36:
2685:
1999:
608:
2680:{\displaystyle {\begin{aligned}e_{1}(X_{1},X_{2},X_{3},X_{4})&=X_{1}+X_{2}+X_{3}+X_{4},\\e_{2}(X_{1},X_{2},X_{3},X_{4})&=X_{1}X_{2}+X_{1}X_{3}+X_{1}X_{4}+X_{2}X_{3}+X_{2}X_{4}+X_{3}X_{4},\\e_{3}(X_{1},X_{2},X_{3},X_{4})&=X_{1}X_{2}X_{3}+X_{1}X_{2}X_{4}+X_{1}X_{3}X_{4}+X_{2}X_{3}X_{4},\\e_{4}(X_{1},X_{2},X_{3},X_{4})&=X_{1}X_{2}X_{3}X_{4}.\,\\\end{aligned}}}
1981:
232:
2977:
1314:
1634:
603:{\displaystyle {\begin{aligned}e_{1}(X_{1},X_{2},\dots ,X_{n})&=\sum _{1\leq j\leq n}X_{j},\\e_{2}(X_{1},X_{2},\dots ,X_{n})&=\sum _{1\leq j<k\leq n}X_{j}X_{k},\\e_{3}(X_{1},X_{2},\dots ,X_{n})&=\sum _{1\leq j<k<l\leq n}X_{j}X_{k}X_{l},\\\end{aligned}}}
3800:
4664:
1616:
907:
3468:
2705:
4244:
3967:
have equal lacunary parts. (This is because every monomial which can appear in a lacunary part must lack at least one variable, and thus can be transformed by a permutation of the variables into a monomial which contains only the variables
1092:
4408:
5406:
of elementary symmetric polynomials are linearly independent, is also easily proved. The lemma shows that all these products have different leading monomials, and this suffices: if a nontrivial linear combination of the
728:
1976:{\displaystyle {\begin{aligned}e_{1}(X_{1},X_{2},X_{3})&=X_{1}+X_{2}+X_{3},\\e_{2}(X_{1},X_{2},X_{3})&=X_{1}X_{2}+X_{1}X_{3}+X_{2}X_{3},\\e_{3}(X_{1},X_{2},X_{3})&=X_{1}X_{2}X_{3}.\,\\\end{aligned}}}
4948:, and also leads to a fairly direct procedure to effectively write a symmetric polynomial as a polynomial in the elementary symmetric ones. Assume the symmetric polynomial to be homogeneous of degree
3632:
4469:
2004:
1639:
1455:
237:
1450:
746:
5341:
is either zero or a symmetric polynomial with a strictly smaller leading monomial. Writing this difference inductively as a polynomial in the elementary symmetric polynomials, and adding back
3299:
2972:{\displaystyle \prod _{j=1}^{n}(\lambda -X_{j})=\lambda ^{n}-e_{1}(X_{1},\ldots ,X_{n})\lambda ^{n-1}+e_{2}(X_{1},\ldots ,X_{n})\lambda ^{n-2}+\cdots +(-1)^{n}e_{n}(X_{1},\ldots ,X_{n}).}
1432:
4099:
3077:. When we substitute these eigenvalues into the elementary symmetric polynomials, we obtain – up to their sign – the coefficients of the characteristic polynomial, which are
5249:. To count the occurrences of the individual variables in the resulting monomial, fill the column of the Young diagram corresponding to the factor concerned with the numbers
5446:) the largest leading monomial; the leading term of this contribution cannot be cancelled by any other contribution of the linear combination, which gives a contradiction.
3160:
3578:
of the homogeneous polynomial. The general case then follows by splitting an arbitrary symmetric polynomial into its homogeneous components (which are again symmetric).
1309:{\displaystyle e_{\lambda }(X_{1},\dots ,X_{n})=e_{\lambda _{1}}(X_{1},\dots ,X_{n})\cdot e_{\lambda _{2}}(X_{1},\dots ,X_{n})\cdots e_{\lambda _{m}}(X_{1},\dots ,X_{n})}
122:
is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of
4287:
116:, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial
65:
4853:) which by the inductive hypothesis can be expressed as a polynomial in the elementary symmetric functions. Combining the representations for
619:
17:
977:
is included among the elementary symmetric polynomials, but excluding it allows generally simpler formulation of results and properties.)
5435:
were zero, one focuses on the contribution in the linear combination with nonzero coefficient and with (as polynomial in the variables
5256:
of the variables, then all boxes in the first row contain 1, those in the second row 2, and so forth, which means the leading term is
5460:
3172:
3054:
their sign – the elementary symmetric polynomials. These relations between the roots and the coefficients of a polynomial are called
1027:
5267:
Now one proves by induction on the leading monomial in lexicographic order, that any nonzero homogeneous symmetric polynomial
3795:{\displaystyle P(X_{1},\ldots ,X_{n})=P_{\text{lacunary}}(X_{1},\ldots ,X_{n})+X_{1}\cdots X_{n}\cdot Q(X_{1},\ldots ,X_{n}).}
4659:{\displaystyle R(X_{1},\ldots ,X_{n-1},0)={\tilde {Q}}(\sigma _{1,n-1},\ldots ,\sigma _{n-1,n-1})=P(X_{1},\ldots ,X_{n-1},0)}
1611:{\displaystyle {\begin{aligned}e_{1}(X_{1},X_{2})&=X_{1}+X_{2},\\e_{2}(X_{1},X_{2})&=X_{1}X_{2}.\,\\\end{aligned}}}
902:{\displaystyle e_{k}(X_{1},\ldots ,X_{n})=\sum _{1\leq j_{1}<j_{2}<\cdots <j_{k}\leq n}X_{j_{1}}\dotsm X_{j_{k}},}
4878:
The uniqueness of the representation can be proved inductively in a similar way. (It is equivalent to the fact that the
5193:. The leading term of the product is the product of the leading terms of each factor (this is true whenever one uses a
5542:
5520:
3463:{\displaystyle P(X_{1},\ldots ,X_{n})=Q{\big (}e_{1}(X_{1},\ldots ,X_{n}),\ldots ,e_{n}(X_{1},\ldots ,X_{n}){\big )}}
87:
58:
5586:
3176:
3136:
variables. More specifically, the ring of symmetric polynomials with integer coefficients equals the integral
5591:
4239:{\displaystyle {\tilde {P}}(X_{1},\ldots ,X_{n-1})={\tilde {Q}}(\sigma _{1,n-1},\ldots ,\sigma _{n-1,n-1})}
1378:
4930:
The following proof is also inductive, but does not involve other polynomials than those symmetric in
5480:
4901:
3062:
48:
5475:
52:
44:
3143:
5485:
5008:, etc. Furthermore parametrize all products of elementary symmetric polynomials that have degree
3119:
5470:
5125:
is present only because traditionally this product is associated to the transpose partition of
3575:
3559:
3555:
3082:
123:
69:
4990:, in other words the dominant term of a polynomial is one with the highest occurring power of
5495:
5092:
arises for exactly one product of elementary symmetric polynomials, which we shall denote by
3008:
4403:{\displaystyle R(X_{1},\ldots ,X_{n}):={\tilde {Q}}(\sigma _{1,n},\ldots ,\sigma _{n-1,n}).}
3098:
is – up to the sign – the constant term of the characteristic polynomial, i.e. the value of
5455:
5378:
The fact that this expression is unique, or equivalently that all the products (monomials)
3588:
the result is trivial because every polynomial in one variable is automatically symmetric.
3127:
3074:
3018:
113:
8:
4969:
3864:
is symmetric, the lacunary part is determined by its terms containing only the variables
3055:
105:
5530:
5490:
3164:
3123:
3069:
is an example of application of Vieta's formulas. The roots of this polynomial are the
2695:
The elementary symmetric polynomials appear when we expand a linear factorization of a
5131:). The essential ingredient of the proof is the following simple property, which uses
5538:
5516:
5015:
3484:
1034:
5465:
3531:
3188:
3168:
3078:
2696:
138:
5285:
is symmetric, its leading monomial has weakly decreasing exponents, so it is some
5508:
3137:
5194:
5560:
5197:, like the lexicographic order used here), and the leading term of the factor
5580:
5065:
3109:. Thus the determinant of a square matrix is the product of the eigenvalues.
3066:
5279:
can be written as polynomial in the elementary symmetric polynomials. Since
5132:
1356:
elementary symmetric polynomials for the first four positive values of
1026:
of variables, that is, taking variables with repetition, one arrives at the
4954:; different homogeneous components can be decomposed separately. Order the
3171:. For another system of symmetric polynomials with the same property see
3095:
3047:
101:
3070:
723:{\displaystyle e_{n}(X_{1},X_{2},\dots ,X_{n})=X_{1}X_{2}\cdots X_{n}.}
4910:.) The fact that the polynomial representation is unique implies that
3175:, and for a system with a similar, but slightly weaker, property see
3591:
Assume now that the theorem has been proved for all polynomials for
4955:
4790:
of all variables, which equals the elementary symmetric polynomial
1022:
variables. (By contrast, if one performs the same operation using
992:
there exists exactly one elementary symmetric polynomial of degree
4093:. By the inductive hypothesis, this polynomial can be written as
1037:(that is, a finite non-increasing sequence of positive integers)
4774:
has no lacunary part, and is therefore divisible by the product
3626:
can be decomposed as a sum of homogeneous symmetric polynomials
3182:
2982:
That is, when we substitute numerical values for the variables
150:, and it is formed by adding together all distinct products of
5064:
boxes, and arrange those columns from left to right to form a
4015:
are precisely the terms that survive the operation of setting
3051:
5369:
to it, one obtains the sought for polynomial expression for
3196:, denote the ring of symmetric polynomials in the variables
5074:
boxes in all. The shape of this diagram is a partition of
4837:
is a homogeneous symmetric polynomial of degree less than
3085:(the sum of the elements of the diagonal) is the value of
5562:
Prelude to Galois Theory: Exploring
Symmetric Polynomials
5058:
come first, then build for each such factor a column of
5024:. Order the individual elementary symmetric polynomials
4727:
before each monomial which contains only the variables
3930:
before each monomial which contains only the variables
3094:, and thus the sum of the eigenvalues. Similarly, the
4472:
4290:
4102:
3635:
3302:
3146:
2708:
2002:
1637:
1453:
1381:
1095:
1086:, also called an elementary symmetric polynomial, by
749:
622:
235:
4999:, and among those the one with the highest power of
4752:. As we know, this shows that the lacunary part of
4050:, which is a symmetric polynomial in the variables
3483:. Another way of saying the same thing is that the
4658:
4402:
4238:
3924:having the same degree, and if the coefficient of
3794:
3462:
3154:
2971:
2679:
1975:
1610:
1426:
1308:
901:
722:
602:
5052:in the product so that those with larger indices
5578:
57:but its sources remain unclear because it lacks
4758:coincides with that of the original polynomial
4271:denote the elementary symmetric polynomials in
3112:The set of elementary symmetric polynomials in
3163:. (See below for a more general statement and
5014:(they are in fact homogeneous) as follows by
4972:, where the individual variables are ordered
3906:are two homogeneous symmetric polynomials in
3455:
3349:
3183:Fundamental theorem of symmetric polynomials
1004:variables. To form the one that has degree
3601:variables and all symmetric polynomials in
3269:This means that every symmetric polynomial
4869:one finds a polynomial representation for
4669:(the first equality holds because setting
3820:which contain only a proper subset of the
3814:is defined as the sum of all monomials in
3173:Complete homogeneous symmetric polynomials
3167:.) This fact is one of the foundations of
1028:complete homogeneous symmetric polynomials
5537:. Cambridge: Cambridge University Press.
5515:(2nd ed.). Oxford: Clarendon Press.
5507:
5461:Complete homogeneous symmetric polynomial
3614:. Every homogeneous symmetric polynomial
3148:
2672:
1968:
1603:
112:are one type of basic building block for
88:Learn how and when to remove this message
5558:
5513:Symmetric Functions and Hall Polynomials
4746:equals the corresponding coefficient of
3949:equals the corresponding coefficient of
3562:with respect to the number of variables
3554:The theorem may be proved for symmetric
164:The elementary symmetric polynomials in
5529:
1060:, one defines the symmetric polynomial
14:
5579:
5303:. Let the coefficient of this term be
4721:). In other words, the coefficient of
1010:, we take the sum of all products of
4925:
3844:, i.e., where at least one variable
29:
3226:. This is a polynomial ring in the
1427:{\displaystyle e_{1}(X_{1})=X_{1}.}
24:
3081:of the matrix. In particular, the
25:
5603:
5559:Trifonov, Martin (5 March 2024).
5552:
5535:Enumerative Combinatorics, Vol. 2
3996:which contain only the variables
3230:elementary symmetric polynomials
980:Thus, for each positive integer
110:elementary symmetric polynomials
34:
5135:for monomials in the variables
4026:to 0, so their sum equals
3549:
3021:are the values substituted for
4653:
4609:
4600:
4538:
4532:
4520:
4476:
4394:
4344:
4338:
4326:
4294:
4233:
4171:
4165:
4153:
4115:
4109:
3786:
3754:
3719:
3687:
3671:
3639:
3450:
3418:
3396:
3364:
3338:
3306:
3177:Power sum symmetric polynomial
2963:
2931:
2912:
2902:
2874:
2842:
2810:
2778:
2749:
2730:
2619:
2567:
2411:
2359:
2197:
2145:
2069:
2017:
1925:
1886:
1793:
1754:
1691:
1652:
1570:
1544:
1494:
1468:
1405:
1392:
1303:
1271:
1248:
1216:
1193:
1161:
1138:
1106:
792:
760:
678:
633:
519:
474:
399:
354:
295:
250:
13:
1:
5501:
4436:is a symmetric polynomial in
4281:Consider now the polynomial
3883:, i.e., which do not contain
2690:
159:
18:Elementary symmetric function
3293:has a unique representation
3155:{\displaystyle \mathbb {Z} }
7:
5449:
4764:. Therefore the difference
1345:
10:
5608:
4255:. Here the doubly indexed
613:and so forth, ending with
4902:algebraically independent
3805:Here the "lacunary part"
3063:characteristic polynomial
4843:(in fact degree at most
4454:, of the same degree as
4069:that we shall denote by
2699:: we have the identity
1350:The following lists the
43:This article includes a
5587:Homogeneous polynomials
5486:MacMahon Master theorem
3556:homogeneous polynomials
1320:Sometimes the notation
72:more precise citations.
5481:Maclaurin's inequality
5152:. The leading term of
4660:
4404:
4240:
3796:
3607:variables with degree
3574:, with respect to the
3464:
3156:
3007:, we obtain the monic
2973:
2729:
2681:
1977:
1612:
1428:
1310:
986:less than or equal to
903:
724:
604:
5496:Representation theory
5476:Newton's inequalities
5080:, and each partition
4661:
4405:
4241:
3894:. More precisely: If
3797:
3465:
3214:with coefficients in
3157:
3128:symmetric polynomials
3009:univariate polynomial
2974:
2709:
2682:
1978:
1613:
1429:
1311:
904:
725:
605:
114:symmetric polynomials
27:Mathematical function
5456:Symmetric polynomial
5133:multi-index notation
4470:
4288:
4100:
3633:
3473:for some polynomial
3300:
3144:
2706:
2000:
1635:
1451:
1379:
1093:
747:
620:
233:
156:distinct variables.
5592:Symmetric functions
5531:Stanley, Richard P.
5471:Newton's identities
1331:is used instead of
137:variables for each
106:commutative algebra
5491:Symmetric function
4656:
4463:, which satisfies
4400:
4236:
3792:
3460:
3152:
2969:
2677:
2675:
1973:
1971:
1608:
1606:
1424:
1306:
899:
858:
720:
600:
598:
562:
436:
326:
104:, specifically in
45:list of references
4970:lexicographically
4958:in the variables
4926:Alternative proof
4916:is isomorphic to
4535:
4341:
4168:
4112:
3990:But the terms of
3684:
3485:ring homomorphism
1035:integer partition
798:
529:
409:
305:
226:, are defined by
98:
97:
90:
16:(Redirected from
5599:
5573:
5571:
5570:
5565:(Video). YouTube
5548:
5526:
5509:Macdonald, I. G.
5466:Schur polynomial
5445:
5434:
5405:
5374:
5368:
5340:
5308:
5302:
5296:
5290:
5284:
5278:
5272:
5262:
5255:
5248:
5224:
5185:
5179:
5145:
5130:
5124:
5118:
5091:
5085:
5079:
5073:
5063:
5057:
5051:
5023:
5013:
5007:
4998:
4989:
4968:
4953:
4947:
4921:
4915:
4909:
4899:
4883:
4874:
4868:
4862:
4852:
4842:
4836:
4830:
4804:
4789:
4773:
4763:
4757:
4751:
4745:
4726:
4720:
4710:
4694:
4679:
4665:
4663:
4662:
4657:
4646:
4645:
4621:
4620:
4599:
4598:
4562:
4561:
4537:
4536:
4528:
4513:
4512:
4488:
4487:
4462:
4453:
4435:
4409:
4407:
4406:
4401:
4393:
4392:
4362:
4361:
4343:
4342:
4334:
4325:
4324:
4306:
4305:
4277:
4270:
4254:
4245:
4243:
4242:
4237:
4232:
4231:
4195:
4194:
4170:
4169:
4161:
4152:
4151:
4127:
4126:
4114:
4113:
4105:
4092:
4068:
4049:
4025:
4014:
3995:
3986:
3966:
3960:
3954:
3948:
3929:
3923:
3905:
3899:
3893:
3882:
3863:
3854:
3843:
3825:
3819:
3813:
3801:
3799:
3798:
3793:
3785:
3784:
3766:
3765:
3747:
3746:
3734:
3733:
3718:
3717:
3699:
3698:
3686:
3685:
3682:
3670:
3669:
3651:
3650:
3625:
3619:
3613:
3606:
3600:
3587:
3573:
3567:
3545:
3539:
3529:
3519:
3495:
3482:
3469:
3467:
3466:
3461:
3459:
3458:
3449:
3448:
3430:
3429:
3417:
3416:
3395:
3394:
3376:
3375:
3363:
3362:
3353:
3352:
3337:
3336:
3318:
3317:
3292:
3265:
3255:
3225:
3219:
3213:
3195:
3189:commutative ring
3169:invariant theory
3162:
3161:
3159:
3158:
3153:
3151:
3135:
3117:
3108:
3093:
3056:Vieta's formulas
3045:
3016:
3006:
2978:
2976:
2975:
2970:
2962:
2961:
2943:
2942:
2930:
2929:
2920:
2919:
2892:
2891:
2873:
2872:
2854:
2853:
2841:
2840:
2828:
2827:
2809:
2808:
2790:
2789:
2777:
2776:
2764:
2763:
2748:
2747:
2728:
2723:
2697:monic polynomial
2686:
2684:
2683:
2678:
2676:
2668:
2667:
2658:
2657:
2648:
2647:
2638:
2637:
2618:
2617:
2605:
2604:
2592:
2591:
2579:
2578:
2566:
2565:
2549:
2548:
2539:
2538:
2529:
2528:
2516:
2515:
2506:
2505:
2496:
2495:
2483:
2482:
2473:
2472:
2463:
2462:
2450:
2449:
2440:
2439:
2430:
2429:
2410:
2409:
2397:
2396:
2384:
2383:
2371:
2370:
2358:
2357:
2341:
2340:
2331:
2330:
2318:
2317:
2308:
2307:
2295:
2294:
2285:
2284:
2272:
2271:
2262:
2261:
2249:
2248:
2239:
2238:
2226:
2225:
2216:
2215:
2196:
2195:
2183:
2182:
2170:
2169:
2157:
2156:
2144:
2143:
2127:
2126:
2114:
2113:
2101:
2100:
2088:
2087:
2068:
2067:
2055:
2054:
2042:
2041:
2029:
2028:
2016:
2015:
1992:
1982:
1980:
1979:
1974:
1972:
1964:
1963:
1954:
1953:
1944:
1943:
1924:
1923:
1911:
1910:
1898:
1897:
1885:
1884:
1868:
1867:
1858:
1857:
1845:
1844:
1835:
1834:
1822:
1821:
1812:
1811:
1792:
1791:
1779:
1778:
1766:
1765:
1753:
1752:
1736:
1735:
1723:
1722:
1710:
1709:
1690:
1689:
1677:
1676:
1664:
1663:
1651:
1650:
1627:
1617:
1615:
1614:
1609:
1607:
1599:
1598:
1589:
1588:
1569:
1568:
1556:
1555:
1543:
1542:
1526:
1525:
1513:
1512:
1493:
1492:
1480:
1479:
1467:
1466:
1443:
1433:
1431:
1430:
1425:
1420:
1419:
1404:
1403:
1391:
1390:
1371:
1361:
1355:
1341:
1330:
1315:
1313:
1312:
1307:
1302:
1301:
1283:
1282:
1270:
1269:
1268:
1267:
1247:
1246:
1228:
1227:
1215:
1214:
1213:
1212:
1192:
1191:
1173:
1172:
1160:
1159:
1158:
1157:
1137:
1136:
1118:
1117:
1105:
1104:
1085:
1059:
1021:
1016:-subsets of the
1015:
1009:
1003:
997:
991:
985:
976:
949:
939:
908:
906:
905:
900:
895:
894:
893:
892:
875:
874:
873:
872:
857:
850:
849:
831:
830:
818:
817:
791:
790:
772:
771:
759:
758:
739:
733:In general, for
729:
727:
726:
721:
716:
715:
703:
702:
693:
692:
677:
676:
658:
657:
645:
644:
632:
631:
609:
607:
606:
601:
599:
592:
591:
582:
581:
572:
571:
561:
518:
517:
499:
498:
486:
485:
473:
472:
456:
455:
446:
445:
435:
398:
397:
379:
378:
366:
365:
353:
352:
336:
335:
325:
294:
293:
275:
274:
262:
261:
249:
248:
225:
215:
187:
169:
155:
149:
139:positive integer
136:
130:
121:
93:
86:
82:
79:
73:
68:this article by
59:inline citations
38:
37:
30:
21:
5607:
5606:
5602:
5601:
5600:
5598:
5597:
5596:
5577:
5576:
5568:
5566:
5555:
5545:
5523:
5504:
5452:
5444:
5436:
5432:
5423:
5416:
5408:
5403:
5394:
5387:
5379:
5370:
5366:
5357:
5350:
5342:
5338:
5329:
5322:
5310:
5304:
5298:
5297:a partition of
5292:
5286:
5280:
5274:
5268:
5257:
5250:
5247:
5238:
5232:
5226:
5222:
5213:
5206:
5198:
5181:
5177:
5168:
5161:
5153:
5144:
5136:
5126:
5120:
5117:
5108:
5101:
5093:
5087:
5081:
5075:
5069:
5059:
5053:
5049:
5040:
5033:
5025:
5019:
5009:
5006:
5000:
4997:
4991:
4988:
4979:
4973:
4967:
4959:
4949:
4946:
4937:
4931:
4928:
4917:
4911:
4905:
4897:
4891:
4885:
4879:
4870:
4864:
4854:
4844:
4838:
4832:
4831:, the quotient
4826:
4806:
4805:. Then writing
4803:
4791:
4787:
4781:
4775:
4765:
4759:
4753:
4747:
4744:
4734:
4728:
4722:
4712:
4709:
4696:
4693:
4681:
4678:
4670:
4635:
4631:
4616:
4612:
4576:
4572:
4545:
4541:
4527:
4526:
4502:
4498:
4483:
4479:
4471:
4468:
4467:
4461:
4455:
4452:
4443:
4437:
4433:
4424:
4414:
4376:
4372:
4351:
4347:
4333:
4332:
4320:
4316:
4301:
4297:
4289:
4286:
4285:
4272:
4269:
4256:
4250:
4209:
4205:
4178:
4174:
4160:
4159:
4141:
4137:
4122:
4118:
4104:
4103:
4101:
4098:
4097:
4090:
4080:
4070:
4067:
4057:
4051:
4047:
4037:
4027:
4024:
4016:
4013:
4003:
3997:
3991:
3985:
3975:
3969:
3962:
3956:
3950:
3947:
3937:
3931:
3925:
3922:
3913:
3907:
3901:
3895:
3892:
3884:
3881:
3871:
3865:
3859:
3853:
3845:
3842:
3833:
3827:
3821:
3815:
3812:
3806:
3780:
3776:
3761:
3757:
3742:
3738:
3729:
3725:
3713:
3709:
3694:
3690:
3681:
3677:
3665:
3661:
3646:
3642:
3634:
3631:
3630:
3621:
3615:
3608:
3602:
3592:
3582:
3569:
3568:and, for fixed
3563:
3552:
3541:
3535:
3521:
3516:
3510:
3502:
3497:
3493:
3488:
3474:
3454:
3453:
3444:
3440:
3425:
3421:
3412:
3408:
3390:
3386:
3371:
3367:
3358:
3354:
3348:
3347:
3332:
3328:
3313:
3309:
3301:
3298:
3297:
3286:
3280:
3270:
3257:
3253:
3244:
3236:
3231:
3221:
3215:
3212:
3203:
3197:
3191:
3185:
3147:
3145:
3142:
3141:
3140:
3138:polynomial ring
3131:
3113:
3107:
3099:
3092:
3086:
3044:
3035:
3028:
3022:
3012:
3011:(with variable
3005:
2996:
2989:
2983:
2957:
2953:
2938:
2934:
2925:
2921:
2915:
2911:
2881:
2877:
2868:
2864:
2849:
2845:
2836:
2832:
2817:
2813:
2804:
2800:
2785:
2781:
2772:
2768:
2759:
2755:
2743:
2739:
2724:
2713:
2707:
2704:
2703:
2693:
2674:
2673:
2663:
2659:
2653:
2649:
2643:
2639:
2633:
2629:
2622:
2613:
2609:
2600:
2596:
2587:
2583:
2574:
2570:
2561:
2557:
2554:
2553:
2544:
2540:
2534:
2530:
2524:
2520:
2511:
2507:
2501:
2497:
2491:
2487:
2478:
2474:
2468:
2464:
2458:
2454:
2445:
2441:
2435:
2431:
2425:
2421:
2414:
2405:
2401:
2392:
2388:
2379:
2375:
2366:
2362:
2353:
2349:
2346:
2345:
2336:
2332:
2326:
2322:
2313:
2309:
2303:
2299:
2290:
2286:
2280:
2276:
2267:
2263:
2257:
2253:
2244:
2240:
2234:
2230:
2221:
2217:
2211:
2207:
2200:
2191:
2187:
2178:
2174:
2165:
2161:
2152:
2148:
2139:
2135:
2132:
2131:
2122:
2118:
2109:
2105:
2096:
2092:
2083:
2079:
2072:
2063:
2059:
2050:
2046:
2037:
2033:
2024:
2020:
2011:
2007:
2003:
2001:
1998:
1997:
1987:
1970:
1969:
1959:
1955:
1949:
1945:
1939:
1935:
1928:
1919:
1915:
1906:
1902:
1893:
1889:
1880:
1876:
1873:
1872:
1863:
1859:
1853:
1849:
1840:
1836:
1830:
1826:
1817:
1813:
1807:
1803:
1796:
1787:
1783:
1774:
1770:
1761:
1757:
1748:
1744:
1741:
1740:
1731:
1727:
1718:
1714:
1705:
1701:
1694:
1685:
1681:
1672:
1668:
1659:
1655:
1646:
1642:
1638:
1636:
1633:
1632:
1622:
1605:
1604:
1594:
1590:
1584:
1580:
1573:
1564:
1560:
1551:
1547:
1538:
1534:
1531:
1530:
1521:
1517:
1508:
1504:
1497:
1488:
1484:
1475:
1471:
1462:
1458:
1454:
1452:
1449:
1448:
1438:
1415:
1411:
1399:
1395:
1386:
1382:
1380:
1377:
1376:
1366:
1357:
1351:
1348:
1340:
1332:
1329:
1321:
1297:
1293:
1278:
1274:
1263:
1259:
1258:
1254:
1242:
1238:
1223:
1219:
1208:
1204:
1203:
1199:
1187:
1183:
1168:
1164:
1153:
1149:
1148:
1144:
1132:
1128:
1113:
1109:
1100:
1096:
1094:
1091:
1090:
1083:
1074:
1066:
1061:
1057:
1048:
1038:
1017:
1011:
1005:
999:
993:
987:
981:
974:
965:
958:
951:
941:
937:
928:
921:
913:
888:
884:
883:
879:
868:
864:
863:
859:
845:
841:
826:
822:
813:
809:
802:
786:
782:
767:
763:
754:
750:
748:
745:
744:
734:
711:
707:
698:
694:
688:
684:
672:
668:
653:
649:
640:
636:
627:
623:
621:
618:
617:
597:
596:
587:
583:
577:
573:
567:
563:
533:
522:
513:
509:
494:
490:
481:
477:
468:
464:
461:
460:
451:
447:
441:
437:
413:
402:
393:
389:
374:
370:
361:
357:
348:
344:
341:
340:
331:
327:
309:
298:
289:
285:
270:
266:
257:
253:
244:
240:
236:
234:
231:
230:
217:
213:
204:
197:
189:
186:
177:
171:
165:
162:
151:
141:
132:
126:
117:
94:
83:
77:
74:
63:
49:related reading
39:
35:
28:
23:
22:
15:
12:
11:
5:
5605:
5595:
5594:
5589:
5575:
5574:
5554:
5553:External links
5551:
5550:
5549:
5543:
5527:
5521:
5503:
5500:
5499:
5498:
5493:
5488:
5483:
5478:
5473:
5468:
5463:
5458:
5451:
5448:
5440:
5428:
5421:
5412:
5399:
5392:
5383:
5362:
5355:
5346:
5334:
5327:
5318:
5265:
5264:
5243:
5236:
5230:
5218:
5211:
5202:
5195:monomial order
5173:
5166:
5157:
5140:
5113:
5106:
5097:
5045:
5038:
5029:
5004:
4995:
4984:
4980:> ... >
4977:
4963:
4942:
4935:
4927:
4924:
4904:over the ring
4895:
4889:
4818:
4795:
4785:
4779:
4739:
4732:
4700:
4685:
4674:
4667:
4666:
4655:
4652:
4649:
4644:
4641:
4638:
4634:
4630:
4627:
4624:
4619:
4615:
4611:
4608:
4605:
4602:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4575:
4571:
4568:
4565:
4560:
4557:
4554:
4551:
4548:
4544:
4540:
4534:
4531:
4525:
4522:
4519:
4516:
4511:
4508:
4505:
4501:
4497:
4494:
4491:
4486:
4482:
4478:
4475:
4459:
4448:
4441:
4429:
4422:
4411:
4410:
4399:
4396:
4391:
4388:
4385:
4382:
4379:
4375:
4371:
4368:
4365:
4360:
4357:
4354:
4350:
4346:
4340:
4337:
4331:
4328:
4323:
4319:
4315:
4312:
4309:
4304:
4300:
4296:
4293:
4260:
4247:
4246:
4235:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4208:
4204:
4201:
4198:
4193:
4190:
4187:
4184:
4181:
4177:
4173:
4167:
4164:
4158:
4155:
4150:
4147:
4144:
4140:
4136:
4133:
4130:
4125:
4121:
4117:
4111:
4108:
4085:
4078:
4062:
4055:
4042:
4035:
4020:
4008:
4001:
3980:
3973:
3942:
3935:
3918:
3911:
3888:
3876:
3869:
3849:
3838:
3831:
3810:
3803:
3802:
3791:
3788:
3783:
3779:
3775:
3772:
3769:
3764:
3760:
3756:
3753:
3750:
3745:
3741:
3737:
3732:
3728:
3724:
3721:
3716:
3712:
3708:
3705:
3702:
3697:
3693:
3689:
3680:
3676:
3673:
3668:
3664:
3660:
3657:
3654:
3649:
3645:
3641:
3638:
3551:
3548:
3514:
3508:
3500:
3491:
3471:
3470:
3457:
3452:
3447:
3443:
3439:
3436:
3433:
3428:
3424:
3420:
3415:
3411:
3407:
3404:
3401:
3398:
3393:
3389:
3385:
3382:
3379:
3374:
3370:
3366:
3361:
3357:
3351:
3346:
3343:
3340:
3335:
3331:
3327:
3324:
3321:
3316:
3312:
3308:
3305:
3284:
3278:
3249:
3242:
3234:
3208:
3201:
3184:
3181:
3150:
3103:
3090:
3040:
3033:
3026:
3001:
2994:
2987:
2980:
2979:
2968:
2965:
2960:
2956:
2952:
2949:
2946:
2941:
2937:
2933:
2928:
2924:
2918:
2914:
2910:
2907:
2904:
2901:
2898:
2895:
2890:
2887:
2884:
2880:
2876:
2871:
2867:
2863:
2860:
2857:
2852:
2848:
2844:
2839:
2835:
2831:
2826:
2823:
2820:
2816:
2812:
2807:
2803:
2799:
2796:
2793:
2788:
2784:
2780:
2775:
2771:
2767:
2762:
2758:
2754:
2751:
2746:
2742:
2738:
2735:
2732:
2727:
2722:
2719:
2716:
2712:
2692:
2689:
2688:
2687:
2671:
2666:
2662:
2656:
2652:
2646:
2642:
2636:
2632:
2628:
2625:
2623:
2621:
2616:
2612:
2608:
2603:
2599:
2595:
2590:
2586:
2582:
2577:
2573:
2569:
2564:
2560:
2556:
2555:
2552:
2547:
2543:
2537:
2533:
2527:
2523:
2519:
2514:
2510:
2504:
2500:
2494:
2490:
2486:
2481:
2477:
2471:
2467:
2461:
2457:
2453:
2448:
2444:
2438:
2434:
2428:
2424:
2420:
2417:
2415:
2413:
2408:
2404:
2400:
2395:
2391:
2387:
2382:
2378:
2374:
2369:
2365:
2361:
2356:
2352:
2348:
2347:
2344:
2339:
2335:
2329:
2325:
2321:
2316:
2312:
2306:
2302:
2298:
2293:
2289:
2283:
2279:
2275:
2270:
2266:
2260:
2256:
2252:
2247:
2243:
2237:
2233:
2229:
2224:
2220:
2214:
2210:
2206:
2203:
2201:
2199:
2194:
2190:
2186:
2181:
2177:
2173:
2168:
2164:
2160:
2155:
2151:
2147:
2142:
2138:
2134:
2133:
2130:
2125:
2121:
2117:
2112:
2108:
2104:
2099:
2095:
2091:
2086:
2082:
2078:
2075:
2073:
2071:
2066:
2062:
2058:
2053:
2049:
2045:
2040:
2036:
2032:
2027:
2023:
2019:
2014:
2010:
2006:
2005:
1984:
1983:
1967:
1962:
1958:
1952:
1948:
1942:
1938:
1934:
1931:
1929:
1927:
1922:
1918:
1914:
1909:
1905:
1901:
1896:
1892:
1888:
1883:
1879:
1875:
1874:
1871:
1866:
1862:
1856:
1852:
1848:
1843:
1839:
1833:
1829:
1825:
1820:
1816:
1810:
1806:
1802:
1799:
1797:
1795:
1790:
1786:
1782:
1777:
1773:
1769:
1764:
1760:
1756:
1751:
1747:
1743:
1742:
1739:
1734:
1730:
1726:
1721:
1717:
1713:
1708:
1704:
1700:
1697:
1695:
1693:
1688:
1684:
1680:
1675:
1671:
1667:
1662:
1658:
1654:
1649:
1645:
1641:
1640:
1619:
1618:
1602:
1597:
1593:
1587:
1583:
1579:
1576:
1574:
1572:
1567:
1563:
1559:
1554:
1550:
1546:
1541:
1537:
1533:
1532:
1529:
1524:
1520:
1516:
1511:
1507:
1503:
1500:
1498:
1496:
1491:
1487:
1483:
1478:
1474:
1470:
1465:
1461:
1457:
1456:
1435:
1434:
1423:
1418:
1414:
1410:
1407:
1402:
1398:
1394:
1389:
1385:
1347:
1344:
1336:
1325:
1318:
1317:
1305:
1300:
1296:
1292:
1289:
1286:
1281:
1277:
1273:
1266:
1262:
1257:
1253:
1250:
1245:
1241:
1237:
1234:
1231:
1226:
1222:
1218:
1211:
1207:
1202:
1198:
1195:
1190:
1186:
1182:
1179:
1176:
1171:
1167:
1163:
1156:
1152:
1147:
1143:
1140:
1135:
1131:
1127:
1124:
1121:
1116:
1112:
1108:
1103:
1099:
1079:
1072:
1064:
1053:
1046:
970:
963:
956:
950:. (Sometimes,
933:
926:
917:
910:
909:
898:
891:
887:
882:
878:
871:
867:
862:
856:
853:
848:
844:
840:
837:
834:
829:
825:
821:
816:
812:
808:
805:
801:
797:
794:
789:
785:
781:
778:
775:
770:
766:
762:
757:
753:
731:
730:
719:
714:
710:
706:
701:
697:
691:
687:
683:
680:
675:
671:
667:
664:
661:
656:
652:
648:
643:
639:
635:
630:
626:
611:
610:
595:
590:
586:
580:
576:
570:
566:
560:
557:
554:
551:
548:
545:
542:
539:
536:
532:
528:
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508:
505:
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497:
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459:
454:
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419:
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412:
408:
405:
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401:
396:
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388:
385:
382:
377:
373:
369:
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343:
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318:
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308:
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53:external links
42:
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5066:Young diagram
5062:
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4756:
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4742:
4738:
4731:
4725:
4719:
4715:
4707:
4703:
4699:
4692:
4688:
4684:
4680:to 0 in
4677:
4673:
4650:
4647:
4642:
4639:
4636:
4632:
4628:
4625:
4622:
4617:
4613:
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4592:
4589:
4586:
4583:
4580:
4577:
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4397:
4389:
4386:
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4317:
4313:
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4307:
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4298:
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4279:
4275:
4267:
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4259:
4253:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4206:
4202:
4199:
4196:
4191:
4188:
4185:
4182:
4179:
4175:
4162:
4156:
4148:
4145:
4142:
4138:
4134:
4131:
4128:
4123:
4119:
4106:
4096:
4095:
4094:
4088:
4084:
4077:
4073:
4065:
4061:
4054:
4045:
4041:
4034:
4030:
4023:
4019:
4011:
4007:
4000:
3994:
3988:
3983:
3979:
3972:
3965:
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3928:
3921:
3917:
3910:
3904:
3898:
3891:
3887:
3879:
3875:
3868:
3862:
3856:
3855:is missing.
3852:
3848:
3841:
3837:
3830:
3824:
3818:
3809:
3789:
3781:
3777:
3773:
3770:
3767:
3762:
3758:
3751:
3748:
3743:
3739:
3735:
3730:
3726:
3722:
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3706:
3703:
3700:
3695:
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3678:
3674:
3666:
3662:
3658:
3655:
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3647:
3643:
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3629:
3628:
3627:
3624:
3618:
3612:
3605:
3599:
3595:
3589:
3585:
3579:
3577:
3572:
3566:
3561:
3557:
3547:
3544:
3538:
3533:
3528:
3524:
3517:
3507:
3503:
3494:
3486:
3481:
3477:
3445:
3441:
3437:
3434:
3431:
3426:
3422:
3413:
3409:
3405:
3402:
3399:
3391:
3387:
3383:
3380:
3377:
3372:
3368:
3359:
3355:
3344:
3341:
3333:
3329:
3325:
3322:
3319:
3314:
3310:
3303:
3296:
3295:
3294:
3291:
3287:
3277:
3273:
3267:
3264:
3260:
3252:
3248:
3241:
3237:
3229:
3224:
3218:
3211:
3207:
3200:
3194:
3190:
3180:
3178:
3174:
3170:
3166:
3139:
3134:
3129:
3125:
3121:
3116:
3110:
3106:
3102:
3097:
3089:
3084:
3080:
3076:
3072:
3068:
3067:square matrix
3064:
3059:
3057:
3053:
3049:
3043:
3039:
3032:
3025:
3020:
3015:
3010:
3004:
3000:
2993:
2986:
2966:
2958:
2954:
2950:
2947:
2944:
2939:
2935:
2926:
2922:
2916:
2908:
2905:
2899:
2896:
2893:
2888:
2885:
2882:
2878:
2869:
2865:
2861:
2858:
2855:
2850:
2846:
2837:
2833:
2829:
2824:
2821:
2818:
2814:
2805:
2801:
2797:
2794:
2791:
2786:
2782:
2773:
2769:
2765:
2760:
2756:
2752:
2744:
2740:
2736:
2733:
2725:
2720:
2717:
2714:
2710:
2702:
2701:
2700:
2698:
2669:
2664:
2660:
2654:
2650:
2644:
2640:
2634:
2630:
2626:
2624:
2614:
2610:
2606:
2601:
2597:
2593:
2588:
2584:
2580:
2575:
2571:
2562:
2558:
2550:
2545:
2541:
2535:
2531:
2525:
2521:
2517:
2512:
2508:
2502:
2498:
2492:
2488:
2484:
2479:
2475:
2469:
2465:
2459:
2455:
2451:
2446:
2442:
2436:
2432:
2426:
2422:
2418:
2416:
2406:
2402:
2398:
2393:
2389:
2385:
2380:
2376:
2372:
2367:
2363:
2354:
2350:
2342:
2337:
2333:
2327:
2323:
2319:
2314:
2310:
2304:
2300:
2296:
2291:
2287:
2281:
2277:
2273:
2268:
2264:
2258:
2254:
2250:
2245:
2241:
2235:
2231:
2227:
2222:
2218:
2212:
2208:
2204:
2202:
2192:
2188:
2184:
2179:
2175:
2171:
2166:
2162:
2158:
2153:
2149:
2140:
2136:
2128:
2123:
2119:
2115:
2110:
2106:
2102:
2097:
2093:
2089:
2084:
2080:
2076:
2074:
2064:
2060:
2056:
2051:
2047:
2043:
2038:
2034:
2030:
2025:
2021:
2012:
2008:
1996:
1995:
1994:
1990:
1965:
1960:
1956:
1950:
1946:
1940:
1936:
1932:
1930:
1920:
1916:
1912:
1907:
1903:
1899:
1894:
1890:
1881:
1877:
1869:
1864:
1860:
1854:
1850:
1846:
1841:
1837:
1831:
1827:
1823:
1818:
1814:
1808:
1804:
1800:
1798:
1788:
1784:
1780:
1775:
1771:
1767:
1762:
1758:
1749:
1745:
1737:
1732:
1728:
1724:
1719:
1715:
1711:
1706:
1702:
1698:
1696:
1686:
1682:
1678:
1673:
1669:
1665:
1660:
1656:
1647:
1643:
1631:
1630:
1629:
1625:
1600:
1595:
1591:
1585:
1581:
1577:
1575:
1565:
1561:
1557:
1552:
1548:
1539:
1535:
1527:
1522:
1518:
1514:
1509:
1505:
1501:
1499:
1489:
1485:
1481:
1476:
1472:
1463:
1459:
1447:
1446:
1445:
1441:
1421:
1416:
1412:
1408:
1400:
1396:
1387:
1383:
1375:
1374:
1373:
1369:
1363:
1360:
1354:
1343:
1339:
1335:
1328:
1324:
1298:
1294:
1290:
1287:
1284:
1279:
1275:
1264:
1260:
1255:
1251:
1243:
1239:
1235:
1232:
1229:
1224:
1220:
1209:
1205:
1200:
1196:
1188:
1184:
1180:
1177:
1174:
1169:
1165:
1154:
1150:
1145:
1141:
1133:
1129:
1125:
1122:
1119:
1114:
1110:
1101:
1097:
1089:
1088:
1087:
1082:
1078:
1071:
1067:
1056:
1052:
1045:
1041:
1036:
1031:
1029:
1025:
1020:
1014:
1008:
1002:
996:
990:
984:
978:
973:
969:
962:
955:
948:
944:
936:
932:
925:
920:
916:
896:
889:
885:
880:
876:
869:
865:
860:
854:
851:
846:
842:
838:
835:
832:
827:
823:
819:
814:
810:
806:
803:
799:
795:
787:
783:
779:
776:
773:
768:
764:
755:
751:
743:
742:
741:
737:
717:
712:
708:
704:
699:
695:
689:
685:
681:
673:
669:
665:
662:
659:
654:
650:
646:
641:
637:
628:
624:
616:
615:
614:
593:
588:
584:
578:
574:
568:
564:
558:
555:
552:
549:
546:
543:
540:
537:
534:
530:
526:
524:
514:
510:
506:
503:
500:
495:
491:
487:
482:
478:
469:
465:
457:
452:
448:
442:
438:
432:
429:
426:
423:
420:
417:
414:
410:
406:
404:
394:
390:
386:
383:
380:
375:
371:
367:
362:
358:
349:
345:
337:
332:
328:
322:
319:
316:
313:
310:
306:
302:
300:
290:
286:
282:
279:
276:
271:
267:
263:
258:
254:
245:
241:
229:
228:
227:
224:
220:
212:
208:
201:
196:
192:
185:
181:
174:
168:
157:
154:
148:
144:
140:
135:
129:
125:
120:
115:
111:
107:
103:
92:
89:
81:
71:
67:
61:
60:
54:
50:
46:
41:
32:
31:
19:
5567:. Retrieved
5561:
5534:
5512:
5441:
5437:
5429:
5425:
5418:
5413:
5409:
5400:
5396:
5389:
5384:
5380:
5377:
5371:
5363:
5359:
5352:
5347:
5343:
5335:
5331:
5324:
5319:
5315:
5311:
5305:
5299:
5293:
5287:
5281:
5275:
5269:
5266:
5258:
5252:
5244:
5240:
5233:
5227:
5219:
5215:
5208:
5203:
5199:
5190:
5182:
5174:
5170:
5163:
5158:
5154:
5149:
5148:
5141:
5137:
5127:
5121:
5114:
5110:
5103:
5098:
5094:
5088:
5082:
5076:
5070:
5060:
5054:
5046:
5042:
5035:
5030:
5026:
5020:
5010:
5001:
4992:
4985:
4981:
4974:
4964:
4960:
4950:
4943:
4939:
4932:
4929:
4918:
4912:
4906:
4893:
4886:
4884:polynomials
4880:
4877:
4871:
4865:
4859:
4855:
4849:
4845:
4839:
4833:
4827:
4823:
4819:
4815:
4811:
4807:
4800:
4796:
4792:
4783:
4776:
4770:
4766:
4760:
4754:
4748:
4740:
4736:
4729:
4723:
4717:
4713:
4705:
4701:
4697:
4690:
4686:
4682:
4675:
4671:
4668:
4456:
4449:
4445:
4438:
4430:
4426:
4419:
4415:
4412:
4280:
4278:variables.
4273:
4265:
4261:
4257:
4251:
4248:
4086:
4082:
4075:
4071:
4063:
4059:
4052:
4043:
4039:
4032:
4028:
4021:
4017:
4009:
4005:
3998:
3992:
3989:
3981:
3977:
3970:
3963:
3957:
3951:
3943:
3939:
3932:
3926:
3919:
3915:
3908:
3902:
3896:
3889:
3885:
3877:
3873:
3866:
3860:
3857:
3850:
3846:
3839:
3835:
3828:
3822:
3816:
3807:
3804:
3622:
3616:
3610:
3603:
3597:
3593:
3590:
3583:
3581:In the case
3580:
3570:
3564:
3558:by a double
3553:
3550:Proof sketch
3542:
3536:
3526:
3522:
3512:
3505:
3498:
3489:
3479:
3475:
3472:
3289:
3282:
3275:
3271:
3268:
3262:
3258:
3250:
3246:
3239:
3232:
3227:
3222:
3216:
3209:
3205:
3198:
3192:
3186:
3132:
3114:
3111:
3104:
3100:
3087:
3060:
3048:coefficients
3041:
3037:
3030:
3023:
3013:
3002:
2998:
2991:
2984:
2981:
2694:
1988:
1985:
1623:
1620:
1439:
1436:
1367:
1364:
1358:
1352:
1349:
1337:
1333:
1326:
1322:
1319:
1080:
1076:
1069:
1062:
1054:
1050:
1043:
1039:
1032:
1023:
1018:
1012:
1006:
1000:
994:
988:
982:
979:
971:
967:
960:
953:
946:
942:
934:
930:
923:
918:
914:
911:
735:
732:
612:
222:
218:
210:
206:
199:
194:
190:
183:
179:
172:
166:
163:
152:
146:
142:
133:
127:
118:
109:
99:
84:
78:January 2017
75:
64:Please help
56:
5225:is clearly
5068:containing
3532:isomorphism
3530:defines an
3487:that sends
3096:determinant
3071:eigenvalues
102:mathematics
70:introducing
5581:Categories
5569:2024-03-26
5502:References
5273:of degree
5016:partitions
4711:, for all
3826:variables
3525:= 1, ...,
3261:= 1, ...,
3118:variables
3079:invariants
3046:and whose
2691:Properties
740:we define
221:= 1, ...,
188:, written
170:variables
160:Definition
4956:monomials
4640:−
4626:…
4593:−
4581:−
4574:σ
4567:…
4556:−
4543:σ
4533:~
4507:−
4493:…
4381:−
4374:σ
4367:…
4349:σ
4339:~
4311:…
4249:for some
4226:−
4214:−
4207:σ
4200:…
4189:−
4176:σ
4166:~
4146:−
4132:…
4110:~
3771:…
3749:⋅
3736:⋯
3704:…
3656:…
3560:induction
3435:…
3403:…
3381:…
3323:…
3120:generates
2948:…
2906:−
2897:⋯
2886:−
2879:λ
2859:…
2822:−
2815:λ
2795:…
2766:−
2757:λ
2737:−
2734:λ
2711:∏
1288:…
1261:λ
1252:⋯
1233:…
1206:λ
1197:⋅
1178:…
1151:λ
1123:…
1102:λ
1033:Given an
1024:multisets
877:⋯
852:≤
836:⋯
807:≤
800:∑
777:…
705:⋯
663:…
556:≤
538:≤
531:∑
504:…
430:≤
418:≤
411:∑
384:…
320:≤
314:≤
307:∑
280:…
5533:(1999).
5511:(1995).
5450:See also
5417: (
5388: (
5351: (
5323: (
5251:1, ...,
5207: (
5162: (
5102: (
4460:lacunary
3858:Because
3811:lacunary
3683:lacunary
3534:between
3187:For any
3017:) whose
1346:Examples
912:so that
5424:, ...,
5395:, ...,
5358:, ...,
5330:, ...,
5309:, then
5261:
5214:, ...,
5169:, ...,
5119:) (the
5109:, ...,
5041:, ...,
4938:, ...,
4892:, ...,
4735:, ...,
4444:, ...,
4425:, ...,
4081:, ...,
4058:, ...,
4038:, ...,
4004:, ...,
3976:, ...,
3955:, then
3938:, ...,
3914:, ...,
3872:, ...,
3834:, ...,
3511:, ...,
3281:, ...,
3245:, ...,
3204:, ...,
3073:of the
3036:, ...,
2997:, ...,
1075:, ...,
1049:, ...,
966:, ...,
929:, ...,
205:, ...,
178:, ...,
66:improve
5541:
5519:
4695:gives
3576:degree
3075:matrix
3050:are –
124:degree
108:, the
5291:with
5191:Proof
5150:Lemma
4716:<
4413:Then
3609:<
3596:<
3165:proof
3083:trace
3065:of a
3052:up to
3019:roots
945:>
938:) = 0
51:, or
5539:ISBN
5517:ISBN
4900:are
4863:and
4048:, 0)
3961:and
3900:and
3540:and
3520:for
3288:) ∈
3256:for
3124:ring
3122:the
3061:The
1986:For
1621:For
1437:For
1365:For
952:1 =
839:<
833:<
820:<
550:<
544:<
424:<
216:for
5239:···
5180:is
5086:of
5018:of
4875:.
4782:···
4743:− 1
4708:− 1
4276:− 1
4268:− 1
4089:− 1
4066:− 1
4046:− 1
4012:− 1
3987:.)
3984:− 1
3946:− 1
3880:− 1
3620:in
3586:= 1
3496:to
3220:by
3130:in
3126:of
1991:= 4
1626:= 3
1444::
1442:= 2
1370:= 1
1042:= (
1030:.)
998:in
940:if
738:≥ 0
131:in
100:In
5583::
5375:.
5344:ce
5316:ce
5314:−
5186:.
5146:.
4922:.
4858:−
4848:−
4814:=
4810:−
4769:−
4330::=
4252:Q̃
4072:P̃
3546:.
3478:∈
3266:.
3179:.
3058:.
3029:,
2990:,
1993::
1628::
1372::
1362:.
1342:.
145:≤
55:,
47:,
5572:.
5547:.
5525:.
5442:i
5438:X
5433:)
5430:n
5426:X
5422:1
5419:X
5414:λ
5410:e
5404:)
5401:n
5397:X
5393:1
5390:X
5385:λ
5381:e
5372:P
5367:)
5364:n
5360:X
5356:1
5353:X
5348:λ
5339:)
5336:n
5332:X
5328:1
5325:X
5320:λ
5312:P
5306:c
5300:d
5294:λ
5288:X
5282:P
5276:d
5270:P
5263:.
5259:X
5253:i
5245:i
5241:X
5237:2
5234:X
5231:1
5228:X
5223:)
5220:n
5216:X
5212:1
5209:X
5204:i
5200:e
5183:X
5178:)
5175:n
5171:X
5167:1
5164:X
5159:λ
5155:e
5142:i
5138:X
5128:λ
5122:t
5115:n
5111:X
5107:1
5104:X
5099:λ
5095:e
5089:d
5083:λ
5077:d
5071:d
5061:i
5055:i
5050:)
5047:n
5043:X
5039:1
5036:X
5034:(
5031:i
5027:e
5021:d
5011:d
5005:2
5002:X
4996:1
4993:X
4986:n
4982:X
4978:1
4975:X
4965:i
4961:X
4951:d
4944:n
4940:X
4936:1
4933:X
4919:A
4913:A
4907:A
4896:n
4894:e
4890:1
4887:e
4881:n
4872:P
4866:R
4860:R
4856:P
4850:n
4846:d
4840:d
4834:Q
4828:Q
4824:n
4822:,
4820:n
4816:σ
4812:R
4808:P
4801:n
4799:,
4797:n
4793:σ
4786:n
4784:X
4780:1
4777:X
4771:R
4767:P
4761:P
4755:R
4749:P
4741:n
4737:X
4733:1
4730:X
4724:R
4718:n
4714:j
4706:n
4704:,
4702:j
4698:σ
4691:n
4689:,
4687:j
4683:σ
4676:n
4672:X
4654:)
4651:0
4648:,
4643:1
4637:n
4633:X
4629:,
4623:,
4618:1
4614:X
4610:(
4607:P
4604:=
4601:)
4596:1
4590:n
4587:,
4584:1
4578:n
4570:,
4564:,
4559:1
4553:n
4550:,
4547:1
4539:(
4530:Q
4524:=
4521:)
4518:0
4515:,
4510:1
4504:n
4500:X
4496:,
4490:,
4485:1
4481:X
4477:(
4474:R
4457:P
4450:n
4446:X
4442:1
4439:X
4434:)
4431:n
4427:X
4423:1
4420:X
4418:(
4416:R
4398:.
4395:)
4390:n
4387:,
4384:1
4378:n
4370:,
4364:,
4359:n
4356:,
4353:1
4345:(
4336:Q
4327:)
4322:n
4318:X
4314:,
4308:,
4303:1
4299:X
4295:(
4292:R
4274:n
4266:n
4264:,
4262:j
4258:σ
4234:)
4229:1
4223:n
4220:,
4217:1
4211:n
4203:,
4197:,
4192:1
4186:n
4183:,
4180:1
4172:(
4163:Q
4157:=
4154:)
4149:1
4143:n
4139:X
4135:,
4129:,
4124:1
4120:X
4116:(
4107:P
4091:)
4087:n
4083:X
4079:1
4076:X
4074:(
4064:n
4060:X
4056:1
4053:X
4044:n
4040:X
4036:1
4033:X
4031:(
4029:P
4022:n
4018:X
4010:n
4006:X
4002:1
3999:X
3993:P
3982:n
3978:X
3974:1
3971:X
3964:B
3958:A
3952:B
3944:n
3940:X
3936:1
3933:X
3927:A
3920:n
3916:X
3912:1
3909:X
3903:B
3897:A
3890:n
3886:X
3878:n
3874:X
3870:1
3867:X
3861:P
3851:j
3847:X
3840:n
3836:X
3832:1
3829:X
3823:n
3817:P
3808:P
3790:.
3787:)
3782:n
3778:X
3774:,
3768:,
3763:1
3759:X
3755:(
3752:Q
3744:n
3740:X
3731:1
3727:X
3723:+
3720:)
3715:n
3711:X
3707:,
3701:,
3696:1
3692:X
3688:(
3679:P
3675:=
3672:)
3667:n
3663:X
3659:,
3653:,
3648:1
3644:X
3640:(
3637:P
3623:A
3617:P
3611:d
3604:n
3598:n
3594:m
3584:n
3571:n
3565:n
3543:A
3537:A
3527:n
3523:k
3518:)
3515:n
3513:X
3509:1
3506:X
3504:(
3501:k
3499:e
3492:k
3490:Y
3480:A
3476:Q
3456:)
3451:)
3446:n
3442:X
3438:,
3432:,
3427:1
3423:X
3419:(
3414:n
3410:e
3406:,
3400:,
3397:)
3392:n
3388:X
3384:,
3378:,
3373:1
3369:X
3365:(
3360:1
3356:e
3350:(
3345:Q
3342:=
3339:)
3334:n
3330:X
3326:,
3320:,
3315:1
3311:X
3307:(
3304:P
3290:A
3285:n
3283:X
3279:1
3276:X
3274:(
3272:P
3263:n
3259:k
3254:)
3251:n
3247:X
3243:1
3240:X
3238:(
3235:k
3233:e
3228:n
3223:A
3217:A
3210:n
3206:X
3202:1
3199:X
3193:A
3149:Z
3133:n
3115:n
3105:n
3101:e
3091:1
3088:e
3042:n
3038:X
3034:2
3031:X
3027:1
3024:X
3014:λ
3003:n
2999:X
2995:2
2992:X
2988:1
2985:X
2967:.
2964:)
2959:n
2955:X
2951:,
2945:,
2940:1
2936:X
2932:(
2927:n
2923:e
2917:n
2913:)
2909:1
2903:(
2900:+
2894:+
2889:2
2883:n
2875:)
2870:n
2866:X
2862:,
2856:,
2851:1
2847:X
2843:(
2838:2
2834:e
2830:+
2825:1
2819:n
2811:)
2806:n
2802:X
2798:,
2792:,
2787:1
2783:X
2779:(
2774:1
2770:e
2761:n
2753:=
2750:)
2745:j
2741:X
2731:(
2726:n
2721:1
2718:=
2715:j
2670:.
2665:4
2661:X
2655:3
2651:X
2645:2
2641:X
2635:1
2631:X
2627:=
2620:)
2615:4
2611:X
2607:,
2602:3
2598:X
2594:,
2589:2
2585:X
2581:,
2576:1
2572:X
2568:(
2563:4
2559:e
2551:,
2546:4
2542:X
2536:3
2532:X
2526:2
2522:X
2518:+
2513:4
2509:X
2503:3
2499:X
2493:1
2489:X
2485:+
2480:4
2476:X
2470:2
2466:X
2460:1
2456:X
2452:+
2447:3
2443:X
2437:2
2433:X
2427:1
2423:X
2419:=
2412:)
2407:4
2403:X
2399:,
2394:3
2390:X
2386:,
2381:2
2377:X
2373:,
2368:1
2364:X
2360:(
2355:3
2351:e
2343:,
2338:4
2334:X
2328:3
2324:X
2320:+
2315:4
2311:X
2305:2
2301:X
2297:+
2292:3
2288:X
2282:2
2278:X
2274:+
2269:4
2265:X
2259:1
2255:X
2251:+
2246:3
2242:X
2236:1
2232:X
2228:+
2223:2
2219:X
2213:1
2209:X
2205:=
2198:)
2193:4
2189:X
2185:,
2180:3
2176:X
2172:,
2167:2
2163:X
2159:,
2154:1
2150:X
2146:(
2141:2
2137:e
2129:,
2124:4
2120:X
2116:+
2111:3
2107:X
2103:+
2098:2
2094:X
2090:+
2085:1
2081:X
2077:=
2070:)
2065:4
2061:X
2057:,
2052:3
2048:X
2044:,
2039:2
2035:X
2031:,
2026:1
2022:X
2018:(
2013:1
2009:e
1989:n
1966:.
1961:3
1957:X
1951:2
1947:X
1941:1
1937:X
1933:=
1926:)
1921:3
1917:X
1913:,
1908:2
1904:X
1900:,
1895:1
1891:X
1887:(
1882:3
1878:e
1870:,
1865:3
1861:X
1855:2
1851:X
1847:+
1842:3
1838:X
1832:1
1828:X
1824:+
1819:2
1815:X
1809:1
1805:X
1801:=
1794:)
1789:3
1785:X
1781:,
1776:2
1772:X
1768:,
1763:1
1759:X
1755:(
1750:2
1746:e
1738:,
1733:3
1729:X
1725:+
1720:2
1716:X
1712:+
1707:1
1703:X
1699:=
1692:)
1687:3
1683:X
1679:,
1674:2
1670:X
1666:,
1661:1
1657:X
1653:(
1648:1
1644:e
1624:n
1601:.
1596:2
1592:X
1586:1
1582:X
1578:=
1571:)
1566:2
1562:X
1558:,
1553:1
1549:X
1545:(
1540:2
1536:e
1528:,
1523:2
1519:X
1515:+
1510:1
1506:X
1502:=
1495:)
1490:2
1486:X
1482:,
1477:1
1473:X
1469:(
1464:1
1460:e
1440:n
1422:.
1417:1
1413:X
1409:=
1406:)
1401:1
1397:X
1393:(
1388:1
1384:e
1368:n
1359:n
1353:n
1338:k
1334:e
1327:k
1323:σ
1316:.
1304:)
1299:n
1295:X
1291:,
1285:,
1280:1
1276:X
1272:(
1265:m
1256:e
1249:)
1244:n
1240:X
1236:,
1230:,
1225:1
1221:X
1217:(
1210:2
1201:e
1194:)
1189:n
1185:X
1181:,
1175:,
1170:1
1166:X
1162:(
1155:1
1146:e
1142:=
1139:)
1134:n
1130:X
1126:,
1120:,
1115:1
1111:X
1107:(
1098:e
1084:)
1081:n
1077:X
1073:1
1070:X
1068:(
1065:λ
1063:e
1058:)
1055:m
1051:λ
1047:1
1044:λ
1040:λ
1019:n
1013:k
1007:k
1001:n
995:k
989:n
983:k
975:)
972:n
968:X
964:1
961:X
959:(
957:0
954:e
947:n
943:k
935:n
931:X
927:1
924:X
922:(
919:k
915:e
897:,
890:k
886:j
881:X
870:1
866:j
861:X
855:n
847:k
843:j
828:2
824:j
815:1
811:j
804:1
796:=
793:)
788:n
784:X
780:,
774:,
769:1
765:X
761:(
756:k
752:e
736:k
718:.
713:n
709:X
700:2
696:X
690:1
686:X
682:=
679:)
674:n
670:X
666:,
660:,
655:2
651:X
647:,
642:1
638:X
634:(
629:n
625:e
594:,
589:l
585:X
579:k
575:X
569:j
565:X
559:n
553:l
547:k
541:j
535:1
527:=
520:)
515:n
511:X
507:,
501:,
496:2
492:X
488:,
483:1
479:X
475:(
470:3
466:e
458:,
453:k
449:X
443:j
439:X
433:n
427:k
421:j
415:1
407:=
400:)
395:n
391:X
387:,
381:,
376:2
372:X
368:,
363:1
359:X
355:(
350:2
346:e
338:,
333:j
329:X
323:n
317:j
311:1
303:=
296:)
291:n
287:X
283:,
277:,
272:2
268:X
264:,
259:1
255:X
251:(
246:1
242:e
223:n
219:k
214:)
211:n
207:X
203:1
200:X
198:(
195:k
191:e
184:n
180:X
176:1
173:X
167:n
153:d
147:n
143:d
134:n
128:d
119:P
91:)
85:(
80:)
76:(
62:.
20:)
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