487:
belong to the ideal generated by the preceding ones. Gröbner basis theory implies that this list is necessarily finite, and is thus a finite basis of the ideal. However, for deciding whether the list is complete, one must consider every element of the infinite sequence, which cannot be done in the finite time allowed to an algorithm.
486:
allow a direct proof that is as constructive as possible: Gröbner bases produce an algorithm for testing whether a polynomial belong to the ideal generated by other polynomials. So, given an infinite sequence of polynomials, one can construct algorithmically the list of those polynomials that do not
122:. For example, the basis theorem asserts that every ideal has a finite generator set, but the original proof does not provide any way to compute it for a specific ideal. This approach was so astonishing for mathematicians of that time that the first version of the article was rejected by
2406:
1167:
3450:
3027:
1646:
1475:
800:
2860:
2538:
1353:
2491:
2450:
3980:
2209:
1238:
126:, the greatest specialist of invariants of that time, with the comment "This is not mathematics. This is theology." Later, he recognized "I have convinced myself that even theology has its merits."
1966:
3501:
880:
1586:
2281:
2930:
3268:
2967:
1720:
607:
3864:
3730:
3178:
3090:
1542:
663:
2711:
1864:
746:
2632:
2075:
1997:
1028:
694:
3912:
3888:
2119:
2021:
1812:
1768:
1744:
962:
3141:
2602:
3312:
2235:
2044:
3659:
2744:
1673:
1505:
938:
911:
4000:
3783:
3761:
3625:
3552:
3524:
3198:
2886:
2652:
2558:
2289:
2095:
1884:
1788:
1261:
1048:
982:
552:
516:
441:
386:
358:
329:
302:
273:
253:
229:
209:
185:
152:
1056:
482:
on the number of variables, and, at each induction step use the non-constructive proof for one variable less. Introduced more than eighty years later,
3323:
2972:
1591:
1364:
751:
4028:
2755:
2496:
1269:
3732:
and further as the locus of its generators, it follows that every affine variety is the locus of finitely many polynomials — i.e. the
91:, where he solved several problems on invariants. In this article, he proved also two other fundamental theorems on polynomials, the
2455:
2414:
4195:
3917:
2127:
1175:
17:
4003:
1889:
4233:
3465:
809:
4168:
1547:
2240:
118:
Another aspect of this article had a great impact on mathematics of the 20th century; this is the systematic use of
4223:
2891:
3209:
2935:
1686:
573:
3790:
3664:
3146:
3032:
1510:
4187:
616:
561:
We will give two proofs, in both only the "left" case is considered; the proof for the right case is similar.
4213:
2663:
1817:
699:
2607:
2049:
1971:
987:
668:
3764:
50:
4218:
3893:
3869:
3733:
2100:
2002:
1793:
1749:
1725:
943:
610:
3102:
2569:
119:
3279:
448:
4228:
4024:
803:
479:
475:
99:(theorem on relations). These three theorems were the starting point of the interpretation of
4114:
4056:
2217:
2026:
3637:
2722:
1651:
1483:
916:
889:
468:
3661:(i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal
3506:
Note that the only reason we had to split into two cases was to ensure that the powers of
2401:{\displaystyle \left\{f_{i},f_{j}^{(k)}\,:\ i<N,\,j<N^{(k)},\,k<d\right\}\!\!\;.}
8:
4160:
452:
104:
46:
38:
2097:, and so are finitely generated by the leading coefficients of finitely many members of
4081:
3985:
3768:
3746:
3569:
3537:
3509:
3183:
2871:
2637:
2543:
2080:
1869:
1790:, and so is finitely generated by the leading coefficients of finitely many members of
1773:
1246:
1033:
967:
528:
501:
460:
456:
391:
371:
334:
314:
278:
258:
238:
214:
194:
161:
155:
137:
100:
65:
1162:{\displaystyle (a_{0})\subset (a_{0},a_{1})\subset (a_{0},a_{1},a_{2})\subset \cdots }
4191:
4164:
4085:
4073:
4065:
3555:
88:
483:
4154:
523:
519:
92:
69:
42:
4150:
4020:
3631:
883:
96:
4207:
4077:
4051:
3445:{\displaystyle h_{0}=\sum _{j}u_{j}X^{\deg(h)-\deg(f_{j}^{(k)})}f_{j}^{(k)},}
464:
232:
108:
84:
4179:
4016:
3737:
112:
76:
are
Noetherian rings. So, the theorem can be generalized and restated as:
123:
31:
4015:
Formal proofs of
Hilbert's basis theorem have been verified through the
4069:
3022:{\displaystyle h-h_{0}\in {\mathfrak {a}}\setminus {\mathfrak {a}}^{*}}
1641:{\displaystyle f_{N}-g\in {\mathfrak {a}}\setminus {\mathfrak {b}}_{N}}
188:
1470:{\displaystyle g=\sum _{i<N}u_{i}X^{\deg(f_{N})-\deg(f_{i})}f_{i},}
3559:
795:{\displaystyle f_{n}\in {\mathfrak {a}}\setminus {\mathfrak {b}}_{n}}
609:
is a non-finitely generated left ideal. Then by recursion (using the
2855:{\displaystyle h_{0}=\sum _{j}u_{j}X^{\deg(h)-\deg(f_{j})}f_{j},}
2533:{\displaystyle h\in {\mathfrak {a}}\setminus {\mathfrak {a}}^{*}}
2493:. Suppose for the sake of contradiction this is not so. Then let
1348:{\displaystyle a_{N}=\sum _{i<N}u_{i}a_{i},\qquad u_{i}\in R.}
1241:
73:
61:
3526:
multiplying the factors were non-negative in the constructions.
78:
every polynomial ring over a
Noetherian ring is also Noetherian
2540:
be of minimal degree, and denote its leading coefficient by
2486:{\displaystyle {\mathfrak {a}}\subseteq {\mathfrak {a}}^{*}}
2445:{\displaystyle {\mathfrak {a}}^{*}\subseteq {\mathfrak {a}}}
3975:{\displaystyle {\mathfrak {a}}=(p_{0},\dotsc ,p_{N-1})}
2204:{\displaystyle f_{0}^{(k)},\ldots ,f_{N^{(k)}-1}^{(k)}}
1233:{\displaystyle {\mathfrak {b}}=(a_{0},\ldots ,a_{N-1})}
107:. In particular, the basis theorem implies that every
4175:
The definitive
English-language biography of Hilbert.
4054:(1890). "Über die Theorie der algebraischen Formen".
3988:
3920:
3896:
3872:
3793:
3771:
3749:
3667:
3640:
3572:
3540:
3512:
3468:
3326:
3282:
3212:
3186:
3149:
3105:
3035:
2975:
2938:
2894:
2874:
2758:
2725:
2666:
2640:
2610:
2572:
2546:
2499:
2458:
2417:
2292:
2243:
2220:
2130:
2103:
2083:
2052:
2029:
2005:
1974:
1961:{\displaystyle \{\deg(f_{0}),\ldots ,\deg(f_{N-1})\}}
1892:
1872:
1820:
1796:
1776:
1752:
1728:
1689:
1654:
1594:
1550:
1513:
1486:
1367:
1272:
1249:
1178:
1059:
1036:
990:
970:
946:
919:
892:
812:
754:
702:
671:
619:
576:
531:
504:
455:) in the course of his proof of finite generation of
394:
374:
337:
317:
281:
261:
241:
217:
197:
164:
140:
447:
Hilbert proved the theorem (for the special case of
3496:{\displaystyle {\mathfrak {a}}={\mathfrak {a}}^{*}}
875:{\displaystyle \{\deg(f_{0}),\deg(f_{1}),\ldots \}}
3994:
3974:
3906:
3882:
3858:
3777:
3755:
3724:
3653:
3619:
3546:
3518:
3495:
3444:
3306:
3262:
3192:
3172:
3135:
3084:
3021:
2961:
2924:
2880:
2854:
2738:
2705:
2646:
2626:
2596:
2552:
2532:
2485:
2444:
2400:
2275:
2229:
2203:
2113:
2089:
2069:
2038:
2015:
1991:
1960:
1878:
1858:
1806:
1782:
1762:
1738:
1714:
1667:
1640:
1580:
1536:
1499:
1469:
1347:
1255:
1232:
1161:
1042:
1022:
976:
956:
932:
905:
874:
794:
740:
688:
657:
601:
546:
510:
435:
380:
352:
323:
296:
267:
247:
223:
203:
179:
146:
4092:
2393:
2392:
1999:be the set of leading coefficients of members of
1746:be the set of leading coefficients of members of
4205:
1581:{\displaystyle f_{N}\notin {\mathfrak {b}}_{N}}
3458:we yield a similar contradiction as in Case 1.
2276:{\displaystyle {\mathfrak {a}}^{*}\subseteq R}
554:is also a left (resp. right) Noetherian ring.
4046:
4044:
3558:. Hilbert's basis theorem has some immediate
3890:is an ideal. The basis theorem implies that
2925:{\displaystyle h_{0}\in {\mathfrak {a}}^{*}}
1955:
1893:
869:
813:
652:
620:
3263:{\displaystyle a=\sum _{j}u_{j}a_{j}^{(k)}}
2962:{\displaystyle h\notin {\mathfrak {a}}^{*}}
68:whose ideals have this property are called
4041:
2394:
1715:{\displaystyle {\mathfrak {a}}\subseteq R}
602:{\displaystyle {\mathfrak {a}}\subseteq R}
111:is the intersection of a finite number of
3859:{\displaystyle A\simeq R/{\mathfrak {a}}}
2377:
2351:
2332:
275:is Noetherian, the same must be true for
255:is "not too large", in the sense that if
3725:{\displaystyle {\mathfrak {a}}\subset R}
3173:{\displaystyle a\in {\mathfrak {b}}_{k}}
3085:{\displaystyle \deg(h-h_{0})<\deg(h)}
2604:. Regardless of this condition, we have
1537:{\displaystyle g\in {\mathfrak {b}}_{N}}
27:Polynomial ideals are finitely generated
4050:
1480:whose leading term is equal to that of
658:{\displaystyle \{f_{0},f_{1},\ldots \}}
14:
4206:
1770:. This is obviously a left ideal over
4178:
4126:
2706:{\displaystyle a=\sum _{j}u_{j}a_{j}}
1859:{\displaystyle f_{0},\ldots ,f_{N-1}}
741:{\displaystyle f_{0},\ldots ,f_{n-1}}
613:) there is a sequence of polynomials
83:The theorem was stated and proved by
4149:
4110:
4098:
2627:{\displaystyle a\in {\mathfrak {b}}}
3923:
3899:
3875:
3851:
3670:
3482:
3471:
3276:of the leading coefficients of the
3159:
3008:
2997:
2948:
2911:
2868:which has the same leading term as
2619:
2519:
2508:
2472:
2461:
2437:
2421:
2247:
2106:
2070:{\displaystyle {\mathfrak {b}}_{k}}
2056:
2008:
1992:{\displaystyle {\mathfrak {b}}_{k}}
1978:
1799:
1755:
1731:
1692:
1627:
1616:
1567:
1523:
1181:
1023:{\displaystyle a_{0},a_{1},\ldots }
949:
781:
770:
689:{\displaystyle {\mathfrak {b}}_{n}}
675:
579:
24:
4136:
1050:is Noetherian the chain of ideals
87:in 1890 in his seminal article on
25:
4245:
4144:Ideals, Varieties, and Algorithms
4104:
3002:
2513:
1621:
775:
4010:
3914:must be finitely generated, say
2283:be the left ideal generated by:
1675:, contradicting the minimality.
882:is a non-decreasing sequence of
459:. The theorem is interpreted in
3907:{\displaystyle {\mathfrak {a}}}
3883:{\displaystyle {\mathfrak {a}}}
3529:
3092:, which contradicts minimality.
2114:{\displaystyle {\mathfrak {a}}}
2016:{\displaystyle {\mathfrak {a}}}
1807:{\displaystyle {\mathfrak {a}}}
1763:{\displaystyle {\mathfrak {a}}}
1739:{\displaystyle {\mathfrak {b}}}
1678:
1325:
957:{\displaystyle {\mathfrak {b}}}
696:is the left ideal generated by
471:of finitely many polynomials.
72:. Every field, and the ring of
4120:
3969:
3931:
3841:
3803:
3719:
3681:
3614:
3576:
3434:
3428:
3413:
3408:
3402:
3389:
3377:
3371:
3299:
3293:
3255:
3249:
3136:{\displaystyle \deg(h)=k<d}
3118:
3112:
3079:
3073:
3061:
3042:
2834:
2821:
2809:
2803:
2585:
2579:
2369:
2363:
2327:
2321:
2270:
2264:
2196:
2190:
2177:
2171:
2147:
2141:
1952:
1933:
1915:
1902:
1709:
1703:
1449:
1436:
1424:
1411:
1227:
1189:
1150:
1111:
1105:
1079:
1073:
1060:
913:be the leading coefficient of
860:
847:
835:
822:
596:
590:
565:
541:
535:
430:
398:
347:
341:
291:
285:
174:
168:
13:
1:
4188:Graduate Texts in Mathematics
4034:
3503:which is finitely generated.
2597:{\displaystyle \deg(h)\geq d}
95:(zero-locus theorem) and the
4190:(Third ed.), Springer,
3200:is a left linear combination
2654:is a left linear combination
129:
7:
3307:{\displaystyle f_{j}^{(k)}}
2719:of the coefficients of the
388:is a Noetherian ring, then
331:is a Noetherian ring, then
57:in Hilbert's terminology).
10:
4250:
4234:Theorems about polynomials
3462:Thus our claim holds, and
1886:be the maximum of the set
474:Hilbert's proof is highly
4142:Cox, Little, and O'Shea,
3566:By induction we see that
611:axiom of dependent choice
467:is the set of the common
4146:, Springer-Verlag, 1997.
3627:will also be Noetherian.
518:is a left (resp. right)
490:
449:multivariate polynomials
309:Hilbert's Basis Theorem.
120:non-constructive methods
4224:Theorems in ring theory
4184:Advanced Linear Algebra
4129:, p. 136 §5 Theorem 5.9
35:Hilbert's basis theorem
4029:ring_theory.polynomial
3996:
3976:
3908:
3884:
3860:
3779:
3757:
3726:
3655:
3621:
3548:
3520:
3497:
3446:
3308:
3264:
3194:
3174:
3137:
3086:
3023:
2963:
2926:
2882:
2856:
2740:
2707:
2648:
2628:
2598:
2554:
2534:
2487:
2446:
2402:
2277:
2231:
2230:{\displaystyle \leq k}
2205:
2115:
2091:
2071:
2040:
2039:{\displaystyle \leq k}
2017:
1993:
1962:
1880:
1860:
1808:
1784:
1764:
1740:
1716:
1669:
1642:
1582:
1538:
1501:
1471:
1349:
1257:
1234:
1163:
1044:
1024:
978:
958:
934:
907:
876:
796:
742:
690:
659:
603:
548:
512:
445:
437:
382:
362:
354:
325:
298:
269:
249:
225:
205:
181:
148:
4057:Mathematische Annalen
3997:
3977:
3909:
3885:
3861:
3780:
3758:
3727:
3656:
3654:{\displaystyle R^{n}}
3622:
3549:
3521:
3498:
3447:
3309:
3265:
3195:
3175:
3138:
3087:
3024:
2964:
2927:
2883:
2857:
2741:
2739:{\displaystyle f_{j}}
2708:
2649:
2629:
2599:
2555:
2535:
2488:
2447:
2403:
2278:
2232:
2206:
2116:
2092:
2077:are left ideals over
2072:
2041:
2018:
1994:
1963:
1881:
1861:
1809:
1785:
1765:
1741:
1722:be a left ideal. Let
1717:
1670:
1668:{\displaystyle f_{N}}
1648:has degree less than
1643:
1583:
1539:
1502:
1500:{\displaystyle f_{N}}
1472:
1350:
1258:
1235:
1172:must terminate. Thus
1164:
1045:
1025:
979:
964:be the left ideal in
959:
935:
933:{\displaystyle f_{n}}
908:
906:{\displaystyle a_{n}}
877:
797:
743:
691:
660:
604:
549:
513:
443:is a Noetherian ring.
438:
383:
363:
360:is a Noetherian ring.
355:
326:
306:
299:
270:
250:
226:
206:
191:in the indeterminate
182:
149:
18:Hilbert Basis Theorem
3986:
3918:
3894:
3870:
3791:
3787:, then we know that
3769:
3747:
3665:
3638:
3570:
3538:
3510:
3466:
3324:
3280:
3210:
3184:
3147:
3103:
3033:
2973:
2936:
2892:
2872:
2756:
2723:
2664:
2638:
2608:
2570:
2544:
2497:
2456:
2415:
2290:
2241:
2218:
2128:
2101:
2081:
2050:
2027:
2003:
1972:
1890:
1870:
1818:
1794:
1774:
1750:
1726:
1687:
1652:
1592:
1548:
1511:
1484:
1365:
1270:
1263:. So in particular,
1247:
1176:
1057:
1034:
988:
968:
944:
917:
890:
810:
806:. By construction,
752:
700:
669:
617:
574:
529:
502:
392:
372:
335:
315:
279:
259:
239:
215:
195:
162:
138:
4214:Commutative algebra
3765:finitely-generated
3438:
3412:
3303:
3259:
2331:
2200:
2151:
1588:, which means that
457:rings of invariants
187:denote the ring of
105:commutative algebra
37:asserts that every
4070:10.1007/BF01208503
4004:finitely presented
3992:
3972:
3904:
3880:
3856:
3775:
3753:
3722:
3651:
3617:
3544:
3516:
3493:
3442:
3418:
3392:
3349:
3304:
3283:
3260:
3239:
3228:
3190:
3170:
3133:
3082:
3019:
2959:
2922:
2878:
2852:
2781:
2736:
2703:
2682:
2644:
2624:
2594:
2550:
2530:
2483:
2442:
2398:
2311:
2273:
2227:
2201:
2161:
2131:
2111:
2087:
2067:
2036:
2023:, whose degree is
2013:
1989:
1958:
1876:
1856:
1804:
1780:
1760:
1736:
1712:
1665:
1638:
1578:
1534:
1497:
1467:
1389:
1345:
1301:
1253:
1230:
1159:
1040:
1020:
974:
954:
930:
903:
872:
792:
738:
686:
655:
599:
544:
508:
463:as follows: every
461:algebraic geometry
433:
378:
350:
321:
294:
265:
245:
221:
201:
177:
144:
101:algebraic geometry
4197:978-0-387-72828-5
3995:{\displaystyle A}
3778:{\displaystyle R}
3756:{\displaystyle A}
3736:of finitely many
3620:{\displaystyle R}
3547:{\displaystyle R}
3519:{\displaystyle X}
3340:
3219:
3193:{\displaystyle a}
2881:{\displaystyle h}
2772:
2673:
2647:{\displaystyle a}
2553:{\displaystyle a}
2338:
2090:{\displaystyle R}
2046:. As before, the
1879:{\displaystyle d}
1783:{\displaystyle R}
1374:
1286:
1256:{\displaystyle N}
1043:{\displaystyle R}
977:{\displaystyle R}
547:{\displaystyle R}
511:{\displaystyle R}
478:: it proceeds by
436:{\displaystyle R}
381:{\displaystyle R}
353:{\displaystyle R}
324:{\displaystyle R}
297:{\displaystyle R}
268:{\displaystyle R}
248:{\displaystyle R}
224:{\displaystyle R}
204:{\displaystyle X}
180:{\displaystyle R}
147:{\displaystyle R}
16:(Redirected from
4241:
4219:Invariant theory
4200:
4174:
4151:Reid, Constance.
4130:
4124:
4118:
4108:
4102:
4096:
4090:
4089:
4048:
4001:
3999:
3998:
3993:
3981:
3979:
3978:
3973:
3968:
3967:
3943:
3942:
3927:
3926:
3913:
3911:
3910:
3905:
3903:
3902:
3889:
3887:
3886:
3881:
3879:
3878:
3865:
3863:
3862:
3857:
3855:
3854:
3848:
3840:
3839:
3815:
3814:
3784:
3782:
3781:
3776:
3762:
3760:
3759:
3754:
3731:
3729:
3728:
3723:
3718:
3717:
3693:
3692:
3674:
3673:
3660:
3658:
3657:
3652:
3650:
3649:
3626:
3624:
3623:
3618:
3613:
3612:
3588:
3587:
3556:commutative ring
3554:be a Noetherian
3553:
3551:
3550:
3545:
3525:
3523:
3522:
3517:
3502:
3500:
3499:
3494:
3492:
3491:
3486:
3485:
3475:
3474:
3451:
3449:
3448:
3443:
3437:
3426:
3417:
3416:
3411:
3400:
3359:
3358:
3348:
3336:
3335:
3313:
3311:
3310:
3305:
3302:
3291:
3269:
3267:
3266:
3261:
3258:
3247:
3238:
3237:
3227:
3199:
3197:
3196:
3191:
3179:
3177:
3176:
3171:
3169:
3168:
3163:
3162:
3142:
3140:
3139:
3134:
3091:
3089:
3088:
3083:
3060:
3059:
3028:
3026:
3025:
3020:
3018:
3017:
3012:
3011:
3001:
3000:
2991:
2990:
2968:
2966:
2965:
2960:
2958:
2957:
2952:
2951:
2931:
2929:
2928:
2923:
2921:
2920:
2915:
2914:
2904:
2903:
2887:
2885:
2884:
2879:
2861:
2859:
2858:
2853:
2848:
2847:
2838:
2837:
2833:
2832:
2791:
2790:
2780:
2768:
2767:
2745:
2743:
2742:
2737:
2735:
2734:
2712:
2710:
2709:
2704:
2702:
2701:
2692:
2691:
2681:
2653:
2651:
2650:
2645:
2633:
2631:
2630:
2625:
2623:
2622:
2603:
2601:
2600:
2595:
2559:
2557:
2556:
2551:
2539:
2537:
2536:
2531:
2529:
2528:
2523:
2522:
2512:
2511:
2492:
2490:
2489:
2484:
2482:
2481:
2476:
2475:
2465:
2464:
2451:
2449:
2448:
2443:
2441:
2440:
2431:
2430:
2425:
2424:
2407:
2405:
2404:
2399:
2391:
2387:
2373:
2372:
2336:
2330:
2319:
2307:
2306:
2282:
2280:
2279:
2274:
2257:
2256:
2251:
2250:
2236:
2234:
2233:
2228:
2210:
2208:
2207:
2202:
2199:
2188:
2181:
2180:
2150:
2139:
2120:
2118:
2117:
2112:
2110:
2109:
2096:
2094:
2093:
2088:
2076:
2074:
2073:
2068:
2066:
2065:
2060:
2059:
2045:
2043:
2042:
2037:
2022:
2020:
2019:
2014:
2012:
2011:
1998:
1996:
1995:
1990:
1988:
1987:
1982:
1981:
1967:
1965:
1964:
1959:
1951:
1950:
1914:
1913:
1885:
1883:
1882:
1877:
1865:
1863:
1862:
1857:
1855:
1854:
1830:
1829:
1813:
1811:
1810:
1805:
1803:
1802:
1789:
1787:
1786:
1781:
1769:
1767:
1766:
1761:
1759:
1758:
1745:
1743:
1742:
1737:
1735:
1734:
1721:
1719:
1718:
1713:
1696:
1695:
1674:
1672:
1671:
1666:
1664:
1663:
1647:
1645:
1644:
1639:
1637:
1636:
1631:
1630:
1620:
1619:
1604:
1603:
1587:
1585:
1584:
1579:
1577:
1576:
1571:
1570:
1560:
1559:
1543:
1541:
1540:
1535:
1533:
1532:
1527:
1526:
1506:
1504:
1503:
1498:
1496:
1495:
1476:
1474:
1473:
1468:
1463:
1462:
1453:
1452:
1448:
1447:
1423:
1422:
1399:
1398:
1388:
1354:
1352:
1351:
1346:
1335:
1334:
1321:
1320:
1311:
1310:
1300:
1282:
1281:
1262:
1260:
1259:
1254:
1239:
1237:
1236:
1231:
1226:
1225:
1201:
1200:
1185:
1184:
1168:
1166:
1165:
1160:
1149:
1148:
1136:
1135:
1123:
1122:
1104:
1103:
1091:
1090:
1072:
1071:
1049:
1047:
1046:
1041:
1029:
1027:
1026:
1021:
1013:
1012:
1000:
999:
983:
981:
980:
975:
963:
961:
960:
955:
953:
952:
939:
937:
936:
931:
929:
928:
912:
910:
909:
904:
902:
901:
881:
879:
878:
873:
859:
858:
834:
833:
801:
799:
798:
793:
791:
790:
785:
784:
774:
773:
764:
763:
747:
745:
744:
739:
737:
736:
712:
711:
695:
693:
692:
687:
685:
684:
679:
678:
664:
662:
661:
656:
645:
644:
632:
631:
608:
606:
605:
600:
583:
582:
553:
551:
550:
545:
517:
515:
514:
509:
476:non-constructive
442:
440:
439:
434:
429:
428:
410:
409:
387:
385:
384:
379:
359:
357:
356:
351:
330:
328:
327:
322:
303:
301:
300:
295:
274:
272:
271:
266:
254:
252:
251:
246:
230:
228:
227:
222:
210:
208:
207:
202:
186:
184:
183:
178:
153:
151:
150:
145:
89:invariant theory
70:Noetherian rings
21:
4249:
4248:
4244:
4243:
4242:
4240:
4239:
4238:
4204:
4203:
4198:
4171:
4139:
4137:Further reading
4134:
4133:
4125:
4121:
4109:
4105:
4097:
4093:
4049:
4042:
4037:
4013:
3987:
3984:
3983:
3957:
3953:
3938:
3934:
3922:
3921:
3919:
3916:
3915:
3898:
3897:
3895:
3892:
3891:
3874:
3873:
3871:
3868:
3867:
3850:
3849:
3844:
3829:
3825:
3810:
3806:
3792:
3789:
3788:
3770:
3767:
3766:
3748:
3745:
3744:
3707:
3703:
3688:
3684:
3669:
3668:
3666:
3663:
3662:
3645:
3641:
3639:
3636:
3635:
3602:
3598:
3583:
3579:
3571:
3568:
3567:
3539:
3536:
3535:
3532:
3511:
3508:
3507:
3487:
3481:
3480:
3479:
3470:
3469:
3467:
3464:
3463:
3427:
3422:
3401:
3396:
3364:
3360:
3354:
3350:
3344:
3331:
3327:
3325:
3322:
3321:
3292:
3287:
3281:
3278:
3277:
3248:
3243:
3233:
3229:
3223:
3211:
3208:
3207:
3185:
3182:
3181:
3164:
3158:
3157:
3156:
3148:
3145:
3144:
3104:
3101:
3100:
3055:
3051:
3034:
3031:
3030:
3013:
3007:
3006:
3005:
2996:
2995:
2986:
2982:
2974:
2971:
2970:
2953:
2947:
2946:
2945:
2937:
2934:
2933:
2916:
2910:
2909:
2908:
2899:
2895:
2893:
2890:
2889:
2873:
2870:
2869:
2843:
2839:
2828:
2824:
2796:
2792:
2786:
2782:
2776:
2763:
2759:
2757:
2754:
2753:
2730:
2726:
2724:
2721:
2720:
2697:
2693:
2687:
2683:
2677:
2665:
2662:
2661:
2639:
2636:
2635:
2618:
2617:
2609:
2606:
2605:
2571:
2568:
2567:
2545:
2542:
2541:
2524:
2518:
2517:
2516:
2507:
2506:
2498:
2495:
2494:
2477:
2471:
2470:
2469:
2460:
2459:
2457:
2454:
2453:
2452:and claim also
2436:
2435:
2426:
2420:
2419:
2418:
2416:
2413:
2412:
2362:
2358:
2320:
2315:
2302:
2298:
2297:
2293:
2291:
2288:
2287:
2252:
2246:
2245:
2244:
2242:
2239:
2238:
2219:
2216:
2215:
2189:
2170:
2166:
2165:
2140:
2135:
2129:
2126:
2125:
2105:
2104:
2102:
2099:
2098:
2082:
2079:
2078:
2061:
2055:
2054:
2053:
2051:
2048:
2047:
2028:
2025:
2024:
2007:
2006:
2004:
2001:
2000:
1983:
1977:
1976:
1975:
1973:
1970:
1969:
1940:
1936:
1909:
1905:
1891:
1888:
1887:
1871:
1868:
1867:
1844:
1840:
1825:
1821:
1819:
1816:
1815:
1798:
1797:
1795:
1792:
1791:
1775:
1772:
1771:
1754:
1753:
1751:
1748:
1747:
1730:
1729:
1727:
1724:
1723:
1691:
1690:
1688:
1685:
1684:
1681:
1659:
1655:
1653:
1650:
1649:
1632:
1626:
1625:
1624:
1615:
1614:
1599:
1595:
1593:
1590:
1589:
1572:
1566:
1565:
1564:
1555:
1551:
1549:
1546:
1545:
1528:
1522:
1521:
1520:
1512:
1509:
1508:
1491:
1487:
1485:
1482:
1481:
1458:
1454:
1443:
1439:
1418:
1414:
1404:
1400:
1394:
1390:
1378:
1366:
1363:
1362:
1330:
1326:
1316:
1312:
1306:
1302:
1290:
1277:
1273:
1271:
1268:
1267:
1248:
1245:
1244:
1215:
1211:
1196:
1192:
1180:
1179:
1177:
1174:
1173:
1144:
1140:
1131:
1127:
1118:
1114:
1099:
1095:
1086:
1082:
1067:
1063:
1058:
1055:
1054:
1035:
1032:
1031:
1008:
1004:
995:
991:
989:
986:
985:
969:
966:
965:
948:
947:
945:
942:
941:
924:
920:
918:
915:
914:
897:
893:
891:
888:
887:
884:natural numbers
854:
850:
829:
825:
811:
808:
807:
786:
780:
779:
778:
769:
768:
759:
755:
753:
750:
749:
726:
722:
707:
703:
701:
698:
697:
680:
674:
673:
672:
670:
667:
666:
640:
636:
627:
623:
618:
615:
614:
578:
577:
575:
572:
571:
568:
530:
527:
526:
524:polynomial ring
520:Noetherian ring
503:
500:
499:
493:
424:
420:
405:
401:
393:
390:
389:
373:
370:
369:
336:
333:
332:
316:
313:
312:
280:
277:
276:
260:
257:
256:
240:
237:
236:
235:proved that if
216:
213:
212:
196:
193:
192:
163:
160:
159:
139:
136:
135:
132:
93:Nullstellensatz
43:polynomial ring
28:
23:
22:
15:
12:
11:
5:
4247:
4237:
4236:
4231:
4226:
4221:
4216:
4202:
4201:
4196:
4180:Roman, Stephen
4176:
4169:
4147:
4138:
4135:
4132:
4131:
4119:
4103:
4091:
4064:(4): 473–534.
4052:Hilbert, David
4039:
4038:
4036:
4033:
4012:
4009:
4008:
4007:
3991:
3971:
3966:
3963:
3960:
3956:
3952:
3949:
3946:
3941:
3937:
3933:
3930:
3925:
3901:
3877:
3853:
3847:
3843:
3838:
3835:
3832:
3828:
3824:
3821:
3818:
3813:
3809:
3805:
3802:
3799:
3796:
3774:
3752:
3741:
3721:
3716:
3713:
3710:
3706:
3702:
3699:
3696:
3691:
3687:
3683:
3680:
3677:
3672:
3648:
3644:
3632:affine variety
3628:
3616:
3611:
3608:
3605:
3601:
3597:
3594:
3591:
3586:
3582:
3578:
3575:
3543:
3531:
3528:
3515:
3490:
3484:
3478:
3473:
3460:
3459:
3455:
3454:
3453:
3452:
3441:
3436:
3433:
3430:
3425:
3421:
3415:
3410:
3407:
3404:
3399:
3395:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3363:
3357:
3353:
3347:
3343:
3339:
3334:
3330:
3316:
3315:
3301:
3298:
3295:
3290:
3286:
3273:
3272:
3271:
3270:
3257:
3254:
3251:
3246:
3242:
3236:
3232:
3226:
3222:
3218:
3215:
3202:
3201:
3189:
3167:
3161:
3155:
3152:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3094:
3093:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3058:
3054:
3050:
3047:
3044:
3041:
3038:
3016:
3010:
3004:
2999:
2994:
2989:
2985:
2981:
2978:
2956:
2950:
2944:
2941:
2919:
2913:
2907:
2902:
2898:
2877:
2865:
2864:
2863:
2862:
2851:
2846:
2842:
2836:
2831:
2827:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2795:
2789:
2785:
2779:
2775:
2771:
2766:
2762:
2748:
2747:
2733:
2729:
2716:
2715:
2714:
2713:
2700:
2696:
2690:
2686:
2680:
2676:
2672:
2669:
2656:
2655:
2643:
2621:
2616:
2613:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2549:
2527:
2521:
2515:
2510:
2505:
2502:
2480:
2474:
2468:
2463:
2439:
2434:
2429:
2423:
2409:
2408:
2397:
2390:
2386:
2383:
2380:
2376:
2371:
2368:
2365:
2361:
2357:
2354:
2350:
2347:
2344:
2341:
2335:
2329:
2326:
2323:
2318:
2314:
2310:
2305:
2301:
2296:
2272:
2269:
2266:
2263:
2260:
2255:
2249:
2226:
2223:
2212:
2211:
2198:
2195:
2192:
2187:
2184:
2179:
2176:
2173:
2169:
2164:
2160:
2157:
2154:
2149:
2146:
2143:
2138:
2134:
2108:
2086:
2064:
2058:
2035:
2032:
2010:
1986:
1980:
1957:
1954:
1949:
1946:
1943:
1939:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1912:
1908:
1904:
1901:
1898:
1895:
1875:
1853:
1850:
1847:
1843:
1839:
1836:
1833:
1828:
1824:
1801:
1779:
1757:
1733:
1711:
1708:
1705:
1702:
1699:
1694:
1680:
1677:
1662:
1658:
1635:
1629:
1623:
1618:
1613:
1610:
1607:
1602:
1598:
1575:
1569:
1563:
1558:
1554:
1531:
1525:
1519:
1516:
1494:
1490:
1478:
1477:
1466:
1461:
1457:
1451:
1446:
1442:
1438:
1435:
1432:
1429:
1426:
1421:
1417:
1413:
1410:
1407:
1403:
1397:
1393:
1387:
1384:
1381:
1377:
1373:
1370:
1356:
1355:
1344:
1341:
1338:
1333:
1329:
1324:
1319:
1315:
1309:
1305:
1299:
1296:
1293:
1289:
1285:
1280:
1276:
1252:
1229:
1224:
1221:
1218:
1214:
1210:
1207:
1204:
1199:
1195:
1191:
1188:
1183:
1170:
1169:
1158:
1155:
1152:
1147:
1143:
1139:
1134:
1130:
1126:
1121:
1117:
1113:
1110:
1107:
1102:
1098:
1094:
1089:
1085:
1081:
1078:
1075:
1070:
1066:
1062:
1039:
1019:
1016:
1011:
1007:
1003:
998:
994:
973:
951:
927:
923:
900:
896:
871:
868:
865:
862:
857:
853:
849:
846:
843:
840:
837:
832:
828:
824:
821:
818:
815:
802:is of minimal
789:
783:
777:
772:
767:
762:
758:
735:
732:
729:
725:
721:
718:
715:
710:
706:
683:
677:
654:
651:
648:
643:
639:
635:
630:
626:
622:
598:
595:
592:
589:
586:
581:
567:
564:
563:
562:
543:
540:
537:
534:
507:
492:
489:
432:
427:
423:
419:
416:
413:
408:
404:
400:
397:
377:
349:
346:
343:
340:
320:
293:
290:
287:
284:
264:
244:
220:
200:
176:
173:
170:
167:
143:
131:
128:
97:syzygy theorem
51:generating set
26:
9:
6:
4:
3:
2:
4246:
4235:
4232:
4230:
4229:David Hilbert
4227:
4225:
4222:
4220:
4217:
4215:
4212:
4211:
4209:
4199:
4193:
4189:
4185:
4181:
4177:
4172:
4170:0-387-94674-8
4166:
4162:
4158:
4157:
4152:
4148:
4145:
4141:
4140:
4128:
4123:
4116:
4112:
4107:
4101:, p. 34.
4100:
4095:
4087:
4083:
4079:
4075:
4071:
4067:
4063:
4059:
4058:
4053:
4047:
4045:
4040:
4032:
4030:
4026:
4022:
4021:HILBASIS file
4018:
4017:Mizar project
4011:Formal proofs
4005:
3989:
3964:
3961:
3958:
3954:
3950:
3947:
3944:
3939:
3935:
3928:
3845:
3836:
3833:
3830:
3826:
3822:
3819:
3816:
3811:
3807:
3800:
3797:
3794:
3786:
3772:
3750:
3742:
3739:
3738:hypersurfaces
3735:
3714:
3711:
3708:
3704:
3700:
3697:
3694:
3689:
3685:
3678:
3675:
3646:
3642:
3633:
3629:
3609:
3606:
3603:
3599:
3595:
3592:
3589:
3584:
3580:
3573:
3565:
3564:
3563:
3561:
3557:
3541:
3527:
3513:
3504:
3488:
3476:
3457:
3456:
3439:
3431:
3423:
3419:
3405:
3397:
3393:
3386:
3383:
3380:
3374:
3368:
3365:
3361:
3355:
3351:
3345:
3341:
3337:
3332:
3328:
3320:
3319:
3318:
3317:
3314:. Considering
3296:
3288:
3284:
3275:
3274:
3252:
3244:
3240:
3234:
3230:
3224:
3220:
3216:
3213:
3206:
3205:
3204:
3203:
3187:
3165:
3153:
3150:
3130:
3127:
3124:
3121:
3115:
3109:
3106:
3099:
3096:
3095:
3076:
3070:
3067:
3064:
3056:
3052:
3048:
3045:
3039:
3036:
3014:
2992:
2987:
2983:
2979:
2976:
2954:
2942:
2939:
2917:
2905:
2900:
2896:
2875:
2867:
2866:
2849:
2844:
2840:
2829:
2825:
2818:
2815:
2812:
2806:
2800:
2797:
2793:
2787:
2783:
2777:
2773:
2769:
2764:
2760:
2752:
2751:
2750:
2749:
2731:
2727:
2718:
2717:
2698:
2694:
2688:
2684:
2678:
2674:
2670:
2667:
2660:
2659:
2658:
2657:
2641:
2614:
2611:
2591:
2588:
2582:
2576:
2573:
2566:
2563:
2562:
2561:
2547:
2525:
2503:
2500:
2478:
2466:
2432:
2427:
2395:
2388:
2384:
2381:
2378:
2374:
2366:
2359:
2355:
2352:
2348:
2345:
2342:
2339:
2333:
2324:
2316:
2312:
2308:
2303:
2299:
2294:
2286:
2285:
2284:
2267:
2261:
2258:
2253:
2224:
2221:
2214:with degrees
2193:
2185:
2182:
2174:
2167:
2162:
2158:
2155:
2152:
2144:
2136:
2132:
2124:
2123:
2122:
2084:
2062:
2033:
2030:
1984:
1947:
1944:
1941:
1937:
1930:
1927:
1924:
1921:
1918:
1910:
1906:
1899:
1896:
1873:
1851:
1848:
1845:
1841:
1837:
1834:
1831:
1826:
1822:
1777:
1706:
1700:
1697:
1676:
1660:
1656:
1633:
1611:
1608:
1605:
1600:
1596:
1573:
1561:
1556:
1552:
1529:
1517:
1514:
1492:
1488:
1464:
1459:
1455:
1444:
1440:
1433:
1430:
1427:
1419:
1415:
1408:
1405:
1401:
1395:
1391:
1385:
1382:
1379:
1375:
1371:
1368:
1361:
1360:
1359:
1358:Now consider
1342:
1339:
1336:
1331:
1327:
1322:
1317:
1313:
1307:
1303:
1297:
1294:
1291:
1287:
1283:
1278:
1274:
1266:
1265:
1264:
1250:
1243:
1222:
1219:
1216:
1212:
1208:
1205:
1202:
1197:
1193:
1186:
1156:
1153:
1145:
1141:
1137:
1132:
1128:
1124:
1119:
1115:
1108:
1100:
1096:
1092:
1087:
1083:
1076:
1068:
1064:
1053:
1052:
1051:
1037:
1017:
1014:
1009:
1005:
1001:
996:
992:
984:generated by
971:
925:
921:
898:
894:
885:
866:
863:
855:
851:
844:
841:
838:
830:
826:
819:
816:
805:
787:
765:
760:
756:
733:
730:
727:
723:
719:
716:
713:
708:
704:
681:
665:such that if
649:
646:
641:
637:
633:
628:
624:
612:
593:
587:
584:
560:
557:
556:
555:
538:
532:
525:
521:
505:
497:
488:
485:
484:Gröbner bases
481:
477:
472:
470:
466:
465:algebraic set
462:
458:
454:
450:
444:
425:
421:
417:
414:
411:
406:
402:
395:
375:
367:
361:
344:
338:
318:
310:
305:
288:
282:
262:
242:
234:
218:
198:
190:
171:
165:
157:
141:
127:
125:
121:
116:
114:
113:hypersurfaces
110:
109:algebraic set
106:
102:
98:
94:
90:
86:
85:David Hilbert
81:
79:
75:
71:
67:
63:
58:
56:
52:
49:has a finite
48:
44:
40:
36:
33:
19:
4183:
4159:. New York:
4155:
4143:
4122:
4106:
4094:
4061:
4055:
4014:
3734:intersection
3533:
3530:Applications
3505:
3461:
3097:
2969:. Therefore
2564:
2410:
2213:
1682:
1679:Second proof
1507:; moreover,
1479:
1357:
1171:
569:
558:
495:
494:
473:
446:
365:
364:
308:
307:
304:. Formally,
133:
117:
103:in terms of
82:
77:
59:
54:
34:
29:
3560:corollaries
2888:; moreover
1544:. However,
566:First proof
522:, then the
189:polynomials
124:Paul Gordan
32:mathematics
4208:Categories
4127:Roman 2008
4113:, p.
4035:References
3630:Since any
2746:. Consider
2237:. Now let
1968:, and let
366:Corollary.
60:In modern
53:(a finite
4111:Reid 1996
4099:Reid 1996
4086:179177713
4078:0025-5831
3962:−
3948:…
3834:−
3820:…
3798:≃
3712:−
3698:…
3676:⊂
3607:−
3593:…
3489:∗
3387:
3381:−
3369:
3342:∑
3221:∑
3154:∈
3110:
3071:
3049:−
3040:
3015:∗
3003:∖
2993:∈
2980:−
2955:∗
2943:∉
2918:∗
2906:∈
2819:
2813:−
2801:
2774:∑
2675:∑
2615:∈
2589:≥
2577:
2526:∗
2514:∖
2504:∈
2479:∗
2467:⊆
2433:⊆
2428:∗
2259:⊆
2254:∗
2222:≤
2183:−
2156:…
2031:≤
1945:−
1931:
1922:…
1900:
1849:−
1835:…
1698:⊆
1622:∖
1612:∈
1606:−
1562:∉
1518:∈
1434:
1428:−
1409:
1376:∑
1337:∈
1288:∑
1240:for some
1220:−
1206:…
1157:⋯
1154:⊂
1109:⊂
1077:⊂
1018:…
867:…
845:
820:
776:∖
766:∈
731:−
717:…
650:…
585:⊆
480:induction
415:…
130:Statement
4182:(2008),
4161:Springer
4153:(1996).
3866:, where
3785:-algebra
2411:We have
1030:. Since
940:and let
570:Suppose
496:Theorem.
74:integers
4156:Hilbert
3982:, i.e.
3143:. Then
3098:Case 2:
2565:Case 1:
1242:integer
559:Remark.
451:over a
233:Hilbert
62:algebra
45:over a
4194:
4167:
4084:
4076:
4023:) and
2932:while
2337:
2121:, say
1866:. Let
1814:; say
886:. Let
804:degree
158:, let
4082:S2CID
4027:(see
4019:(see
3763:is a
3634:over
2634:, so
748:then
491:Proof
469:zeros
453:field
211:over
154:is a
66:rings
55:basis
47:field
41:of a
39:ideal
4192:ISBN
4165:ISBN
4074:ISSN
4025:Lean
3534:Let
3128:<
3065:<
3029:and
2382:<
2356:<
2343:<
1683:Let
1383:<
1295:<
156:ring
4066:doi
4031:).
4002:is
3743:If
3384:deg
3366:deg
3180:so
3107:deg
3068:deg
3037:deg
2816:deg
2798:deg
2574:deg
1928:deg
1897:deg
1431:deg
1406:deg
842:deg
817:deg
498:If
368:If
311:If
134:If
30:In
4210::
4186:,
4163:.
4115:37
4080:.
4072:.
4062:36
4060:.
4043:^
3562:.
2560:.
231:.
115:.
80:.
64:,
4173:.
4117:.
4088:.
4068::
4006:.
3990:A
3970:)
3965:1
3959:N
3955:p
3951:,
3945:,
3940:0
3936:p
3932:(
3929:=
3924:a
3900:a
3876:a
3852:a
3846:/
3842:]
3837:1
3831:n
3827:X
3823:,
3817:,
3812:0
3808:X
3804:[
3801:R
3795:A
3773:R
3751:A
3740:.
3720:]
3715:1
3709:n
3705:X
3701:,
3695:,
3690:0
3686:X
3682:[
3679:R
3671:a
3647:n
3643:R
3615:]
3610:1
3604:n
3600:X
3596:,
3590:,
3585:0
3581:X
3577:[
3574:R
3542:R
3514:X
3483:a
3477:=
3472:a
3440:,
3435:)
3432:k
3429:(
3424:j
3420:f
3414:)
3409:)
3406:k
3403:(
3398:j
3394:f
3390:(
3378:)
3375:h
3372:(
3362:X
3356:j
3352:u
3346:j
3338:=
3333:0
3329:h
3300:)
3297:k
3294:(
3289:j
3285:f
3256:)
3253:k
3250:(
3245:j
3241:a
3235:j
3231:u
3225:j
3217:=
3214:a
3188:a
3166:k
3160:b
3151:a
3131:d
3125:k
3122:=
3119:)
3116:h
3113:(
3080:)
3077:h
3074:(
3062:)
3057:0
3053:h
3046:h
3043:(
3009:a
2998:a
2988:0
2984:h
2977:h
2949:a
2940:h
2912:a
2901:0
2897:h
2876:h
2850:,
2845:j
2841:f
2835:)
2830:j
2826:f
2822:(
2810:)
2807:h
2804:(
2794:X
2788:j
2784:u
2778:j
2770:=
2765:0
2761:h
2732:j
2728:f
2699:j
2695:a
2689:j
2685:u
2679:j
2671:=
2668:a
2642:a
2620:b
2612:a
2592:d
2586:)
2583:h
2580:(
2548:a
2520:a
2509:a
2501:h
2473:a
2462:a
2438:a
2422:a
2396:.
2389:}
2385:d
2379:k
2375:,
2370:)
2367:k
2364:(
2360:N
2353:j
2349:,
2346:N
2340:i
2334::
2328:)
2325:k
2322:(
2317:j
2313:f
2309:,
2304:i
2300:f
2295:{
2271:]
2268:X
2265:[
2262:R
2248:a
2225:k
2197:)
2194:k
2191:(
2186:1
2178:)
2175:k
2172:(
2168:N
2163:f
2159:,
2153:,
2148:)
2145:k
2142:(
2137:0
2133:f
2107:a
2085:R
2063:k
2057:b
2034:k
2009:a
1985:k
1979:b
1956:}
1953:)
1948:1
1942:N
1938:f
1934:(
1925:,
1919:,
1916:)
1911:0
1907:f
1903:(
1894:{
1874:d
1852:1
1846:N
1842:f
1838:,
1832:,
1827:0
1823:f
1800:a
1778:R
1756:a
1732:b
1710:]
1707:X
1704:[
1701:R
1693:a
1661:N
1657:f
1634:N
1628:b
1617:a
1609:g
1601:N
1597:f
1574:N
1568:b
1557:N
1553:f
1530:N
1524:b
1515:g
1493:N
1489:f
1465:,
1460:i
1456:f
1450:)
1445:i
1441:f
1437:(
1425:)
1420:N
1416:f
1412:(
1402:X
1396:i
1392:u
1386:N
1380:i
1372:=
1369:g
1343:.
1340:R
1332:i
1328:u
1323:,
1318:i
1314:a
1308:i
1304:u
1298:N
1292:i
1284:=
1279:N
1275:a
1251:N
1228:)
1223:1
1217:N
1213:a
1209:,
1203:,
1198:0
1194:a
1190:(
1187:=
1182:b
1151:)
1146:2
1142:a
1138:,
1133:1
1129:a
1125:,
1120:0
1116:a
1112:(
1106:)
1101:1
1097:a
1093:,
1088:0
1084:a
1080:(
1074:)
1069:0
1065:a
1061:(
1038:R
1015:,
1010:1
1006:a
1002:,
997:0
993:a
972:R
950:b
926:n
922:f
899:n
895:a
870:}
864:,
861:)
856:1
852:f
848:(
839:,
836:)
831:0
827:f
823:(
814:{
788:n
782:b
771:a
761:n
757:f
734:1
728:n
724:f
720:,
714:,
709:0
705:f
682:n
676:b
653:}
647:,
642:1
638:f
634:,
629:0
625:f
621:{
597:]
594:X
591:[
588:R
580:a
542:]
539:X
536:[
533:R
506:R
431:]
426:n
422:X
418:,
412:,
407:1
403:X
399:[
396:R
376:R
348:]
345:X
342:[
339:R
319:R
292:]
289:X
286:[
283:R
263:R
243:R
219:R
199:X
175:]
172:X
169:[
166:R
142:R
20:)
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