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