a natural output for a decomposition algorithm, or, in other words, for an algorithm solving a system of equations in
2059:
626:
2248:
113:. In both cases, each of these simpler systems has a triangular shape and remarkable properties, which justifies the terminology.
2207:. Proceedings of 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC 2010), ACM Press, pp. 187--194, 2010.
771:
2069:
1193:
625:
Deriving a practical implementation of this method, however, was and remains a difficult problem. In the 1980s, when the
110:
2238:
980:
2168:
D. Wang. Computing
Triangular Systems and Regular Systems. Journal of Symbolic Computation 30(2) (2000) 221–236
2116:
1082:
284:
2137:
S.C. Chou and X.S. Gao. On the dimension of an arbitrary ascending chain. Chinese Bull. of Sci., 38:799--804, 1991.
329:
139:
2243:
660:
2088:
Wu, W. T. (1987). A zero structure theorem for polynomial equations solving. MM Research
Preprints, 1, 2–12
2146:
Michael
Kalkbrener. Algorithmic properties of polynomial rings. J. Symb. Comput.}, 26(5):525--581, 1998.
589:
547:
507:
2233:
2156:
1973:
1920:
1867:
1814:
1479:
1313:{\displaystyle Z_{\mathbf {k} }(S)=Z_{\mathbf {k} }(S_{1})\cup \cdots \cup Z_{\mathbf {k} }(S_{e})}
82:
2223:
1429:
1181:
837:
165:
78:
28:
2117:
Searching dependency between algebraic equations: an algorithm applied to automated reasoning
2228:
1595:
775:
20:
819:
2203:
Changbo Chen, James H. Davenport, John P. May, Marc Moreno-Maza, Bican Xia, Rong Xiao.
1388:
2217:
2194:
M. Moreno Maza: On triangular decomposition of algebraic varieties. MEGA 2000 (2000).
2064:
1168:
1072:
967:
847:
755:
743:
718:
710:
613:
500:
is a non-empty and irreducible variety then one can compute a reduced triangular set
86:
2119:. Artificial Intelligence in Mathematics, pp. 14715, Oxford University Press.
1584:{\displaystyle S={\begin{cases}x^{2}+y+z=1\\x+y^{2}+z=1\\x+y+z^{2}=1\end{cases}}}
731:
There are various algorithms available for obtaining triangular decomposition of
945:{\displaystyle {\sqrt {(F)}}=\bigcap _{i=1}^{e}{\sqrt {\mathrm {sat} (T_{i})}}.}
619:
480:
971:
747:
706:
126:
832:, regarded as a system of polynomial equations, there are two notions of a
2159:. Journal of Symbolic Computation, 28(1–2):105–124, 1999.
2097:
Ritt, J. (1966). Differential
Algebra. New York, Dover Publications
846:. The first one is to decompose lazily, by representing only the
763:
1796:
283:. A first algebraic variety decomposition result is the famous
1791:
One possible triangular decompositions of the solution set of
770:
are available in various computer algebra systems, including
2036:
1961:
1908:
1855:
1577:
304:
there exist finitely many irreducible algebraic varieties
955:
The second is to describe explicitly all the points of
705:
Different concepts and algorithms extended the work of
724:
Regular chains have been investigated in many papers.
54:
if and only if it is a solution of one of the systems
1808:
1467:
1432:
1391:
1220:
1085:
983:
870:
663:
592:
550:
510:
332:
742:
both in the sense of
Kalkbrener and in the sense of
73:
When the purpose is to describe the solution set of
640:relies on the following variant of Ritt's Theorem.
109:can be obtained by a triangular decomposition into
2205:Triangular decomposition of semi-algebraic systems
2042:
1583:
1450:
1418:
1312:
1119:
1044:
944:
681:
604:
562:
522:
373:
1045:{\displaystyle V(F)=\bigcup _{i=1}^{e}W(T_{i}).}
89:. If the coefficients of the polynomial systems
1120:{\displaystyle {\sqrt {\mathrm {sat} (T_{i})}}}
374:{\displaystyle V=V_{1}\cup \cdots \cup V_{e}.}
105:are real numbers, then the real solutions of
1458:, consider the following polynomial system:
8:
1188:a semi-algebraic system with polynomials in
676:
670:
599:
593:
557:
551:
517:
511:
657:, one can compute a reduced triangular set
2179:Solving zero-dimensional algebraic systems
682:{\displaystyle C\subset \langle F\rangle }
1980:
1968:
1915:
1862:
1809:
1807:
1562:
1524:
1486:
1474:
1466:
1431:
1390:
1301:
1287:
1286:
1264:
1250:
1249:
1226:
1225:
1219:
1106:
1088:
1086:
1084:
1030:
1014:
1003:
982:
928:
910:
908:
902:
891:
871:
869:
728:Zhang, appears in Dongming Wang's paper.
697:reduces to zero by pseudo-division w.r.t
662:
591:
570:reduces to zero by pseudo-division w.r.t
549:
509:
362:
343:
331:
2081:
34:is a set of simpler polynomial systems
861:in the so-called sense of Kalkbrener.
709:. In the early 1990s, the notion of a
2155:P. Aubry, D. Lazard, M. Moreno Maza.
1349:. The regular semi-algebraic systems
7:
616:for the notion of a triangular set.
462:in the above Theorem are called the
808:be ordered variables. We denote by
50:such that a point is a solution of
16:Algorithm on systems of polynomials
2181:. Journal of Symbolic Computation
2157:On the theories of triangular sets
1171:for definitions of these notions.
1095:
1092:
1089:
917:
914:
911:
14:
2060:Wu's method of characteristic set
605:{\displaystyle \langle F\rangle }
563:{\displaystyle \langle F\rangle }
523:{\displaystyle \langle F\rangle }
174:if there exists a polynomial set
1288:
1251:
1227:
217:if for all algebraic varieties
1413:
1395:
1385:the rational number field. In
1307:
1294:
1270:
1257:
1239:
1233:
1194:regular semi-algebraic systems
1112:
1099:
1036:
1023:
993:
987:
934:
921:
879:
873:
647:For any finite polynomial set
540:and such that all polynomials
483:through the following result.
111:regular semi-algebraic systems
1:
2070:Regular semi-algebraic system
1369:of the semi-algebraic system
966:in the so-called sense of in
287:which implies the following.
2115:Yang, L., Zhang, J. (1994).
1192:. There exist finitely many
85:, those simpler systems are
1451:{\displaystyle x>y>z}
1127:denotes the radical of the
294:For each algebraic variety
292:Theorem (Lasker - Noether).
2265:
689:such that all polynomial
140:algebraically closed field
768:Characteristic Set Method
638:Characteristic Set Method
436:irreducible decomposition
123:Characteristic Set Method
2249:Computer algebra systems
1600:
1367:triangular decomposition
1174:Assume from now on that
834:triangular decomposition
434:is unique and forms the
25:triangular decomposition
1426:with variable ordering
752:Lextriangular Algorithm
584:Ritt characteristic set
504:contained in the ideal
184:such that the zero set
2044:
1585:
1452:
1420:
1339:denotes the points of
1314:
1121:
1046:
1019:
946:
907:
683:
645:Theorem (Wen-Tsun Wu).
606:
564:
524:
464:irreducible components
375:
285:Lasker–Noether theorem
2045:
1586:
1453:
1421:
1315:
1122:
1047:
999:
947:
887:
850:of the algebraic set
684:
607:
565:
525:
376:
1806:
1465:
1430:
1389:
1218:
1083:
981:
868:
661:
590:
582:in Ritt's Theorem a
548:
508:
330:
81:of its coefficient
2239:Algebraic geometry
2040:
2035:
1960:
1907:
1854:
1581:
1576:
1448:
1416:
1310:
1211:such that we have
1167:. Please refer to
1117:
1071:are finitely many
1042:
942:
818:the corresponding
782:Formal definitions
766:together with the
679:
627:Characteristic set
612:. Please refer to
602:
560:
520:
371:
324:such that we have
1594:According to the
1419:{\displaystyle Q}
1182:real closed field
1115:
937:
882:
838:algebraic closure
166:algebraic variety
148:be a subfield of
79:algebraic closure
29:polynomial system
2256:
2244:Computer algebra
2208:
2201:
2195:
2192:
2186:
2175:
2169:
2166:
2160:
2153:
2147:
2144:
2138:
2135:
2129:
2126:
2120:
2113:
2107:
2104:
2098:
2095:
2089:
2086:
2049:
2047:
2046:
2041:
2039:
2038:
1985:
1984:
1964:
1963:
1911:
1910:
1858:
1857:
1794:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1590:
1588:
1587:
1582:
1580:
1579:
1567:
1566:
1529:
1528:
1491:
1490:
1457:
1455:
1454:
1449:
1425:
1423:
1422:
1417:
1384:
1372:
1364:
1348:
1344:
1338:
1319:
1317:
1316:
1311:
1306:
1305:
1293:
1292:
1291:
1269:
1268:
1256:
1255:
1254:
1232:
1231:
1230:
1210:
1191:
1187:
1179:
1166:
1153:
1139:
1126:
1124:
1123:
1118:
1116:
1111:
1110:
1098:
1087:
1078:
1070:
1051:
1049:
1048:
1043:
1035:
1034:
1018:
1013:
965:
951:
949:
948:
943:
938:
933:
932:
920:
909:
906:
901:
883:
872:
860:
845:
831:
817:
807:
791:
764:Marc Moreno Maza
760:Triade Algorithm
741:
700:
696:
692:
688:
686:
685:
680:
656:
611:
609:
608:
603:
581:
578:We call the set
573:
569:
567:
566:
561:
543:
539:
533:
529:
527:
526:
521:
503:
499:
475:
469:
461:
441:
433:
415:
400:
380:
378:
377:
372:
367:
366:
348:
347:
323:
303:
282:
269:
256:
236:
212:
205:
201:
197:
183:
173:
163:
153:
147:
137:
108:
104:
76:
69:
53:
49:
33:
21:computer algebra
2264:
2263:
2259:
2258:
2257:
2255:
2254:
2253:
2214:
2213:
2212:
2211:
2202:
2198:
2193:
2189:
2176:
2172:
2167:
2163:
2154:
2150:
2145:
2141:
2136:
2132:
2127:
2123:
2114:
2110:
2105:
2101:
2096:
2092:
2087:
2083:
2078:
2056:
2034:
2033:
2021:
2020:
2008:
2007:
1976:
1969:
1959:
1958:
1946:
1945:
1933:
1932:
1916:
1906:
1905:
1893:
1892:
1880:
1879:
1863:
1853:
1852:
1840:
1839:
1827:
1826:
1810:
1804:
1803:
1792:
1789:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1575:
1574:
1558:
1543:
1542:
1520:
1511:
1510:
1482:
1475:
1463:
1462:
1428:
1427:
1387:
1386:
1382:
1379:
1370:
1362:
1356:
1350:
1346:
1340:
1332:
1324:
1297:
1282:
1260:
1245:
1221:
1216:
1215:
1208:
1202:
1196:
1189:
1185:
1175:
1164:
1159:
1156:quasi-component
1150:
1141:
1137:
1132:
1129:saturated ideal
1102:
1081:
1080:
1076:
1068:
1062:
1056:
1026:
979:
978:
956:
924:
866:
865:
851:
841:
823:
820:polynomial ring
809:
805:
799:
793:
792:be a field and
787:
784:
732:
698:
694:
690:
659:
658:
648:
588:
587:
579:
571:
546:
545:
541:
535:
531:
506:
505:
501:
491:
488:Theorem (Ritt).
471:
467:
459:
453:
447:
439:
430:
424:
417:
402:
398:
391:
386:
358:
339:
328:
327:
317:
311:
305:
295:
281:
271:
268:
258:
257:implies either
255:
248:
238:
231:
224:
218:
210:
203:
199:
185:
175:
169:
164:is an (affine)
155:
149:
143:
133:
119:
106:
102:
96:
90:
74:
67:
61:
55:
51:
47:
41:
35:
31:
17:
12:
11:
5:
2262:
2260:
2252:
2251:
2246:
2241:
2236:
2231:
2226:
2216:
2215:
2210:
2209:
2196:
2187:
2170:
2161:
2148:
2139:
2130:
2121:
2108:
2099:
2090:
2080:
2079:
2077:
2074:
2073:
2072:
2067:
2062:
2055:
2052:
2051:
2050:
2037:
2032:
2029:
2026:
2023:
2022:
2019:
2016:
2013:
2010:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1983:
1979:
1975:
1974:
1972:
1967:
1962:
1957:
1954:
1951:
1948:
1947:
1944:
1941:
1938:
1935:
1934:
1931:
1928:
1925:
1922:
1921:
1919:
1914:
1909:
1904:
1901:
1898:
1895:
1894:
1891:
1888:
1885:
1882:
1881:
1878:
1875:
1872:
1869:
1868:
1866:
1861:
1856:
1851:
1848:
1845:
1842:
1841:
1838:
1835:
1832:
1829:
1828:
1825:
1822:
1819:
1816:
1815:
1813:
1624:PolynomialRing
1601:
1592:
1591:
1578:
1573:
1570:
1565:
1561:
1557:
1554:
1551:
1548:
1545:
1544:
1541:
1538:
1535:
1532:
1527:
1523:
1519:
1516:
1513:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1489:
1485:
1481:
1480:
1478:
1473:
1470:
1447:
1444:
1441:
1438:
1435:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1378:
1375:
1360:
1354:
1328:
1321:
1320:
1309:
1304:
1300:
1296:
1290:
1285:
1281:
1278:
1275:
1272:
1267:
1263:
1259:
1253:
1248:
1244:
1241:
1238:
1235:
1229:
1224:
1206:
1200:
1162:
1148:
1135:
1114:
1109:
1105:
1101:
1097:
1094:
1091:
1073:regular chains
1066:
1060:
1055:In both cases
1053:
1052:
1041:
1038:
1033:
1029:
1025:
1022:
1017:
1012:
1009:
1006:
1002:
998:
995:
992:
989:
986:
953:
952:
941:
936:
931:
927:
923:
919:
916:
913:
905:
900:
897:
894:
890:
886:
881:
878:
875:
848:generic points
803:
800:< ... <
797:
783:
780:
719:Regular chains
703:
702:
678:
675:
672:
669:
666:
601:
598:
595:
576:
575:
559:
556:
553:
519:
516:
513:
457:
451:
446:The varieties
444:
443:
428:
422:
396:
389:
385:Moreover, if
383:
382:
381:
370:
365:
361:
357:
354:
351:
346:
342:
338:
335:
315:
309:
279:
266:
253:
246:
229:
222:
118:
115:
100:
94:
87:regular chains
65:
59:
45:
39:
15:
13:
10:
9:
6:
4:
3:
2:
2261:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2232:
2230:
2227:
2225:
2222:
2221:
2219:
2206:
2200:
2197:
2191:
2188:
2184:
2180:
2174:
2171:
2165:
2162:
2158:
2152:
2149:
2143:
2140:
2134:
2131:
2125:
2122:
2118:
2112:
2109:
2103:
2100:
2094:
2091:
2085:
2082:
2075:
2071:
2068:
2066:
2065:Regular chain
2063:
2061:
2058:
2057:
2053:
2030:
2027:
2024:
2017:
2014:
2011:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1981:
1977:
1970:
1965:
1955:
1952:
1949:
1942:
1939:
1936:
1929:
1926:
1923:
1917:
1912:
1902:
1899:
1896:
1889:
1886:
1883:
1876:
1873:
1870:
1864:
1859:
1849:
1846:
1843:
1836:
1833:
1830:
1823:
1820:
1817:
1811:
1802:
1801:
1800:
1798:
1797:RegularChains
1741:Triangularize
1609:RegularChains
1599:
1597:
1571:
1568:
1563:
1559:
1555:
1552:
1549:
1546:
1539:
1536:
1533:
1530:
1525:
1521:
1517:
1514:
1507:
1504:
1501:
1498:
1495:
1492:
1487:
1483:
1476:
1471:
1468:
1461:
1460:
1459:
1445:
1442:
1439:
1436:
1433:
1410:
1407:
1404:
1401:
1398:
1392:
1376:
1374:
1368:
1363:
1353:
1343:
1336:
1331:
1327:
1302:
1298:
1283:
1279:
1276:
1273:
1265:
1261:
1246:
1242:
1236:
1222:
1214:
1213:
1212:
1209:
1199:
1195:
1183:
1178:
1172:
1170:
1169:regular chain
1165:
1157:
1151:
1144:
1138:
1130:
1107:
1103:
1074:
1069:
1059:
1039:
1031:
1027:
1020:
1015:
1010:
1007:
1004:
1000:
996:
990:
984:
977:
976:
975:
973:
969:
963:
959:
939:
929:
925:
903:
898:
895:
892:
888:
884:
876:
864:
863:
862:
858:
854:
849:
844:
839:
835:
830:
826:
821:
816:
812:
806:
796:
790:
781:
779:
777:
773:
769:
765:
761:
757:
756:Daniel Lazard
753:
749:
745:
739:
735:
729:
725:
722:
720:
714:
712:
711:regular chain
708:
673:
667:
664:
655:
651:
646:
643:
642:
641:
639:
634:
630:
628:
623:
621:
617:
615:
614:regular chain
596:
586:of the ideal
585:
554:
538:
530:generated by
514:
498:
494:
489:
486:
485:
484:
482:
477:
474:
465:
460:
450:
437:
431:
421:
416:then the set
414:
410:
406:
399:
392:
384:
368:
363:
359:
355:
352:
349:
344:
340:
336:
333:
326:
325:
322:
318:
308:
302:
298:
293:
290:
289:
288:
286:
278:
274:
265:
261:
252:
245:
241:
237:the relation
235:
228:
221:
216:
207:
196:
192:
188:
182:
178:
172:
167:
162:
158:
152:
146:
141:
136:
130:
128:
124:
116:
114:
112:
103:
93:
88:
84:
80:
71:
68:
58:
48:
38:
30:
26:
22:
2199:
2190:
2182:
2178:
2173:
2164:
2151:
2142:
2133:
2124:
2111:
2102:
2093:
2084:
1799:library is:
1790:
1593:
1380:
1366:
1358:
1351:
1345:which solve
1341:
1334:
1329:
1325:
1322:
1204:
1197:
1176:
1173:
1160:
1155:
1154:denotes the
1146:
1142:
1133:
1128:
1064:
1057:
1054:
961:
957:
954:
856:
852:
842:
833:
828:
824:
814:
810:
801:
794:
788:
785:
767:
759:
751:
737:
733:
730:
726:
723:
715:
704:
653:
649:
644:
637:
635:
631:
624:
618:
583:
577:
536:
496:
492:
487:
478:
472:
463:
455:
448:
445:
435:
426:
419:
412:
408:
404:
394:
387:
320:
313:
306:
300:
296:
291:
276:
272:
263:
259:
250:
243:
239:
233:
226:
219:
214:
209:Recall that
208:
194:
190:
186:
180:
176:
170:
160:
156:
150:
144:
134:
131:
122:
120:
98:
91:
72:
63:
56:
43:
36:
24:
18:
2234:Polynomials
2177:D. Lazard,
1795:with using
1184:. Consider
972:Wen-Tsun Wu
748:Wen-Tsun Wu
707:Wen-Tsun Wu
620:Joseph Ritt
481:Joseph Ritt
215:irreducible
154:. A subset
127:Wen-Tsun Wu
2218:Categories
2076:References
401:holds for
2224:Equations
1996:−
1966:∪
1913:∪
1860:∪
1768:Equations
1280:∪
1277:⋯
1274:∪
1001:⋃
889:⋂
836:over the
677:⟩
671:⟨
668:⊂
600:⟩
594:⟨
558:⟩
552:⟨
518:⟩
512:⟨
356:∪
353:⋯
350:∪
2054:See also
1377:Examples
758:and the
213:is said
2229:Algebra
1381:Denote
1365:form a
1357:, ...,
1203:, ...,
1063:, ...,
454:, ...,
425:, ...,
312:, ...,
202:equals
117:History
97:, ...,
77:in the
62:, ...,
42:, ...,
2185:, 1992
1598:code:
1323:where
1140:while
968:Lazard
822:. For
750:. The
744:Lazard
138:be an
1596:Maple
1180:is a
776:Maple
772:Axiom
407:<
168:over
83:field
27:of a
1603:with
1443:>
1437:>
1079:and
970:and
786:Let
774:and
746:and
636:The
403:1 ≤
193:) ⊂
142:and
132:Let
121:The
23:, a
1762:map
1747:sys
1633:sys
1158:of
1131:of
1075:of
840:of
762:by
754:by
693:in
544:in
534:in
490:If
466:of
438:of
270:or
198:of
19:In
2220::
2183:13
1738::=
1636::=
1627:()
1621::=
1373:.
974:.
827:⊂
813:=
778:.
652:⊂
495:⊂
476:.
432:}
411:≤
393:⊈
319:⊂
299:⊂
275:=
262:=
249:∪
242:=
232:⊂
225:,
206:.
179:⊂
159:⊂
70:.
2031:z
2028:=
2025:x
2018:z
2015:=
2012:y
2005:0
2002:=
1999:1
1993:z
1990:2
1987:+
1982:2
1978:z
1971:{
1956:0
1953:=
1950:x
1943:0
1940:=
1937:y
1930:1
1927:=
1924:z
1918:{
1903:1
1900:=
1897:x
1890:0
1887:=
1884:y
1877:0
1874:=
1871:z
1865:{
1850:0
1847:=
1844:x
1837:1
1834:=
1831:y
1824:0
1821:=
1818:z
1812:{
1793:S
1786:;
1783:)
1780:R
1777:,
1774:l
1771:,
1765:(
1759::
1756:)
1753:R
1750:,
1744:(
1735:l
1732::
1729:}
1726:1
1723:-
1720:2
1717:^
1714:z
1711:+
1708:y
1705:+
1702:x
1699:,
1696:1
1693:-
1690:z
1687:+
1684:2
1681:^
1678:y
1675:+
1672:x
1669:,
1666:1
1663:-
1660:z
1657:+
1654:y
1651:+
1648:2
1645:^
1642:x
1639:{
1630::
1618:R
1615::
1612:)
1606:(
1572:1
1569:=
1564:2
1560:z
1556:+
1553:y
1550:+
1547:x
1540:1
1537:=
1534:z
1531:+
1526:2
1522:y
1518:+
1515:x
1508:1
1505:=
1502:z
1499:+
1496:y
1493:+
1488:2
1484:x
1477:{
1472:=
1469:S
1446:z
1440:y
1434:x
1414:]
1411:z
1408:,
1405:y
1402:,
1399:x
1396:[
1393:Q
1383:Q
1371:S
1361:e
1359:S
1355:1
1352:S
1347:S
1342:k
1337:)
1335:S
1333:(
1330:k
1326:Z
1308:)
1303:e
1299:S
1295:(
1289:k
1284:Z
1271:)
1266:1
1262:S
1258:(
1252:k
1247:Z
1243:=
1240:)
1237:S
1234:(
1228:k
1223:Z
1207:e
1205:S
1201:1
1198:S
1190:R
1186:S
1177:k
1163:i
1161:T
1152:)
1149:i
1147:T
1145:(
1143:W
1136:i
1134:T
1113:)
1108:i
1104:T
1100:(
1096:t
1093:a
1090:s
1077:R
1067:e
1065:T
1061:1
1058:T
1040:.
1037:)
1032:i
1028:T
1024:(
1021:W
1016:e
1011:1
1008:=
1005:i
997:=
994:)
991:F
988:(
985:V
964:)
962:F
960:(
958:V
940:.
935:)
930:i
926:T
922:(
918:t
915:a
912:s
904:e
899:1
896:=
893:i
885:=
880:)
877:F
874:(
859:)
857:F
855:(
853:V
843:k
829:R
825:F
815:k
811:R
804:n
802:x
798:1
795:x
789:k
740:)
738:F
736:(
734:V
701:.
699:C
695:F
691:g
674:F
665:C
654:k
650:F
597:F
580:C
574:.
572:C
555:F
542:g
537:k
532:F
515:F
502:C
497:K
493:V
473:k
468:V
458:e
456:V
452:1
449:V
442:.
440:V
429:e
427:V
423:1
420:V
418:{
413:e
409:j
405:i
397:j
395:V
390:i
388:V
369:.
364:e
360:V
345:1
341:V
337:=
334:V
321:K
316:e
314:V
310:1
307:V
301:K
297:V
280:2
277:V
273:V
267:1
264:V
260:V
254:2
251:V
247:1
244:V
240:V
234:K
230:2
227:V
223:1
220:V
211:V
204:V
200:F
195:K
191:F
189:(
187:V
181:k
177:F
171:k
161:K
157:V
151:K
145:k
135:K
107:S
101:e
99:S
95:1
92:S
75:S
66:e
64:S
60:1
57:S
52:S
46:e
44:S
40:1
37:S
32:S
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