1499:
784:
416:
196:
491:
1173:
687:
1775:
253:
895:
1361:
1076:
1251:
587:
1205:
1646:
536:
102:
1369:
973:
Warning proved another theorem, known as
Warning's second theorem, which states that if the system of polynomial equations has the trivial solution, then it has at least
1119:
1004:
1286:
1693:
1604:
1529:
314:
283:
1806:
1713:
1666:
1624:
1309:
1024:
915:
831:
808:
627:
607:
514:
692:
322:
286:
1841:
107:
429:
1880:
1551:
in 1935. The motivation behind Artin's conjecture was his observation that quasi-algebraically closed fields have trivial
17:
1127:
639:
2046:
1725:
67:
1556:
204:
836:
2084:
1821:
2089:
1314:
1029:
1210:
541:
1544:
1181:
689:, that is, if the polynomials have no constant terms, then the system also has a non-trivial solution
1864:
1504:
because each term is 1 for a solution and 0 otherwise. If the sum of the degrees of the polynomials
1494:{\displaystyle \sum _{x\in \mathbb {F} ^{n}}(1-f_{1}^{q-1}(x))\cdot \ldots \cdot (1-f_{r}^{q-1}(x))}
494:
1629:
519:
316:. The theorems are statements about the solutions of the following system of polynomial equations
85:
2079:
63:
1092:
976:
2038:
2031:
1256:
1921:
1890:
1671:
1582:
1507:
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261:
2019:
2011:
1960:
1952:
8:
1785:
1781:
1791:
1698:
1651:
1609:
1294:
1009:
900:
816:
793:
790:
Chevalley's theorem is an immediate consequence of the
Chevalley–Warning theorem since
612:
592:
499:
2042:
2026:
1876:
1809:
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and only the trivial solution. Alternatively, using just one polynomial, we can take
2015:
2007:
1999:
1979:
1956:
1948:
1940:
1928:
1909:
1716:
779:{\displaystyle (a_{1},\dots ,a_{n})\in \mathbb {F} ^{n}\backslash \{(0,\dots ,0)\}}
51:
27:
Certain polynomial equations in enough variables over a finite field have solutions
1917:
1886:
1872:
1990:
Warning, Ewald (1935), "Bemerkung zur vorstehenden Arbeit von Herrn
Chevalley",
1784:
as a divisibility result for the (reciprocals of) the zeroes and poles of the
2073:
1967:
1846:
1576:
411:{\displaystyle f_{j}(x_{1},\dots ,x_{n})=0\quad {\text{for}}\,j=1,\ldots ,r.}
1552:
39:
1555:, together with the fact that finite fields have trivial Brauer group by
2003:
1944:
1860:
1548:
929:
59:
35:
2062:
1983:
1913:
198:
be a set of polynomials such that the number of variables satisfies
1992:
1933:
1897:
1572:
1543:
It is a consequence of
Chevalley's theorem that finite fields are
813:
Both theorems are best possible in the sense that, given any
538:. Or in other words, the cardinality of the vanishing set of
191:{\displaystyle \{f_{j}\}_{j=1}^{r}\subseteq \mathbb {F} }
42:
have solutions. It was proved by Ewald Warning (
486:{\displaystyle (a_{1},\dots ,a_{n})\in \mathbb {F} ^{n}}
1078:. Chevalley's theorem also follows directly from this.
746:
46:) and a slightly weaker form of the theorem, known as
1931:(1935), "Démonstration d'une hypothèse de M. Artin",
1842:"Number of Solutions to Polynomials in Finite Fields"
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110:
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1900:(1964), "Zeros of polynomials over finite fields",
636:states that if the system has the trivial solution
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1168:{\displaystyle \sum _{x\in \mathbb {F} }x^{i}=0}
682:{\displaystyle (0,\dots ,0)\in \mathbb {F} ^{n}}
1770:{\displaystyle {\frac {n-\sum _{j}d_{j}}{d}}.}
1081:
1780:The Ax–Katz result has an interpretation in
1291:The total number of common solutions modulo
773:
749:
559:
545:
125:
111:
77:
1648:dividing the number of solutions; here, if
426:states that the number of common solutions
966:form a basis of the finite field of order
2063:"Proofs of the Chevalley-Warning theorem"
1927:
1634:
1386:
1187:
1143:
736:
669:
524:
473:
383:
248:{\displaystyle n>\sum _{j=1}^{r}d_{j}}
149:
90:
55:
1535:then this vanishes by the remark above.
890:{\displaystyle f_{j}=x_{j},j=1,\dots ,n}
1989:
43:
14:
2072:
1562:
38:in sufficiently many variables over a
2025:
1859:
1579:, determines more accurately a power
1538:
1356:{\displaystyle f_{1},\ldots ,f_{r}=0}
1071:{\displaystyle d:=d_{1}+\dots +d_{r}}
71:
1966:
1026:is the size of the finite field and
1246:{\displaystyle x_{1},\ldots ,x_{n}}
582:{\displaystyle \{f_{j}\}_{j=1}^{r}}
24:
1896:
66:conjecture that finite fields are
25:
2101:
2055:
68:quasi-algebraically closed fields
1200:{\displaystyle \mathbb {F} ^{n}}
1902:American Journal of Mathematics
1547:. This had been conjectured by
377:
58:). Chevalley's theorem implied
1970:(1971), "On a theorem of Ax",
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13:
1:
1827:
1822:Combinatorial Nullstellensatz
1788:. Namely, the same power of
1641:{\displaystyle \mathbb {F} }
531:{\displaystyle \mathbb {F} }
97:{\displaystyle \mathbb {F} }
7:
1815:
10:
2106:
1545:quasi-algebraically closed
1082:Proof of Warning's theorem
424:Chevalley–Warning theorem
78:Statement of the theorems
32:Chevalley–Warning theorem
1114:{\displaystyle i<q-1}
928:polynomial given by the
999:{\displaystyle q^{n-d}}
30:In number theory, the
2033:A course in arithmetic
1863:(1982), Lang, Serge.;
1808:divides each of these
1802:
1771:
1709:
1689:
1668:is the largest of the
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1642:
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1281:{\displaystyle n(q-1)}
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104:be a finite field and
98:
1803:
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1710:
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1688:{\displaystyle d_{j}}
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1643:
1621:
1601:
1599:{\displaystyle q^{b}}
1526:
1524:{\displaystyle f_{i}}
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1207:of any polynomial in
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309:{\displaystyle f_{j}}
280:
278:{\displaystyle d_{j}}
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214:
193:
99:
34:implies that certain
2085:Diophantine geometry
1871:, Berlin, New York:
1792:
1726:
1715:can be taken as the
1699:
1695:, then the exponent
1672:
1652:
1630:
1610:
1583:
1557:Wedderburn's theorem
1508:
1370:
1315:
1295:
1257:
1253:of degree less than
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493:is divisible by the
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323:
293:
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205:
108:
86:
36:polynomial equations
2090:Theorems in algebra
1786:local zeta-function
1606:of the cardinality
1563:The Ax–Katz theorem
1478:
1427:
962:where the elements
578:
144:
48:Chevalley's theorem
2027:Serre, Jean-Pierre
2004:10.1007/BF02940715
1945:10.1007/BF02940714
1810:algebraic integers
1798:
1767:
1747:
1705:
1685:
1658:
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1539:Artin's conjecture
1521:
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188:
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1968:Katz, Nicholas M.
1929:Chevalley, Claude
1882:978-0-387-90686-7
1801:{\displaystyle q}
1762:
1738:
1708:{\displaystyle b}
1661:{\displaystyle d}
1619:{\displaystyle q}
1373:
1304:{\displaystyle p}
1131:
1019:{\displaystyle q}
924:to be the degree
910:{\displaystyle n}
897:has total degree
826:{\displaystyle n}
803:{\displaystyle p}
634:Chevalley theorem
622:{\displaystyle p}
602:{\displaystyle 0}
509:{\displaystyle p}
381:
16:(Redirected from
2097:
2066:
2051:
2036:
2022:
1986:
1963:
1924:
1893:
1869:Collected papers
1852:
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1807:
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1804:
1799:
1782:étale cohomology
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1717:ceiling function
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1178:so the sum over
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1006:solutions where
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50:, was proved by
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2104:
2100:
2099:
2098:
2096:
2095:
2094:
2070:
2069:
2061:
2058:
2049:
1984:10.2307/2373389
1914:10.2307/2373163
1883:
1873:Springer-Verlag
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1569:Ax–Katz theorem
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1367:
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1337:
1322:
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1316:
1313:
1312:
1296:
1293:
1292:
1288:also vanishes.
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1255:
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1185:
1183:
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1125:
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1084:
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1058:
1043:
1039:
1031:
1028:
1027:
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1007:
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978:
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974:
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938:
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902:
899:
898:
857:
853:
844:
840:
838:
835:
834:
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815:
814:
810:is at least 2.
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105:
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22:
18:Ax–Katz theorem
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2056:External links
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1987:
1978:(2): 485–499,
1972:Amer. J. Math.
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1571:, named after
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2080:Finite fields
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2048:0-387-90040-3
2044:
2040:
2035:
2034:
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2017:
2013:
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2001:
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1994:(in German),
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1935:(in French),
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1847:StackExchange
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1702:
1680:
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1613:
1591:
1587:
1578:
1577:Nicholas Katz
1574:
1570:
1560:
1558:
1554:
1550:
1546:
1536:
1534:
1531:is less than
1516:
1512:
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1471:
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1440:
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1334:
1331:
1328:
1323:
1319:
1298:
1289:
1272:
1269:
1266:
1260:
1238:
1234:
1230:
1227:
1224:
1219:
1215:
1192:
1162:
1159:
1154:
1150:
1139:
1136:
1132:
1124:
1123:
1122:
1108:
1105:
1102:
1099:
1096:
1088:
1079:
1063:
1059:
1055:
1052:
1049:
1044:
1040:
1036:
1033:
1013:
991:
988:
985:
981:
971:
969:
965:
960:
956:
952:
948:
941:
935:
931:
927:
920:
904:
884:
881:
878:
875:
872:
869:
866:
863:
858:
854:
850:
845:
841:
820:
811:
797:
767:
764:
761:
758:
755:
741:
731:
723:
719:
715:
712:
709:
704:
700:
674:
664:
658:
655:
652:
649:
646:
635:
631:
616:
596:
574:
569:
566:
563:
553:
549:
503:
496:
478:
468:
460:
456:
452:
449:
446:
441:
437:
425:
421:
420:
405:
402:
399:
396:
393:
390:
387:
384:
374:
371:
363:
359:
355:
352:
349:
344:
340:
331:
327:
319:
318:
317:
301:
297:
288:
270:
266:
240:
236:
230:
225:
222:
219:
215:
211:
208:
201:
200:
199:
180:
176:
172:
169:
166:
161:
157:
145:
140:
135:
132:
129:
119:
115:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
2032:
1995:
1991:
1975:
1971:
1936:
1932:
1905:
1901:
1868:
1845:
1836:
1779:
1568:
1566:
1553:Brauer group
1542:
1532:
1503:
1363:is equal to
1290:
1177:
1086:
1085:
972:
967:
963:
958:
954:
950:
946:
939:
933:
925:
918:
812:
789:
633:
423:
287:total degree
257:
81:
47:
40:finite field
31:
29:
2037:, pp.
1908:: 255–261,
1861:Artin, Emil
833:, the list
74:, page x).
2074:Categories
2020:0011.14601
2012:61.1043.02
1961:0011.14504
1953:61.1043.01
1865:Tate, John
1828:References
1549:Emil Artin
72:Artin 1982
1998:: 76–83,
1939:: 73–75,
1898:Ax, James
1740:∑
1736:−
1472:−
1456:−
1447:⋅
1444:…
1441:⋅
1421:−
1405:−
1382:∈
1375:∑
1332:…
1270:−
1228:…
1140:∈
1133:∑
1106:−
1053:⋯
989:−
879:…
762:…
747:∖
732:∈
713:…
665:∈
653:…
469:∈
450:…
397:…
353:…
216:∑
170:…
146:⊆
64:Dickson's
52:Chevalley
2029:(1973),
1867:(eds.),
1816:See also
1573:James Ax
945:+ ... +
1922:0160775
1891:0671416
1087:Remark:
609:modulo
285:is the
60:Artin's
54: (
2045:
2018:
2010:
1959:
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1920:
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2043:ISBN
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930:norm
632:The
422:The
212:>
82:Let
62:and
56:1935
44:1935
2039:5–6
2016:Zbl
2008:JFM
2000:doi
1980:doi
1957:Zbl
1949:JFM
1941:doi
1910:doi
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1626:of
1311:of
1089:If
932:of
589:is
516:of
380:for
289:of
2076::
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2014:,
2006:,
1996:11
1976:93
1974:,
1955:,
1947:,
1937:11
1918:MR
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1906:86
1904:,
1887:MR
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2065:.
2002::
1982::
1943::
1912::
1850:.
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