25:
600:
385:
165:
The
Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. As of 2018, there are no plausible claims to have proved it. Even the two-variable case has resisted all efforts. There are currently no known compelling reasons for believing the
1625:
The strong real
Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map, in which case it is a covering map of a simply connected manifold, hence
1621:
Michiel de Bondt and Arno van den Essen and Ludwik Drużkowski independently showed that it is enough to prove the
Jacobian Conjecture for complex maps of cubic homogeneous type with a symmetric Jacobian matrix, and further showed that the conjecture holds for maps of cubic linear type with a
595:{\displaystyle J_{F}=\left|{\begin{matrix}{\frac {\partial f_{1}}{\partial X_{1}}}&\cdots &{\frac {\partial f_{1}}{\partial X_{N}}}\\\vdots &\ddots &\vdots \\{\frac {\partial f_{N}}{\partial X_{1}}}&\cdots &{\frac {\partial f_{N}}{\partial X_{N}}}\end{matrix}}\right|,}
1530:
Edwin
Connell and Lou van den Dries proved that if the Jacobian conjecture is false, then it has a counterexample with integer coefficients and Jacobian determinant 1. In consequence, the Jacobian conjecture is true either for all fields of characteristic 0 or for none. For fixed dimension
166:
conjecture to be true, and according to van den Essen there are some suspicions that the conjecture is in fact false for large numbers of variables (indeed, there is equally also no compelling evidence to support these suspicions). The
Jacobian conjecture is number 16 in
1011:
1328:
1649:
dimensions. A self-contained and purely algebraic proof of the last implication is also given by
Kossivi Adjamagbo and Arno van den Essen who also proved in the same paper that these two conjectures are equivalent to the Poisson conjecture.
1467:
2. Hyman Bass, Edwin
Connell, and David Wright showed that the general case follows from the special case where the polynomials are of degree 3, or even more specifically, of cubic homogeneous type, meaning of the form
1618:) was proved by Andrew Campbell for complex maps and in general by Michael Razar and, independently, by David Wright. Tzuong-Tsieng Moh checked the conjecture for polynomials of degree at most 100 in two variables.
1522:
are cubes of homogeneous linear polynomials. It seems that Drużkowski's reduction is one most promising way to go forward. These reductions introduce additional variables and so are not available for fixed
1370:
1104:
884:
2054:
833:
1207:
1156:
1196:
873:
1409:
1049:
146:-dimensional space to itself has Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse. It was first conjectured in 1939 by
2295:
787:
is simply a set of functions linear in the variables, because then the inverse will also be a set of linear functions. A simple non-linear example is given by
1513:
is either zero or a homogeneous cubic. Ludwik Drużkowski showed that one may further assume that the map is of cubic linear type, meaning that the nonzero
1659:
707:
According to van den Essen, the problem was first conjectured by Keller in 1939 for the limited case of two variables and integer coefficients.
2451:
1766:
1820:
Bass, Hyman; Connell, Edwin H.; Wright, David (1982), "The
Jacobian conjecture: reduction of degree and formal expansion of the inverse",
46:
718: > 0 even for one variable. The characteristic of a field, if it is not zero, must be prime, so at least 2. The polynomial
68:
700:
332:
2203:
Drużkowski, Ludwik M. (2005), "The
Jacobian conjecture: symmetric reduction and solution in the symmetric cubic linear case",
1633:
implies the
Jacobian conjecture. Conversely, it is shown by Yoshifumi Tsuchimoto and independently by Alexei Belov-Kanel and
1976:
2325:
Belov-Kanel, Alexei; Kontsevich, Maxim (2007), "The Jacobian conjecture is stably equivalent to the Dixmier conjecture",
2167:
2441:
39:
33:
1747:
Adjamagbo, Kossivi (1995), "On separable algebras over a U.F.D. and the Jacobian conjecture in any characteristic",
661:
has a polynomial reciprocal, so is a nonzero constant. The Jacobian conjecture is the following partial converse:
50:
2119:
de Bondt, Michiel; van den Essen, Arno (2005), "A reduction of the Jacobian conjecture to the symmetric case",
1429:
673:
2165:
de Bondt, Michiel; van den Essen, Arno (2005), "The Jacobian conjecture for symmetric Drużkowski mappings",
1464:
1433:
223:
139:
1436:. In fact for smooth functions (and so in particular for polynomials) a smooth local inverse function to
745:
1866:
740:
1719:
1333:
which is not constant, and the Jacobian conjecture does not apply. The function still has an inverse:
1006:{\displaystyle J_{F}=\left|{\begin{matrix}1+2x&1\\2x&1\end{matrix}}\right|=(1+2x)(1)-(1)2x=1.}
2446:
2436:
2344:
1339:
1902:
Connell, Edwin; van den Dries, Lou (1983), "Injective polynomial maps and the Jacobian conjecture",
1055:
1626:
invertible. Sergey Pinchuk constructed two variable counterexamples of total degree 35 and higher.
1323:{\displaystyle J_{F}=\left|{\begin{matrix}4x&1\\2x&1\end{matrix}}\right|=(4x)(1)-2x(1)=2x,}
167:
147:
100:
2368:
2334:
2087:
1642:
1630:
793:
362:
325:
155:
151:
118:
90:
1119:
1162:
839:
2308:
2071:
1839:
1762:
1698:
2049:
1376:
2352:
2248:
2212:
2176:
2138:
2128:
2063:
2021:
1985:
1947:
1911:
1875:
1829:
1798:
1752:
1688:
1634:
2405:
2364:
2262:
2226:
2190:
2152:
2083:
2035:
1999:
1961:
1925:
1889:
1851:
1776:
1734:
1022:
2401:
2360:
2258:
2222:
2186:
2148:
2079:
2031:
1995:
1957:
1921:
1885:
1847:
1772:
1730:
1535:, it is true if it holds for at least one algebraically closed field of characteristic 0.
358:
2356:
2348:
2280:
1864:
Drużkowski, Ludwik M. (1983), "An effective approach to Keller's Jacobian conjecture",
2430:
2091:
1916:
1803:
2372:
1834:
2239:
Pinchuk, Sergey (1994), "A counterexample to the strong real Jacobian conjecture",
2386:
2133:
2276:
1938:
Campbell, L. Andrew (1973), "A condition for a polynomial map to be invertible",
2421:
1757:
347:
215:
127:
1789:
Wang, Stuart Sui-Sheng (August 1980), "A Jacobian criterion for separability",
2067:
135:
2387:"A proof of the equivalence of the Dixmier, Jacobian and Poisson conjectures"
2312:
2075:
2026:
1843:
1702:
1586:. Keller (1939) proved the birational case, that is, where the two fields
2106:
On the global Jacobian conjecture for polynomials of degree less than 100
159:
1729:, Sémin. Congr., vol. 2, Paris: Soc. Math. France, pp. 55–81,
1463:
Stuart Sui-Sheng Wang proved the Jacobian conjecture for polynomials of
2253:
1990:
1952:
1880:
1727:
Algèbre non commutative, groupes quantiques et invariants (Reims, 1995)
1693:
168:
Stephen Smale's 1998 list of Mathematical Problems for the Next Century
2143:
2339:
2217:
2181:
1974:
Razar, Michael (1979), "Polynomial maps with constant Jacobian",
1455:
has a smooth global inverse, but the inverse is not polynomial.
1679:
Keller, Ott-Heinrich (1939), "Ganze Cremona-Transformationen",
1622:
symmetric Jacobian matrix, over any field of characteristic 0.
750:
suggested extending the Jacobian conjecture to characteristic
18:
2050:"On the Jacobian conjecture and the configurations of roots"
158:
that can be understood using little beyond a knowledge of
710:
The obvious analogue of the Jacobian conjecture fails if
2385:
Adjamagbo, Pascal Kossivi; van den Essen, Arno (2007),
1720:"Polynomial automorphisms and the Jacobian conjecture"
1229:
906:
407:
2283:
1379:
1342:
1210:
1165:
1122:
1058:
1025:
887:
842:
796:
638:
It follows from the multivariable chain rule that if
388:
2012:
Wright, David (1981), "On the Jacobian conjecture",
1576:, the Jacobian conjecture is true if, and only if,
783:
The existence of a polynomial inverse is obvious if
182:> 1 be a fixed integer and consider polynomials
1016:In this case the inverse exists as the polynomials
114:
106:
96:
86:
2289:
1751:, Dordrecht: Kluwer Acad. Publ., pp. 89–103,
1403:
1364:
1322:
1190:
1150:
1098:
1043:
1005:
867:
827:
761:does not divide the degree of the field extension
594:
142:. It states that if a polynomial function from an
16:On invertibility of polynomial maps (mathematics)
2121:Proceedings of the American Mathematical Society
2422:Web page of Tzuong-Tsieng Moh on the conjecture
2055:Journal für die reine und angewandte Mathematik
739:is 0) but it has no inverse function. However,
1749:Automorphisms of affine spaces (Curaçao, 1994)
1822:Bulletin of the American Mathematical Society
8:
81:
154:, as an example of a difficult question in
80:
2338:
2282:
2252:
2216:
2180:
2142:
2132:
2025:
1989:
1951:
1915:
1879:
1833:
1815:
1813:
1802:
1756:
1713:
1711:
1692:
1378:
1349:
1341:
1228:
1215:
1209:
1176:
1164:
1136:
1121:
1087:
1057:
1024:
905:
892:
886:
853:
841:
807:
795:
703:, meaning its components are polynomials.
572:
557:
547:
531:
516:
506:
476:
461:
451:
435:
420:
410:
406:
393:
387:
69:Learn how and when to remove this message
1660:List of unsolved problems in mathematics
134:is a famous unsolved problem concerning
32:This article includes a list of general
1671:
612:is itself a polynomial function of the
1447:is non-zero. For example, the map x →
1681:Monatshefte für Mathematik und Physik
7:
878:so that the Jacobian determinant is
2357:10.17323/1609-4514-2007-7-2-209-218
2277:"Endomorphisms of Weyl algebra and
1904:Journal of Pure and Applied Algebra
642:has a polynomial inverse function
565:
550:
524:
509:
469:
454:
428:
413:
38:it lacks sufficient corresponding
14:
1637:that the Jacobian conjecture for
324:arising in this way is called a
23:
2014:Illinois Journal of Mathematics
1835:10.1090/S0273-0979-1982-15032-7
1365:{\displaystyle x={\sqrt {u-v}}}
2275:Tsuchimoto, Yoshifumi (2005),
1305:
1299:
1287:
1281:
1278:
1269:
1099:{\displaystyle y=v-(u-v)^{2}.}
1084:
1071:
988:
982:
976:
970:
967:
952:
757:by adding the hypothesis that
1:
2452:Unsolved problems in geometry
2134:10.1090/S0002-9939-05-07570-2
1977:Israel Journal of Mathematics
683:is a non-zero constant, then
634:Formulation of the conjecture
2301:Osaka Journal of Mathematics
2205:Annales Polonici Mathematici
2168:Annales Polonici Mathematici
1917:10.1016/0022-4049(83)90094-4
1804:10.1016/0021-8693(80)90233-1
1718:van den Essen, Arno (1997),
1602:) are equal. The case where
1440:exists at every point where
2394:Acta Mathematica Vietnamica
2327:Moscow Mathematical Journal
2048:Moh, Tzuong-Tsieng (1983),
1758:10.1007/978-94-015-8555-2_5
1610:) is a Galois extension of
1542:denote the polynomial ring
828:{\displaystyle u=x^{2}+y+x}
150:, and widely publicized by
2468:
1629:It is well known that the
1151:{\displaystyle u=2x^{2}+y}
2241:Mathematische Zeitschrift
2068:10.1515/crll.1983.340.140
1556:-subalgebra generated by
1191:{\displaystyle v=x^{2}+y}
868:{\displaystyle v=x^{2}+y}
1430:inverse function theorem
1201:then the determinant is
687:has an inverse function
174:The Jacobian determinant
1414:but the expression for
1404:{\displaystyle y=2v-u,}
53:more precise citations.
2291:
2027:10.1215/ijm/1256047158
1641:variables implies the
1434:multivariable calculus
1428:≠ 0 is related to the
1405:
1366:
1324:
1192:
1152:
1100:
1045:
1007:
869:
829:
705:
596:
224:vector-valued function
2292:
1940:Mathematische Annalen
1867:Mathematische Annalen
1418:is not a polynomial.
1406:
1367:
1325:
1193:
1153:
1101:
1046:
1044:{\displaystyle x=u-v}
1008:
870:
830:
663:
597:
2281:
2104:Moh, Tzuong-Tsieng,
1377:
1340:
1208:
1163:
1120:
1056:
1023:
885:
840:
794:
735:which is 1 (because
666:Jacobian conjecture:
386:
346:, is defined as the
333:Jacobian determinant
2349:2005math.....12171B
714:has characteristic
363:partial derivatives
222:. Then we define a
148:Ott-Heinrich Keller
132:Jacobian conjecture
101:Ott-Heinrich Keller
83:
82:Jacobian conjecture
2442:Algebraic geometry
2287:
2254:10.1007/bf02571929
1991:10.1007/bf02764906
1953:10.1007/bf01349234
1881:10.1007/bf01459126
1791:Journal of Algebra
1694:10.1007/BF01695502
1643:Dixmier conjecture
1631:Dixmier conjecture
1401:
1362:
1320:
1260:
1188:
1148:
1096:
1041:
1003:
943:
865:
825:
592:
583:
361:consisting of the
326:polynomial mapping
156:algebraic geometry
152:Shreeram Abhyankar
119:Dixmier conjecture
91:Algebraic geometry
2290:{\displaystyle p}
1768:978-90-481-4566-9
1486:, ...,
1360:
1109:But if we modify
741:Kossivi Adjamagbo
579:
538:
483:
442:
124:
123:
79:
78:
71:
2459:
2409:
2408:
2391:
2382:
2376:
2375:
2342:
2322:
2316:
2315:
2296:
2294:
2293:
2288:
2272:
2266:
2265:
2256:
2236:
2230:
2229:
2220:
2218:10.4064/ap87-0-7
2200:
2194:
2193:
2184:
2182:10.4064/ap86-1-5
2162:
2156:
2155:
2146:
2136:
2127:(8): 2201–2205,
2116:
2110:
2109:
2101:
2095:
2094:
2062:(340): 140–212,
2045:
2039:
2038:
2029:
2009:
2003:
2002:
1993:
1971:
1965:
1964:
1955:
1935:
1929:
1928:
1919:
1899:
1893:
1892:
1883:
1861:
1855:
1854:
1837:
1817:
1808:
1807:
1806:
1786:
1780:
1779:
1760:
1744:
1738:
1737:
1724:
1715:
1706:
1705:
1696:
1676:
1635:Maxim Kontsevich
1585:
1547:
1410:
1408:
1407:
1402:
1371:
1369:
1368:
1363:
1361:
1350:
1329:
1327:
1326:
1321:
1265:
1261:
1220:
1219:
1197:
1195:
1194:
1189:
1181:
1180:
1157:
1155:
1154:
1149:
1141:
1140:
1105:
1103:
1102:
1097:
1092:
1091:
1050:
1048:
1047:
1042:
1012:
1010:
1009:
1004:
948:
944:
897:
896:
874:
872:
871:
866:
858:
857:
834:
832:
831:
826:
812:
811:
779:
756:
749:
734:
727:
601:
599:
598:
593:
588:
584:
580:
578:
577:
576:
563:
562:
561:
548:
539:
537:
536:
535:
522:
521:
520:
507:
484:
482:
481:
480:
467:
466:
465:
452:
443:
441:
440:
439:
426:
425:
424:
411:
398:
397:
372:with respect to
84:
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
2467:
2466:
2462:
2461:
2460:
2458:
2457:
2456:
2427:
2426:
2418:
2413:
2412:
2389:
2384:
2383:
2379:
2324:
2323:
2319:
2279:
2278:
2274:
2273:
2269:
2238:
2237:
2233:
2202:
2201:
2197:
2164:
2163:
2159:
2118:
2117:
2113:
2103:
2102:
2098:
2047:
2046:
2042:
2011:
2010:
2006:
1984:(2–3): 97–106,
1973:
1972:
1968:
1937:
1936:
1932:
1901:
1900:
1896:
1863:
1862:
1858:
1819:
1818:
1811:
1788:
1787:
1783:
1769:
1746:
1745:
1741:
1722:
1717:
1716:
1709:
1678:
1677:
1673:
1668:
1656:
1577:
1572:. For a given
1571:
1562:
1543:
1521:
1512:
1503:
1494:
1485:
1478:
1461:
1445:
1426:
1375:
1374:
1338:
1337:
1259:
1258:
1253:
1244:
1243:
1238:
1224:
1211:
1206:
1205:
1172:
1161:
1160:
1132:
1118:
1117:
1083:
1054:
1053:
1021:
1020:
942:
941:
936:
927:
926:
921:
901:
888:
883:
882:
849:
838:
837:
803:
792:
791:
762:
751:
743:
729:
728:has derivative
719:
681:
659:
636:
628:
622:
610:
582:
581:
568:
564:
553:
549:
545:
540:
527:
523:
512:
508:
503:
502:
497:
492:
486:
485:
472:
468:
457:
453:
449:
444:
431:
427:
416:
412:
402:
389:
384:
383:
377:
370:
359:Jacobian matrix
344:
307:
298:
291:
282:
273:
266:
259:
250:
213:
204:
197:
188:
176:
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
2465:
2463:
2455:
2454:
2449:
2444:
2439:
2429:
2428:
2425:
2424:
2417:
2416:External links
2414:
2411:
2410:
2377:
2333:(2): 209–218,
2317:
2307:(2): 435–452,
2286:
2267:
2231:
2195:
2157:
2111:
2096:
2040:
2020:(3): 423–440,
2004:
1966:
1946:(3): 243–248,
1930:
1910:(3): 235–239,
1894:
1874:(3): 303–313,
1856:
1828:(2): 287–330,
1824:, New Series,
1809:
1797:(2): 453–494,
1781:
1767:
1739:
1707:
1687:(1): 299–306,
1670:
1669:
1667:
1664:
1663:
1662:
1655:
1652:
1567:
1560:
1517:
1508:
1504:), where each
1499:
1490:
1483:
1476:
1472: = (
1460:
1457:
1443:
1424:
1421:The condition
1412:
1411:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1372:
1359:
1356:
1353:
1348:
1345:
1331:
1330:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1264:
1257:
1254:
1252:
1249:
1246:
1245:
1242:
1239:
1237:
1234:
1231:
1230:
1227:
1223:
1218:
1214:
1199:
1198:
1187:
1184:
1179:
1175:
1171:
1168:
1158:
1147:
1144:
1139:
1135:
1131:
1128:
1125:
1107:
1106:
1095:
1090:
1086:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1051:
1040:
1037:
1034:
1031:
1028:
1014:
1013:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
947:
940:
937:
935:
932:
929:
928:
925:
922:
920:
917:
914:
911:
908:
907:
904:
900:
895:
891:
876:
875:
864:
861:
856:
852:
848:
845:
835:
824:
821:
818:
815:
810:
806:
802:
799:
679:
674:characteristic
657:
635:
632:
626:
620:
608:
603:
602:
591:
587:
575:
571:
567:
560:
556:
552:
546:
544:
541:
534:
530:
526:
519:
515:
511:
505:
504:
501:
498:
496:
493:
491:
488:
487:
479:
475:
471:
464:
460:
456:
450:
448:
445:
438:
434:
430:
423:
419:
415:
409:
408:
405:
401:
396:
392:
375:
368:
342:
310:
309:
303:
296:
287:
278:
271:
264:
255:
248:
209:
202:
193:
186:
175:
172:
122:
121:
116:
112:
111:
108:
107:Conjectured in
104:
103:
98:
97:Conjectured by
94:
93:
88:
77:
76:
59:September 2020
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
2464:
2453:
2450:
2448:
2445:
2443:
2440:
2438:
2435:
2434:
2432:
2423:
2420:
2419:
2415:
2407:
2403:
2399:
2395:
2388:
2381:
2378:
2374:
2370:
2366:
2362:
2358:
2354:
2350:
2346:
2341:
2336:
2332:
2328:
2321:
2318:
2314:
2310:
2306:
2302:
2298:
2284:
2271:
2268:
2264:
2260:
2255:
2250:
2246:
2242:
2235:
2232:
2228:
2224:
2219:
2214:
2210:
2206:
2199:
2196:
2192:
2188:
2183:
2178:
2174:
2170:
2169:
2161:
2158:
2154:
2150:
2145:
2140:
2135:
2130:
2126:
2122:
2115:
2112:
2107:
2100:
2097:
2093:
2089:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2056:
2051:
2044:
2041:
2037:
2033:
2028:
2023:
2019:
2015:
2008:
2005:
2001:
1997:
1992:
1987:
1983:
1979:
1978:
1970:
1967:
1963:
1959:
1954:
1949:
1945:
1941:
1934:
1931:
1927:
1923:
1918:
1913:
1909:
1905:
1898:
1895:
1891:
1887:
1882:
1877:
1873:
1869:
1868:
1860:
1857:
1853:
1849:
1845:
1841:
1836:
1831:
1827:
1823:
1816:
1814:
1810:
1805:
1800:
1796:
1792:
1785:
1782:
1778:
1774:
1770:
1764:
1759:
1754:
1750:
1743:
1740:
1736:
1732:
1728:
1721:
1714:
1712:
1708:
1704:
1700:
1695:
1690:
1686:
1682:
1675:
1672:
1665:
1661:
1658:
1657:
1653:
1651:
1648:
1644:
1640:
1636:
1632:
1627:
1623:
1619:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1584:
1580:
1575:
1570:
1566:
1559:
1555:
1551:
1546:
1541:
1536:
1534:
1528:
1526:
1520:
1516:
1511:
1507:
1502:
1498:
1495: +
1493:
1489:
1482:
1479: +
1475:
1471:
1466:
1458:
1456:
1454:
1451: +
1450:
1446:
1439:
1435:
1431:
1427:
1419:
1417:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1373:
1357:
1354:
1351:
1346:
1343:
1336:
1335:
1334:
1317:
1314:
1311:
1308:
1302:
1296:
1293:
1290:
1284:
1275:
1272:
1266:
1262:
1255:
1250:
1247:
1240:
1235:
1232:
1225:
1221:
1216:
1212:
1204:
1203:
1202:
1185:
1182:
1177:
1173:
1169:
1166:
1159:
1145:
1142:
1137:
1133:
1129:
1126:
1123:
1116:
1115:
1114:
1113:slightly, to
1112:
1093:
1088:
1080:
1077:
1074:
1068:
1065:
1062:
1059:
1052:
1038:
1035:
1032:
1029:
1026:
1019:
1018:
1017:
1000:
997:
994:
991:
985:
979:
973:
964:
961:
958:
955:
949:
945:
938:
933:
930:
923:
918:
915:
912:
909:
902:
898:
893:
889:
881:
880:
879:
862:
859:
854:
850:
846:
843:
836:
822:
819:
816:
813:
808:
804:
800:
797:
790:
789:
788:
786:
781:
777:
773:
769:
765:
760:
754:
747:
742:
738:
733:
726:
722:
717:
713:
708:
704:
702:
698:
694:
690:
686:
682:
675:
671:
667:
662:
660:
653:
649:
645:
641:
633:
631:
629:
619:
615:
611:
589:
585:
573:
569:
558:
554:
542:
532:
528:
517:
513:
499:
494:
489:
477:
473:
462:
458:
446:
436:
432:
421:
417:
403:
399:
394:
390:
382:
381:
380:
378:
371:
364:
360:
357:
353:
349:
345:
339:, denoted by
338:
334:
329:
327:
323:
319:
315:
306:
302:
295:
290:
286:
281:
277:
270:
263:
258:
254:
247:
243:
240:
239:
238:
236:
232:
228:
225:
221:
217:
212:
208:
201:
198:in variables
196:
192:
185:
181:
173:
171:
169:
163:
161:
157:
153:
149:
145:
141:
137:
133:
129:
120:
117:
115:Equivalent to
113:
109:
105:
102:
99:
95:
92:
89:
85:
73:
70:
62:
52:
48:
42:
41:
35:
30:
21:
20:
2397:
2393:
2380:
2340:math/0512171
2330:
2326:
2320:
2304:
2300:
2297:-curvatures"
2270:
2244:
2240:
2234:
2208:
2204:
2198:
2175:(1): 43–46,
2172:
2166:
2160:
2124:
2120:
2114:
2105:
2099:
2059:
2053:
2043:
2017:
2013:
2007:
1981:
1975:
1969:
1943:
1939:
1933:
1907:
1903:
1897:
1871:
1865:
1859:
1825:
1821:
1794:
1790:
1784:
1748:
1742:
1726:
1684:
1680:
1674:
1646:
1638:
1628:
1624:
1620:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1582:
1578:
1573:
1568:
1564:
1557:
1553:
1549:
1544:
1539:
1537:
1532:
1529:
1524:
1518:
1514:
1509:
1505:
1500:
1496:
1491:
1487:
1480:
1473:
1469:
1462:
1452:
1448:
1441:
1437:
1422:
1420:
1415:
1413:
1332:
1200:
1110:
1108:
1015:
877:
784:
782:
775:
771:
767:
763:
758:
752:
736:
731:
724:
720:
715:
711:
709:
706:
696:
692:
688:
684:
677:
669:
665:
664:
655:
651:
647:
643:
639:
637:
624:
617:
613:
606:
604:
373:
366:
355:
351:
340:
336:
330:
321:
317:
313:
311:
304:
300:
293:
288:
284:
279:
275:
268:
261:
256:
252:
245:
241:
237:by setting:
234:
230:
226:
219:
216:coefficients
210:
206:
199:
194:
190:
183:
179:
177:
164:
143:
131:
125:
65:
56:
37:
2447:Conjectures
2437:Polynomials
2400:: 205–214,
1552:denote the
744: [
348:determinant
218:in a field
138:in several
136:polynomials
128:mathematics
51:introducing
2431:Categories
2247:(1): 1–4,
2144:2066/33302
2108:, preprint
1666:References
616:variables
34:references
2313:0030-6126
2211:: 83–92,
2092:116143599
2076:0075-4102
1844:1088-9485
1703:0026-9255
1393:−
1355:−
1291:−
1078:−
1069:−
1036:−
980:−
699:which is
566:∂
551:∂
543:⋯
525:∂
510:∂
500:⋮
495:⋱
490:⋮
470:∂
455:∂
447:⋯
429:∂
414:∂
140:variables
2373:15150838
1654:See also
312:Any map
160:calculus
2406:2368008
2365:2337879
2345:Bibcode
2263:1292168
2227:2208537
2191:2183036
2153:2138860
2084:0691964
2036:0620428
2000:0531253
1962:0324062
1926:0701351
1890:0714105
1852:0663785
1777:1352692
1735:1601194
1563:, ...,
1459:Results
701:regular
654:, then
623:, ...,
350:of the
283:),...,
251:, ...,
205:, ...,
189:, ...,
47:improve
2404:
2371:
2363:
2311:
2261:
2225:
2189:
2151:
2090:
2082:
2074:
2034:
1998:
1960:
1924:
1888:
1850:
1842:
1775:
1765:
1733:
1701:
1594:) and
1465:degree
755:> 0
676:0. If
274:, ...,
130:, the
36:, but
2390:(PDF)
2369:S2CID
2335:arXiv
2088:S2CID
1723:(PDF)
748:]
672:have
605:then
299:,...,
260:) = (
214:with
87:Field
2309:ISSN
2072:ISSN
2060:1983
1840:ISSN
1763:ISBN
1699:ISSN
1548:and
1538:Let
770:) /
730:1 −
668:Let
331:The
178:Let
110:1939
2353:doi
2249:doi
2245:217
2213:doi
2177:doi
2139:hdl
2129:doi
2125:133
2064:doi
2022:doi
1986:doi
1948:doi
1944:205
1912:doi
1876:doi
1872:264
1830:doi
1799:doi
1753:doi
1689:doi
1645:in
1432:in
732:p x
365:of
335:of
308:)).
126:In
2433::
2402:MR
2398:32
2396:,
2392:,
2367:,
2361:MR
2359:,
2351:,
2343:,
2329:,
2305:42
2303:,
2299:,
2259:MR
2257:,
2243:,
2223:MR
2221:,
2209:87
2207:,
2187:MR
2185:,
2173:86
2171:,
2149:MR
2147:,
2137:,
2123:,
2086:,
2080:MR
2078:,
2070:,
2058:,
2052:,
2032:MR
2030:,
2018:25
2016:,
1996:MR
1994:,
1982:32
1980:,
1958:MR
1956:,
1942:,
1922:MR
1920:,
1908:28
1906:,
1886:MR
1884:,
1870:,
1848:MR
1846:,
1838:,
1812:^
1795:65
1793:,
1773:MR
1771:,
1761:,
1731:MR
1725:,
1710:^
1697:,
1685:47
1683:,
1639:2N
1581:=
1527:.
1001:1.
780:.
746:ht
737:px
723:−
695:→
691::
650:→
646::
630:.
379::
354:×
328:.
320:→
316::
233:→
229::
170:.
162:.
2355::
2347::
2337::
2331:7
2285:p
2251::
2215::
2179::
2141::
2131::
2066::
2024::
1988::
1950::
1914::
1878::
1832::
1826:7
1801::
1755::
1691::
1647:N
1616:F
1614:(
1612:k
1608:X
1606:(
1604:k
1600:F
1598:(
1596:k
1592:X
1590:(
1588:k
1583:k
1579:k
1574:F
1569:n
1565:f
1561:1
1558:f
1554:k
1550:k
1545:k
1540:k
1533:N
1525:N
1519:i
1515:H
1510:i
1506:H
1501:n
1497:H
1492:n
1488:X
1484:1
1481:H
1477:1
1474:X
1470:F
1453:x
1449:x
1444:F
1442:J
1438:F
1425:F
1423:J
1416:x
1399:,
1396:u
1390:v
1387:2
1384:=
1381:y
1358:v
1352:u
1347:=
1344:x
1318:,
1315:x
1312:2
1309:=
1306:)
1303:1
1300:(
1297:x
1294:2
1288:)
1285:1
1282:(
1279:)
1276:x
1273:4
1270:(
1267:=
1263:|
1256:1
1251:x
1248:2
1241:1
1236:x
1233:4
1226:|
1222:=
1217:F
1213:J
1186:y
1183:+
1178:2
1174:x
1170:=
1167:v
1146:y
1143:+
1138:2
1134:x
1130:2
1127:=
1124:u
1111:F
1094:.
1089:2
1085:)
1081:v
1075:u
1072:(
1066:v
1063:=
1060:y
1039:v
1033:u
1030:=
1027:x
998:=
995:x
992:2
989:)
986:1
983:(
977:)
974:1
971:(
968:)
965:x
962:2
959:+
956:1
953:(
950:=
946:|
939:1
934:x
931:2
924:1
919:x
916:2
913:+
910:1
903:|
899:=
894:F
890:J
863:y
860:+
855:2
851:x
847:=
844:v
823:x
820:+
817:y
814:+
809:2
805:x
801:=
798:u
785:F
778:)
776:F
774:(
772:k
768:X
766:(
764:k
759:p
753:p
725:x
721:x
716:p
712:k
697:k
693:k
689:G
685:F
680:F
678:J
670:k
658:F
656:J
652:k
648:k
644:G
640:F
627:N
625:X
621:1
618:X
614:N
609:F
607:J
590:,
586:|
574:N
570:X
559:N
555:f
533:1
529:X
518:N
514:f
478:N
474:X
463:1
459:f
437:1
433:X
422:1
418:f
404:|
400:=
395:F
391:J
376:j
374:X
369:i
367:f
356:N
352:N
343:F
341:J
337:F
322:k
318:k
314:F
305:N
301:X
297:1
294:X
292:(
289:N
285:f
280:N
276:X
272:1
269:X
267:(
265:1
262:f
257:N
253:X
249:1
246:X
244:(
242:F
235:k
231:k
227:F
220:k
211:N
207:X
203:1
200:X
195:N
191:f
187:1
184:f
180:N
144:n
72:)
66:(
61:)
57:(
43:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.