314:
over the complex numbers) and specific linear actions on polynomial rings, i.e. actions coming from finite-dimensional representations of the Lie-group. This finiteness result was later extended by
177:
189:
Some results were obtained confirming
Hilbert's conjecture in special cases and for certain classes of rings (in particular the conjecture was proved unconditionally for
808:
205:
found a counterexample to
Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants for the action of a
647:, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 31, Bombay: Tata Institute of Fundamental Research,
763:Éfendiev, F. F. (1992). "Explicit construction of elements of the ring S(n, r) of invariants of n-ary forms of degree R".
265:
which are invariant under the action. A classical example in nineteenth century was the extensive study (in particular by
801:
944:
933:
938:
928:
388:Éfendiev F.F. (Fuad Efendi) provided symmetric algorithm generating basis of invariants of n-ary forms of degree r.
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913:
888:
382:
966:
923:
903:
898:
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607:
878:
883:
858:
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Several authors have reduced the sizes of the group and the vector space in Nagata's example. For example,
863:
843:
833:
659:
Totaro, Burt (2008), "Hilbert's 14th problem over finite fields and a conjecture on the cone of curves",
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853:
848:
838:
828:
615:
359:
240:(or more generally, on a finitely generated algebra defined over a field). In this situation the field
36:
971:
299:) on it. Hilbert himself proved the finite generation of invariant rings in the case of the field of
109:
489:
is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on
338:
270:
226:
206:
817:
319:
28:
698:
652:
623:
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8:
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to the class of all semi-simple Lie-groups. A major ingredient in
Hilbert's proof is the
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682:
377:
Zariski's formulation was shown to be equivalent to the original problem, for
960:
690:
355:
334:
266:
257:
which are invariant under the given action of the algebraic group, the ring
427:=1, 2, 3 that are algebraically independent over the prime field. The ring
315:
706:
Interpretations algebrico-geometriques du quatorzieme probleme de
Hilbert
304:
278:
20:
776:
182:
Hilbert conjectured that all such algebras are finitely generated over
726:
Winkelmann, Jörg (2003), "Invariant rings and quasiaffine quotients",
740:
307:
627:
225:
is given as a (suitably defined) ring of polynomial invariants of a
337:'s formulation of Hilbert's fourteenth problem asks whether, for a
708:, Bulletin des Sciences Mathematiques 78 (1954), pp. 155–168.
673:
399:
gave the following counterexample to
Hilbert's problem. The field
198:
556:
showed that over any field there is an action of the sum
326:
inside the polynomial ring generated by the invariants.
248:
functions (quotients of polynomials) in the variables
112:
171:
958:
285:in two variables with the natural action of the
816:
802:
641:Lectures on the fourteenth problem of Hilbert
809:
795:
725:
391:
217:The problem originally arose in algebraic
739:
672:
568:of three copies of the additive group on
461:orthogonal to each of the three vectors (
762:
541:invariant under the action of the group
329:
42:The setting is as follows: Assume that
31:proposed in 1900, asks whether certain
16:Are certain algebras finitely generated
959:
658:
634:
608:"On the fourteenth problem of Hilbert"
602:
553:
439:is a 13-dimensional vector space over
396:
790:
612:Proc. Internat. Congress Math. 1958
281:and also Hilbert) of invariants of
13:
435:in 32 variables. The vector space
403:is a field containing 48 elements
14:
983:
537:. Then the ring of elements of
756:
719:
160:
128:
54:be a subfield of the field of
1:
591:
172:{\displaystyle R:=K\cap k\ .}
586:Locally nilpotent derivation
545:is not a finitely generated
383:Zariski's finiteness theorem
103:defined as the intersection
25:Hilbert's fourteenth problem
7:
579:
576:is not finitely generated.
485:=1, 2, 3. The vector space
443:consisting of all vectors (
370:is finitely generated over
10:
988:
616:Cambridge University Press
233:acting algebraically on a
212:
824:
750:10.1007/s00209-002-0484-9
683:10.1112/S0010437X08003667
27:, that is, number 14 of
493:by fixing all elements
431:is the polynomial ring
392:Nagata's counterexample
201:in 1954). Then in 1959
661:Compositio Mathematica
227:linear algebraic group
207:linear algebraic group
173:
330:Zariski's formulation
320:Hilbert basis theorem
174:
618:, pp. 459–462,
351:, possibly assuming
312:general linear group
287:special linear group
110:
381:normal. (See also:
310:(in particular the
303:for some classical
967:Hilbert's problems
818:Hilbert's problems
777:10.1007/BF02102130
765:Mathematical Notes
574:ring of invariants
169:
56:rational functions
37:finitely generated
29:Hilbert's problems
954:
953:
636:Nagata, Masayoshi
604:Nagata, Masayoshi
364:regular functions
342:algebraic variety
165:
93:Consider now the
979:
972:Invariant theory
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676:
667:(5): 1176–1198,
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626:, archived from
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244:is the field of
221:. Here the ring
219:invariant theory
203:Masayoshi Nagata
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510:
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322:applied to the
301:complex numbers
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261:is the ring of
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235:polynomial ring
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75:
17:
12:
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5:
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771:(2): 204–207.
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734:(1): 163–174,
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362:, the ring of
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630:on 2011-07-17
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554:Totaro (2008)
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397:Nagata (1960)
389:
386:
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375:
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365:
361:
357:
354:
350:
347:over a field
346:
343:
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336:
327:
325:
321:
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309:
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288:
284:
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243:
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229:over a field
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49:
45:
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38:
34:
30:
26:
22:
893:
768:
764:
758:
741:math/0007076
731:
727:
721:
705:
704:O. Zariski,
664:
660:
640:
628:the original
611:
597:Bibliography
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344:
339:quasi-affine
333:
316:Hermann Weyl
296:
289:
283:binary forms
262:
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245:
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194:
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181:
100:
95:
92:
86:
81:
77:
70:
66:
59:
51:
43:
41:
24:
18:
502:and taking
305:semi-simple
279:Paul Gordan
263:polynomials
62:variables,
21:mathematics
961:Categories
592:References
549:-algebra.
308:Lie groups
713:Footnotes
691:0010-437X
674:0808.0695
606:(1960) ,
271:Sylvester
145:…
123:∩
728:Math. Z.
638:(1965),
580:See also
246:rational
193:= 1 and
98:-algebra
50:and let
33:algebras
699:2457523
653:0215828
624:0116056
423:, for
335:Zariski
275:Clebsch
213:History
199:Zariski
197:= 2 by
85:) over
76:, ...,
697:
689:
651:
622:
572:whose
481:) for
471:, ...,
413:, ...,
360:smooth
356:normal
267:Cayley
164:
736:arXiv
669:arXiv
645:(PDF)
457:) in
450:,...,
324:ideal
48:field
46:is a
687:ISSN
35:are
773:doi
746:doi
732:244
679:doi
665:144
511:to
385:.)
366:on
358:or
58:in
19:In
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889:13
884:12
879:11
874:10
769:51
767:.
744:,
730:,
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693:,
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614:,
610:,
520:+
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290:SL
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