216:
who proved that any compact smooth manifold is diffeomorphic to some affine real algebraic manifold; actually, any Nash manifold is Nash diffeomorphic to an affine real algebraic manifold. These results exemplify the fact that the Nash category is somewhat intermediate between the smooth and the
211:
to some Nash manifold. More generally, a smooth manifold admits a Nash manifold structure if and only if it is diffeomorphic to the interior of some compact smooth manifold possibly with boundary. Nash's result was later (1973) completed by
611:
715:
162:
459:
510:
375:
351:
323:
295:
408:
736:
instead of the field of real numbers, and the above statements still hold. Abstract Nash functions can also be defined on the real spectrum of any commutative ring.
486:
526:
630:
199:. A Nash mapping between Nash manifolds is then an analytic mapping with semialgebraic graph. Nash functions and manifolds are named after
265:
was proved independently (1973) by Jean-Jacques Risler and
Gustave Efroymson. Nash manifolds have properties similar to but weaker than
261:
The global properties are more difficult to obtain. The fact that the ring of Nash functions on a Nash manifold (even noncompact) is
748:
M. Coste, J.M. Ruiz and M. Shiota: Global problems on Nash functions. Revista Matemática
Complutense 17 (2004), 83--115.
776:
757:
J-J. Risler: Sur l'anneau des fonctions de Nash globales. C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1513--A1516.
175:-th eigenvalue (in increasing order) is Nash on the open subset of symmetric matrices with no multiple eigenvalue.
115:
is a polynomial, by taking finite unions, finite intersections and complements). Some examples of Nash functions:
124:
421:
266:
491:
356:
332:
304:
276:
204:
17:
380:
200:
464:
8:
241:
variables (i.e., those series satisfying a nontrivial polynomial equation), which is the
763:
A. Tognoli: Su una congettura di Nash. Ann. Scuola Norm. Sup. Pisa 27 (1973), 167--185.
246:
226:
754:
J.F. Nash : Real algebraic manifolds. Annals of
Mathematics 56 (1952), 405--421.
751:
G. Efroymson: A Nullstellensatz for Nash rings. Pacific J. Math. 54 (1974), 101--112.
733:
242:
180:
72:
33:
606:{\displaystyle H^{0}(M,{\mathcal {N}})\to H^{0}(M,{\mathcal {N}}/{\mathcal {I}})}
262:
213:
326:
270:
770:
208:
745:
J. Bochnak, M. Coste and M-F. Roy: Real algebraic geometry. Springer, 1998.
234:
710:{\displaystyle H^{1}(M,{\mathcal {N}})\neq 0,\ {\text{if}}\ \dim(M)>0,}
225:
The local properties of Nash functions are well understood. The ring of
377:
is finite, i.e., there exists a finite open semialgebraic covering
245:
of the ring of germs of rational functions. In particular, it is a
179:
Nash functions are those functions needed in order to have an
229:
of Nash functions at a point of a Nash manifold of dimension
171:
the function which associates to a real symmetric matrix its
119:
Polynomial and regular rational functions are Nash functions.
297:
denote the sheaf of Nash function germs on a Nash manifold
195:, which are semialgebraic analytic submanifolds of some
732:
Nash functions and manifolds can be defined over any
633:
529:
494:
467:
424:
383:
359:
335:
307:
279:
127:
709:
605:
504:
480:
453:
402:
369:
345:
317:
289:
156:
768:
79:is a subset obtained from subsets of the form {
397:
384:
47:satisfying a nontrivial polynomial equation
724:contrarily to the case of Stein manifolds.
512:is globally generated by Nash functions on
207:admits a Nash manifold structure, i.e., is
760:M. Shiota: Nash manifolds. Springer, 1987.
157:{\displaystyle x\mapsto {\sqrt {1+x^{2}}}}
454:{\displaystyle {\mathcal {I}}|_{U_{i}}}
233:is isomorphic to the ring of algebraic
769:
191:Along with Nash functions one defines
203:, who proved (1952) that any compact
256:
183:theorem in real algebraic geometry.
220:
13:
727:
655:
595:
583:
551:
497:
461:is generated by Nash functions on
427:
362:
338:
310:
282:
14:
788:
186:
24:on an open semialgebraic subset
695:
689:
660:
644:
600:
572:
559:
556:
540:
505:{\displaystyle {\mathcal {I}}}
434:
370:{\displaystyle {\mathcal {I}}}
346:{\displaystyle {\mathcal {N}}}
318:{\displaystyle {\mathcal {I}}}
290:{\displaystyle {\mathcal {N}}}
131:
1:
7:
10:
793:
739:
403:{\displaystyle \{U_{i}\}}
267:Cartan's theorems A and B
777:Real algebraic geometry
620:is surjective. However
18:real algebraic geometry
711:
607:
516:, and the natural map
506:
482:
455:
404:
371:
347:
319:
291:
217:algebraic categories.
158:
712:
608:
507:
483:
481:{\displaystyle U_{i}}
456:
405:
372:
348:
320:
292:
201:John Forbes Nash, Jr.
159:
631:
527:
492:
465:
422:
414:such that, for each
381:
357:
333:
305:
277:
125:
73:semialgebraic subset
707:
603:
502:
478:
451:
400:
367:
343:
315:
287:
247:regular local ring
154:
734:real closed field
682:
678:
674:
257:Global properties
181:implicit function
152:
111:) > 0}, where
34:analytic function
784:
716:
714:
713:
708:
680:
679:
676:
672:
659:
658:
643:
642:
612:
610:
609:
604:
599:
598:
592:
587:
586:
571:
570:
555:
554:
539:
538:
511:
509:
508:
503:
501:
500:
487:
485:
484:
479:
477:
476:
460:
458:
457:
452:
450:
449:
448:
447:
437:
431:
430:
409:
407:
406:
401:
396:
395:
376:
374:
373:
368:
366:
365:
353:-ideals. Assume
352:
350:
349:
344:
342:
341:
324:
322:
321:
316:
314:
313:
296:
294:
293:
288:
286:
285:
221:Local properties
163:
161:
160:
155:
153:
151:
150:
135:
792:
791:
787:
786:
785:
783:
782:
781:
767:
766:
742:
730:
728:Generalizations
675:
654:
653:
638:
634:
632:
629:
628:
594:
593:
588:
582:
581:
566:
562:
550:
549:
534:
530:
528:
525:
524:
496:
495:
493:
490:
489:
472:
468:
466:
463:
462:
443:
439:
438:
433:
432:
426:
425:
423:
420:
419:
391:
387:
382:
379:
378:
361:
360:
358:
355:
354:
337:
336:
334:
331:
330:
309:
308:
306:
303:
302:
281:
280:
278:
275:
274:
271:Stein manifolds
259:
223:
214:Alberto Tognoli
205:smooth manifold
189:
146:
142:
134:
126:
123:
122:
63:)) = 0 for all
12:
11:
5:
790:
780:
779:
765:
764:
761:
758:
755:
752:
749:
746:
741:
738:
729:
726:
722:
721:
720:
719:
718:
717:
706:
703:
700:
697:
694:
691:
688:
685:
671:
668:
665:
662:
657:
652:
649:
646:
641:
637:
618:
617:
616:
615:
614:
613:
602:
597:
591:
585:
580:
577:
574:
569:
565:
561:
558:
553:
548:
545:
542:
537:
533:
499:
475:
471:
446:
442:
436:
429:
399:
394:
390:
386:
364:
340:
327:coherent sheaf
312:
284:
258:
255:
222:
219:
193:Nash manifolds
188:
187:Nash manifolds
185:
177:
176:
169:
149:
145:
141:
138:
133:
130:
120:
9:
6:
4:
3:
2:
789:
778:
775:
774:
772:
762:
759:
756:
753:
750:
747:
744:
743:
737:
735:
725:
704:
701:
698:
692:
686:
683:
669:
666:
663:
650:
647:
639:
635:
627:
626:
625:
624:
623:
622:
621:
589:
578:
575:
567:
563:
546:
543:
535:
531:
523:
522:
521:
520:
519:
518:
517:
515:
473:
469:
444:
440:
417:
413:
392:
388:
328:
300:
272:
268:
264:
254:
252:
249:of dimension
248:
244:
243:henselization
240:
236:
232:
228:
218:
215:
210:
209:diffeomorphic
206:
202:
198:
194:
184:
182:
174:
170:
167:
147:
143:
139:
136:
128:
121:
118:
117:
116:
114:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
35:
31:
27:
23:
22:Nash function
19:
731:
723:
619:
513:
415:
411:
298:
260:
250:
238:
235:power series
230:
224:
196:
192:
190:
178:
172:
165:
112:
108:
104:
100:
96:
92:
88:
84:
80:
76:
68:
64:
60:
56:
52:
48:
44:
40:
36:
29:
25:
21:
15:
164:is Nash on
263:noetherian
687:
664:≠
560:→
132:↦
95:)=0} or {
771:Category
103: :
87: :
740:Sources
488:. Then
681:
673:
301:, and
273:. Let
32:is an
325:be a
227:germs
699:>
20:, a
684:dim
410:of
329:of
269:on
237:in
99:in
83:in
75:of
71:(A
67:in
16:In
773::
677:if
418:,
253:.
43:→
39::
28:⊂
705:,
702:0
696:)
693:M
690:(
670:,
667:0
661:)
656:N
651:,
648:M
645:(
640:1
636:H
601:)
596:I
590:/
584:N
579:,
576:M
573:(
568:0
564:H
557:)
552:N
547:,
544:M
541:(
536:0
532:H
514:M
498:I
474:i
470:U
445:i
441:U
435:|
428:I
416:i
412:M
398:}
393:i
389:U
385:{
363:I
339:N
311:I
299:M
283:N
251:n
239:n
231:n
197:R
173:i
168:.
166:R
148:2
144:x
140:+
137:1
129:x
113:P
109:x
107:(
105:P
101:R
97:x
93:x
91:(
89:P
85:R
81:x
77:R
69:U
65:x
61:x
59:(
57:f
55:,
53:x
51:(
49:P
45:R
41:U
37:f
30:R
26:U
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