740:
701:
662:
623:
501:. The geometric content of this statement is that the projection of a d-constructible set is d-constructible. It also eliminates imaginaries, is complete, and model complete.
477:
A d-constructible set is a finite union of closed and open sets in the
Kolchin topology. Equivalently, a d-constructible set is the set of solutions to a quantifier-free, or
348:
Unlike the complex numbers in the theory of algebraically closed fields, there is no natural example of a differentially closed field. Any differentially perfect field
902:
837:
364:. Shelah also showed that the prime differentially closed field of characteristic 0 (the differential closure of the rationals) is not
58:. Differentially closed fields are the analogues for differential equations of algebraically closed fields for polynomial equations.
985:
578:
This correspondence maps a ā-closed subset to the ideal of elements vanishing on it, and maps an ideal to its set of zeros.
360:
extension, which is differentially closed. Shelah showed that the differential closure is unique up to isomorphism over
368:; this was a rather surprising result, as it is not what one would expect by analogy with algebraically closed fields.
438:
The number of differentially closed fields of some infinite cardinality Īŗ is 2; for Īŗ uncountable this was proved by
17:
750:
711:
672:
633:
1011:
945:
869:
75:
471:
775:
365:
183:
832:
799:, Math. Sci. Res. Inst. Publ., vol. 39, Cambridge: Cambridge Univ. Press, pp. 53ā63,
498:
411:. It is the model completion of the theory of differentially perfect fields of characteristic
789:
1016:
43:
995:
933:
890:
860:
824:
804:
334:
341:. In characteristic 0 this implies that it is algebraically closed, but in characteristic
8:
71:
921:
67:
32:
916:
528:
The differential
Nullstellensatz is the analogue in differential algebra of Hilbert's
981:
972:, Lecture Notes in Mathematics, vol. 1696, Berlin: Springer, pp. 129ā141,
851:
512:
eliminates quantifiers in the language of differential fields with a unary function
973:
954:
911:
878:
846:
586:
In characteristic 0 Blum showed that the theory of differentially closed fields is
467:
338:
991:
929:
886:
856:
820:
800:
529:
458:
is defined by taking sets of solutions of systems of differential equations over
404:
378:
478:
382:
811:
Robinson, Abraham (1959), "On the concept of a differentially closed field.",
1005:
965:
940:
897:
603:
198:
598:
showed that the theory of differentially closed fields is not Ļ-stable, and
555:
is a differentially closed field of characteristic 0. . Then
Seidenberg's
900:(1973), "The model theory of differential dields of characteristic pā 0",
607:
591:
357:
28:
739:
700:
661:
622:
587:
431:=0 is the same as the theory of differentially perfect fields so has DCF
977:
958:
925:
882:
613:
563:
Radical differential ideals in the ring of differential polynomials in
129:
occurs in it, or −1 if the differential polynomial is a constant.
520:
th root of all constants, and is 0 on elements that are not constant.
345:>0 differentially closed fields are never algebraically closed.
313:
is the theory of differentially closed fields of characteristic
493:
Like the theory of algebraically closed fields, the theory DCF
427:>0 does not have a model completion, and in characteristic
943:(1976), "The model theory of differential fields revisited",
61:
423:>0. The theory of differential fields of characteristic
209:
if it is either of characteristic 0, or of characteristic
867:
Shelah, Saharon (1973), "Differentially closed fields",
407:
of the theory of differential fields of characteristic
813:
Bulletin of the
Research Council of Israel (Section F)
46:
with a solution in some differential field extending
614:
The structure of definable sets: Zilber's trichotomy
497:
of differentially closed fields of characteristic 0
82:
be a differential field with derivation operator ā.
442:, and for Īŗ countable by Hrushovski and Sokolovic.
833:"The differential closure of a differential field"
193:th powers are constants. It follows that neither
1003:
903:Proceedings of the American Mathematical Society
415:if one adds to the language a symbol giving the
523:
228:is a differentially perfect differential field
337:shows that any differentially closed field is
838:Bulletin of the American Mathematical Society
285:)ā 0. (Some authors add the condition that
466:variables as basic closed sets. Like the
94:is a polynomial in the formal expressions
62:The theory of differentially closed fields
915:
850:
547:if it contains all roots of its elements.
488:
240:are differential polynomials such that
117:of a non-zero differential polynomial in
968:(1998), "Differentially closed fields",
810:
55:
445:
389:=0 this was shown by Robinson, and for
14:
1004:
866:
787:
652:
599:
540:or ā-ideal is an ideal closed under ā.
439:
144:of a differential polynomial of order
830:
790:"Model theory of differential fields"
559:states there is a bijection between
964:
939:
896:
734:
695:
656:
617:
595:
394:
289:has characteristic 0, in which case
970:Model theory and algebraic geometry
797:Model theory, algebra, and geometry
691:
24:
581:
25:
1028:
917:10.1090/S0002-9939-1973-0329887-1
602:showed more precisely that it is
54:. This concept was introduced by
738:
699:
660:
621:
852:10.1090/S0002-9904-1972-12969-0
730:
298:is automatically non-zero, and
257:has order greater than that of
201:, unless ā is trivial. A field
594:Ļ. In non-zero characteristic
197:nor its field of constants is
13:
1:
946:Israel Journal of Mathematics
870:Israel Journal of Mathematics
781:
481:, formula with parameters in
557:differential nullstellensatz
524:Differential Nullstellensatz
205:with derivation ā is called
167:is the subfield of elements
7:
769:
226:differentially closed field
106:, ... with coefficients in
10:
1033:
776:Differential Galois theory
470:, the Kolchin topology is
419:th root of constants when
302:is automatically perfect.)
217:th power of an element of
50:already has a solution in
42:if every finite system of
831:Sacks, Gerald E. (1972),
435:as its model completion.
213:and every constant is a
178:In a differential field
148:ā„0 is the derivative of
88:differential polynomial
788:Marker, David (2000),
747:This section is empty.
708:This section is empty.
669:This section is empty.
630:This section is empty.
499:eliminates quantifiers
489:Quantifier elimination
207:differentially perfect
44:differential equations
508:>0, the theory DCF
261:, then there is some
40:differentially closed
1012:Differential algebra
570:ā-closed subsets of
446:The Kolchin topology
354:differential closure
335:separable polynomial
653:Decidability issues
543:An ideal is called
978:10.1007/BFb0094671
959:10.1007/BF02757008
883:10.1007/BF02756711
538:differential ideal
516:added that is the
504:In characteristic
371:The theory of DCF
161:field of constants
68:differential field
33:differential field
987:978-3-540-64863-5
767:
766:
728:
727:
689:
688:
650:
649:
397:). The theory DCF
321:is 0 or a prime).
152:with respect to ā
66:We recall that a
16:(Redirected from
1024:
998:
961:
953:(3ā4): 331ā352,
936:
919:
893:
863:
854:
827:
807:
794:
762:
759:
749:You can help by
742:
735:
723:
720:
710:You can help by
703:
696:
692:The Manin kernel
684:
681:
671:You can help by
664:
657:
645:
642:
632:You can help by
625:
618:
468:Zariski topology
452:Kolchin topology
339:separably closed
74:equipped with a
21:
18:Kolchin topology
1032:
1031:
1027:
1026:
1025:
1023:
1022:
1021:
1002:
1001:
988:
792:
784:
772:
763:
757:
754:
733:
724:
718:
715:
694:
685:
679:
676:
655:
646:
640:
637:
616:
584:
582:Omega stability
530:nullstellensatz
526:
511:
496:
491:
448:
434:
405:model companion
402:
376:
311:
297:
248:
143:
121:is the largest
64:
56:Robinson (1959)
23:
22:
15:
12:
11:
5:
1030:
1020:
1019:
1014:
1000:
999:
986:
962:
937:
910:(2): 577ā584,
894:
877:(3): 314ā328,
864:
845:(5): 629ā634,
828:
808:
783:
780:
779:
778:
771:
768:
765:
764:
745:
743:
732:
729:
726:
725:
706:
704:
693:
690:
687:
686:
667:
665:
654:
651:
648:
647:
628:
626:
615:
612:
583:
580:
576:
575:
568:
567:variables, and
549:
548:
541:
525:
522:
509:
494:
490:
487:
447:
444:
432:
398:
383:model complete
372:
323:
322:
307:
303:
293:
244:
222:
184:characteristic
176:
157:
139:
130:
111:
78:operator. Let
63:
60:
9:
6:
4:
3:
2:
1029:
1018:
1015:
1013:
1010:
1009:
1007:
997:
993:
989:
983:
979:
975:
971:
967:
963:
960:
956:
952:
948:
947:
942:
938:
935:
931:
927:
923:
918:
913:
909:
905:
904:
899:
895:
892:
888:
884:
880:
876:
872:
871:
865:
862:
858:
853:
848:
844:
840:
839:
834:
829:
826:
822:
818:
814:
809:
806:
802:
798:
791:
786:
785:
777:
774:
773:
761:
752:
748:
744:
741:
737:
736:
722:
713:
709:
705:
702:
698:
697:
683:
674:
670:
666:
663:
659:
658:
644:
635:
631:
627:
624:
620:
619:
611:
609:
605:
601:
600:Shelah (1973)
597:
593:
589:
579:
573:
569:
566:
562:
561:
560:
558:
554:
551:Suppose that
546:
542:
539:
535:
534:
533:
531:
521:
519:
515:
507:
502:
500:
486:
484:
480:
475:
473:
469:
465:
461:
457:
453:
443:
441:
440:Shelah (1973)
436:
430:
426:
422:
418:
414:
410:
406:
401:
396:
392:
388:
384:
380:
375:
369:
367:
363:
359:
355:
351:
346:
344:
340:
336:
333:any ordinary
332:
328:
320:
316:
312:
310:
304:
301:
296:
292:
288:
284:
280:
276:
272:
268:
264:
260:
256:
252:
247:
243:
239:
235:
232:such that if
231:
227:
223:
220:
216:
212:
208:
204:
200:
196:
192:
188:
185:
181:
177:
174:
170:
166:
162:
158:
155:
151:
147:
142:
138:
135:
131:
128:
124:
120:
116:
112:
109:
105:
101:
97:
93:
89:
85:
84:
83:
81:
77:
73:
69:
59:
57:
53:
49:
45:
41:
37:
34:
30:
19:
1017:Model theory
969:
950:
944:
907:
901:
874:
868:
842:
836:
816:
812:
796:
755:
751:adding to it
746:
731:Applications
716:
712:adding to it
707:
677:
673:adding to it
668:
638:
634:adding to it
629:
585:
577:
571:
564:
556:
552:
550:
544:
537:
527:
517:
513:
505:
503:
492:
482:
476:
463:
459:
455:
451:
449:
437:
428:
424:
420:
416:
412:
408:
399:
390:
386:
373:
370:
361:
353:
349:
347:
342:
330:
326:
324:
318:
314:
308:
305:
299:
294:
290:
286:
282:
278:
274:
270:
266:
262:
258:
254:
250:
245:
241:
237:
233:
229:
225:
218:
214:
210:
206:
202:
194:
190:
186:
179:
172:
168:
164:
160:
153:
149:
145:
140:
136:
133:
126:
122:
118:
114:
107:
103:
99:
95:
91:
87:
79:
65:
51:
47:
39:
35:
26:
966:Wood, Carol
941:Wood, Carol
898:Wood, Carol
819:: 113ā128,
608:superstable
596:Wood (1973)
592:Morley rank
395:Wood (1973)
358:prime model
182:of nonzero
125:such that ā
29:mathematics
1006:Categories
782:References
472:Noetherian
76:derivation
758:July 2010
719:July 2010
680:July 2010
641:July 2010
590:and has
393:>0 by
770:See also
606:but not
588:Ļ-stable
379:complete
277:)=0 and
253:ā 0 and
249:ā 0 and
134:separant
996:1678539
934:0329887
926:2039417
891:0344116
861:0299466
825:0125016
805:1773702
545:radical
403:is the
366:minimal
329:=1 and
325:Taking
317:(where
199:perfect
994:
984:
932:
924:
889:
859:
823:
803:
604:stable
479:atomic
352:has a
189:, all
171:with ā
922:JSTOR
793:(PDF)
385:(for
356:, a
269:with
115:order
72:field
70:is a
982:ISBN
450:The
381:and
236:and
159:The
132:The
113:The
31:, a
974:doi
955:doi
912:doi
879:doi
847:doi
753:.
714:.
675:.
636:.
462:in
454:on
377:is
306:DCF
265:in
175:=0.
163:of
102:, ā
98:, ā
90:in
38:is
27:In
1008::
992:MR
990:,
980:,
951:25
949:,
930:MR
928:,
920:,
908:40
906:,
887:MR
885:,
875:16
873:,
857:MR
855:,
843:78
841:,
835:,
821:MR
817:8F
815:,
801:MR
795:,
610:.
536:A
532:.
485:.
474:.
224:A
86:A
976::
957::
914::
881::
849::
760:)
756:(
721:)
717:(
682:)
678:(
643:)
639:(
574:.
572:K
565:n
553:K
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506:p
495:0
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425:p
421:p
417:p
413:p
409:p
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391:p
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362:K
350:K
343:p
331:f
327:g
319:p
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