Symmetric polynomial

2438: 1765: 2433:{\displaystyle {\begin{aligned}a_{n-1}&=-x_{1}-x_{2}-\cdots -x_{n}\\a_{n-2}&=x_{1}x_{2}+x_{1}x_{3}+\cdots +x_{2}x_{3}+\cdots +x_{n-1}x_{n}=\textstyle \sum _{1\leq i<j\leq n}x_{i}x_{j}\\&{}\ \,\vdots \\a_{1}&=(-1)^{n-1}(x_{2}x_{3}\cdots x_{n}+x_{1}x_{3}x_{4}\cdots x_{n}+\cdots +x_{1}x_{2}\cdots x_{n-2}x_{n}+x_{1}x_{2}\cdots x_{n-1})=\textstyle (-1)^{n-1}\sum _{i=1}^{n}\prod _{j\neq i}x_{j}\\a_{0}&=(-1)^{n}x_{1}x_{2}\cdots x_{n}.\end{aligned}}} 6362: 7045:) are both equal to 1, one can isolate either the first or the last term of these summations; the former gives a set of equations that allows one to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the 5286: 5802: 3766: 4858: 5746: 6357:{\displaystyle {\begin{aligned}h_{3}(X_{1},X_{2},X_{3})&=m_{(3)}(X_{1},X_{2},X_{3})+m_{(2,1)}(X_{1},X_{2},X_{3})+m_{(1,1,1)}(X_{1},X_{2},X_{3})\\&=(X_{1}^{3}+X_{2}^{3}+X_{3}^{3})+(X_{1}^{2}X_{2}+X_{1}^{2}X_{3}+X_{1}X_{2}^{2}+X_{1}X_{3}^{2}+X_{2}^{2}X_{3}+X_{2}X_{3}^{2})+(X_{1}X_{2}X_{3}).\\\end{aligned}}} 1267: 2736:

elementary symmetric polynomials are building blocks for all symmetric polynomials in these variables: as mentioned above, any symmetric polynomial in the variables considered can be obtained from these elementary symmetric polynomials using multiplications and additions only. In fact one has the

2569:

in which those roots may lie. In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are

4847: 4476: 3423: 5281:{\displaystyle {\begin{aligned}m_{(2,1)}(X_{1},X_{2},X_{3})&=X_{1}^{2}X_{2}+X_{1}X_{2}^{2}+X_{1}^{2}X_{3}+X_{1}X_{3}^{2}+X_{2}^{2}X_{3}+X_{2}X_{3}^{2}\\&=p_{1}(X_{1},X_{2},X_{3})p_{2}(X_{1},X_{2},X_{3})-p_{3}(X_{1},X_{2},X_{3}).\end{aligned}}} 3416: 1754: 5424: 6792: 3085: 6985: 989: 2555:

variables can be given by a polynomial expression in terms of these elementary symmetric polynomials. It follows that any symmetric polynomial expression in the roots of a monic polynomial can be expressed as a polynomial in the

1000: 450: 4142: 2564:

that contains those coefficients. Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than

3952: 1494: 4618: 602: 200:

Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as

4629: 4270: 689: 5807: 4863: 1770: 3761:{\displaystyle m_{(3,2,1)}(X_{1},X_{2},X_{3})=X_{1}^{3}X_{2}^{2}X_{3}+X_{1}^{3}X_{2}X_{3}^{2}+X_{1}^{2}X_{2}^{3}X_{3}+X_{1}^{2}X_{2}X_{3}^{3}+X_{1}X_{2}^{3}X_{3}^{2}+X_{1}X_{2}^{2}X_{3}^{3}.} 1314:

roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the

304: 5741:{\displaystyle h_{3}(X_{1},X_{2},X_{3})=X_{1}^{3}+X_{1}^{2}X_{2}+X_{1}^{2}X_{3}+X_{1}X_{2}^{2}+X_{1}X_{2}X_{3}+X_{1}X_{3}^{2}+X_{2}^{3}+X_{2}^{2}X_{3}+X_{2}X_{3}^{2}+X_{3}^{3}.} 3220: 1558: 7078:: one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones. 6596: 2895: 843: 746: 2519: 6831: 803: 776: 6825:

An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be expressed as the identities

607:

There are many ways to make specific symmetric polynomials in any number of variables (see the various types below). An example of a somewhat different flavor is

6822:), allowing them to be expressed in terms of the ones up to that point; again the resulting identities become invalid when the number of variables is increased. 994:

has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial. However, the following is symmetric:

1262:{\displaystyle X_{1}^{4}X_{2}^{2}X_{3}+X_{1}X_{2}^{4}X_{3}^{2}+X_{1}^{2}X_{2}X_{3}^{4}+X_{1}^{4}X_{2}X_{3}^{2}+X_{1}X_{2}^{2}X_{3}^{4}+X_{1}^{2}X_{2}^{4}X_{3}} 7092:

Another class of symmetric polynomials is that of the Schur polynomials, which are of fundamental importance in the applications of symmetric polynomials to

3095: 2543: 1315: 851: 183: 178:, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the 310: 1336:

This yields the approach to solving polynomial equations by inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the

4022: 5357: 202: 5336:, which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a 3847: 7214: 1373: 4842:{\displaystyle m_{(2,1)}(X_{1},X_{2})=\textstyle {\frac {1}{2}}p_{1}(X_{1},X_{2})^{3}-{\frac {1}{2}}p_{2}(X_{1},X_{2})p_{1}(X_{1},X_{2}).} 4499: 482: 4489:

coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials. For an example, for

4471:{\displaystyle p_{3}(X_{1},X_{2})=\textstyle {\frac {3}{2}}p_{2}(X_{1},X_{2})p_{1}(X_{1},X_{2})-{\frac {1}{2}}p_{1}(X_{1},X_{2})^{3}.} 3110:

building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of

5309: = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first 7275: 7257: 7096:. They are however not as easy to describe as the other kinds of special symmetric polynomials; see the main article for details. 613: 6797:

As in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond

2614: 2522: 1337: 179: 3980: 206: 186:

states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials. This implies that every

7206: 2466:

Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing

6367:

All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in

7295: 2479:

then become just the particular symmetric polynomials given by the above equations. Those polynomials, without the sign

3822:

of the monomial symmetric polynomials. To do this it suffices to separate the different types of monomial occurring in

3106:

Powers and products of elementary symmetric polynomials work out to rather complicated expressions. If one seeks basic

698:

renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its

7121: 1326:

roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots

214: 31: 7180: 3807: 4481:

In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in

5341: 3833:

The elementary symmetric polynomials are particular cases of monomial symmetric polynomials: for 0 ≤

7185: 4175:

can be expressed as a polynomial expression with rational coefficients in the power sum symmetric polynomials

7133: 2470:, and considering them as indeterminates rather than as constants in an appropriate field; the coefficients 694:

where first a polynomial is constructed that changes sign under every exchange of variables, and taking the

253: 7175: 7154: 7139: 5396: 3811: 3411:{\displaystyle m_{(3,1,1)}(X_{1},X_{2},X_{3})=X_{1}^{3}X_{2}X_{3}+X_{1}X_{2}^{3}X_{3}+X_{1}X_{2}X_{3}^{3}} 2574:, which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials. 2571: 1749:{\displaystyle P=t^{n}+a_{n-1}t^{n-1}+\cdots +a_{2}t^{2}+a_{1}t+a_{0}=(t-x_{1})(t-x_{2})\cdots (t-x_{n}).} 1296: 84:

variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally,

7147: 7109: 7093: 2762: 2448: 1340:

in the roots), how can one recover the roots? This leads to studying solutions of polynomials using the

222: 190: 7161:

of the ring of symmetric polynomials: the Vandermonde polynomial is a square root of the discriminant.

6787:{\displaystyle X_{1}^{3}+X_{2}^{3}-7=-2h_{1}(X_{1},X_{2})^{3}+3h_{1}(X_{1},X_{2})h_{2}(X_{1},X_{2})-7.} 7290: 171: 3080:{\displaystyle X_{1}^{3}+X_{2}^{3}-7=e_{1}(X_{1},X_{2})^{3}-3e_{2}(X_{1},X_{2})e_{1}(X_{1},X_{2})-7} 213:

play important roles alongside the elementary ones. The resulting structures, and in particular the

7158: 5337: 2447:. They show that all coefficients of the polynomial are given in terms of the roots by a symmetric 2444: 1345: 1307: 30:

This article is about individual symmetric polynomials. For the ring of symmetric polynomials, see

7263: 7170: 5345: 3819: 3091: 808: 711: 2482: 6980:{\displaystyle \sum _{i=0}^{k}(-1)^{i}e_{i}(X_{1},\ldots ,X_{n})h_{k-i}(X_{1},\ldots ,X_{n})=0} 7271: 7253: 7210: 5329: 1341: 197:

can alternatively be given as a polynomial expression in the coefficients of the polynomial.

7245: 7228: 7087: 1289: 695: 210: 194: 7224: 781: 754: 7232: 7220: 4152: 2455:

there may be qualitative differences between the roots (like lying in the base field

6495:

has integral coefficients, then the polynomial expression also has integral coefficients.

7105: 3153: 2463:

or multiple roots), none of this affects the way the roots occur in these expressions.

1545: 1327: 984:{\displaystyle X_{1}^{4}X_{2}^{2}X_{3}+X_{1}X_{2}^{4}X_{3}^{2}+X_{1}^{2}X_{2}X_{3}^{4}} 7284: 7117: 7113: 5291:

The corresponding expression was valid for two variables as well (it suffices to set

4151:

power sum symmetric polynomials by additions and multiplications, possibly involving

2666: 1349: 1283: 218: 17: 699: 445:{\displaystyle 4X_{1}^{2}X_{2}^{2}+X_{1}^{3}X_{2}+X_{1}X_{2}^{3}+(X_{1}+X_{2})^{4}} 5321:

needed to express elementary symmetric polynomials (except the constant ones, and

5328:

which coincides with the first power sum) in terms of power sum polynomials. The

4137:{\displaystyle p_{k}(X_{1},\ldots ,X_{n})=X_{1}^{k}+X_{2}^{k}+\cdots +X_{n}^{k}.} 7124:, which avoids having to carry around a fixed number of variables all the time. 2460: 1541: 175: 95: 38: 4016:) is of special interest. It is the power sum symmetric polynomial, defined as 170:

Symmetric polynomials arise naturally in the study of the relation between the

7198: 5332:

provide an explicit method to do this; it involves division by integers up to

1292: 46: 1288:

One context in which symmetric polynomial functions occur is in the study of

182:

are the most fundamental symmetric polynomials. Indeed, a theorem called the

6590:. The first polynomial in the list of examples above can then be written as 2889:. The first polynomial in the list of examples above can then be written as 2812:

coefficients, then the polynomial expression also has integral coefficients.

5776:) is also the sum of all distinct monomial symmetric polynomials of degree 3947:{\displaystyle e_{k}(X_{1},\ldots ,X_{n})=m_{\alpha }(X_{1},\ldots ,X_{n})} 2560:

of the polynomial, and in particular that its value lies in the base field

5392: 3111: 2809: 2620: 3114:, with only those copies required to obtain symmetry. Any monomial in 2801:

this expression is unique up to equivalence of polynomial expressions;

1489:{\displaystyle P=t^{n}+a_{n-1}t^{n-1}+\cdots +a_{2}t^{2}+a_{1}t+a_{0}} 7209:, vol. 211 (Revised third ed.), New York: Springer-Verlag, 6383:

can be obtained from the complete homogeneous symmetric polynomials

4613:{\displaystyle m_{(2,1)}(X_{1},X_{2})=X_{1}^{2}X_{2}+X_{1}X_{2}^{2}} 5313:

power sum polynomials involves rational coefficients may depend on

597:{\displaystyle X_{1}X_{2}X_{3}-2X_{1}X_{2}-2X_{1}X_{3}-2X_{2}X_{3}} 3785:

when β is a permutation of α, so one usually considers only those

6503:= 2, the relevant complete homogeneous symmetric polynomials are 2582:

There are a few types of symmetric polynomials in the variables

3830:

has integer coefficients, then so will the linear combination.

1548:); some of the roots might be equal, but the fact that one has 6455:

can be written as a polynomial expression in the polynomials

7242:. Oxford Mathematical Monographs. Oxford: Clarendon Press. 684:{\displaystyle \prod _{1\leq i<j\leq n}(X_{i}-X_{j})^{2}} 3810:. These monomial symmetric polynomials form a vector space 7146:

under permutation of the entries, change according to the

4147:

All symmetric polynomials can be obtained from the first

1356:

Relation with the roots of a monic univariate polynomial

5305:, it could not be used to illustrate the statement for 2820:= 2, the relevant elementary symmetric polynomials are 4852:

Using three variables one gets a different expression

4687: 4316: 2276: 1978: 6834: 6599: 6431:) via multiplications and additions. More precisely: 5805: 5427: 4861: 4632: 4502: 4273: 4025: 3850: 3426: 3223: 2898: 2485: 1768: 1561: 1376: 1003: 854: 811: 784: 757: 714: 616: 485: 313: 256: 6991: > 0, and any number of variables 7270:, Vol. 2. Cambridge: Cambridge University Press. 4227:In particular, the remaining power sum polynomials 705:On the other hand, the polynomial in two variables 6979: 6786: 6356: 5740: 5280: 4841: 4612: 4470: 4136: 3989: ≥ 1, the monomial symmetric polynomial 3946: 3760: 3410: 3079: 2513: 2432: 1748: 1488: 1261: 983: 837: 797: 770: 740: 683: 596: 455:as is the following polynomial in three variables 444: 298: 2655:distinct variables. (Some authors denote it by σ 5366:, the complete homogeneous symmetric polynomial 5344:; however, it is valid with coefficients in any 27:Polynomial invariant under variable permutations 1536:in some possibly larger field (for instance if 2732:) = 0 in these cases. The remaining 8: 7120:, where they are mostly studied through the 3096:fundamental theorem of symmetric polynomials 2544:fundamental theorem of symmetric polynomials 1316:fundamental theorem of symmetric polynomials 184:fundamental theorem of symmetric polynomials 233:The following polynomials in two variables 5352:Complete homogeneous symmetric polynomials 6962: 6943: 6924: 6911: 6892: 6879: 6869: 6850: 6839: 6833: 6769: 6756: 6743: 6730: 6717: 6704: 6688: 6678: 6665: 6652: 6627: 6622: 6609: 6604: 6598: 6338: 6328: 6318: 6299: 6294: 6284: 6271: 6261: 6256: 6243: 6238: 6228: 6215: 6210: 6200: 6187: 6177: 6172: 6159: 6149: 6144: 6125: 6120: 6107: 6102: 6089: 6084: 6058: 6045: 6032: 6001: 5985: 5972: 5959: 5934: 5918: 5905: 5892: 5873: 5853: 5840: 5827: 5814: 5806: 5804: 5729: 5724: 5711: 5706: 5696: 5683: 5673: 5668: 5655: 5650: 5637: 5632: 5622: 5609: 5599: 5589: 5576: 5571: 5561: 5548: 5538: 5533: 5520: 5510: 5505: 5492: 5487: 5471: 5458: 5445: 5432: 5426: 5358:Complete homogeneous symmetric polynomial 5262: 5249: 5236: 5223: 5207: 5194: 5181: 5168: 5155: 5142: 5129: 5116: 5096: 5091: 5081: 5068: 5058: 5053: 5040: 5035: 5025: 5012: 5002: 4997: 4984: 4979: 4969: 4956: 4946: 4941: 4921: 4908: 4895: 4870: 4862: 4860: 4826: 4813: 4800: 4787: 4774: 4761: 4747: 4738: 4728: 4715: 4702: 4688: 4675: 4662: 4637: 4631: 4604: 4599: 4589: 4576: 4566: 4561: 4545: 4532: 4507: 4501: 4493: = 2, the symmetric polynomial 4458: 4448: 4435: 4422: 4408: 4396: 4383: 4370: 4357: 4344: 4331: 4317: 4304: 4291: 4278: 4272: 4125: 4120: 4101: 4096: 4083: 4078: 4062: 4043: 4030: 4024: 3935: 3916: 3903: 3887: 3868: 3855: 3849: 3749: 3744: 3734: 3729: 3719: 3706: 3701: 3691: 3686: 3676: 3663: 3658: 3648: 3638: 3633: 3620: 3610: 3605: 3595: 3590: 3577: 3572: 3562: 3552: 3547: 3534: 3524: 3519: 3509: 3504: 3488: 3475: 3462: 3431: 3425: 3402: 3397: 3387: 3377: 3364: 3354: 3349: 3339: 3326: 3316: 3306: 3301: 3285: 3272: 3259: 3228: 3222: 3196:) is defined as the sum of all monomials 3156:(possibly zero); writing α = (α 3062: 3049: 3036: 3023: 3010: 2997: 2981: 2971: 2958: 2945: 2926: 2921: 2908: 2903: 2897: 2651:) is the sum of all distinct products of 2499: 2484: 2417: 2404: 2394: 2384: 2358: 2343: 2327: 2317: 2306: 2290: 2258: 2245: 2235: 2222: 2206: 2193: 2183: 2164: 2151: 2141: 2131: 2118: 2105: 2095: 2076: 2050: 2038: 2033: 2021: 2011: 1983: 1969: 1953: 1934: 1924: 1905: 1895: 1882: 1872: 1849: 1835: 1816: 1803: 1777: 1769: 1767: 1759:By comparing coefficients one finds that 1734: 1709: 1687: 1665: 1649: 1636: 1626: 1601: 1585: 1572: 1560: 1480: 1464: 1451: 1441: 1416: 1400: 1387: 1375: 1253: 1243: 1238: 1228: 1223: 1210: 1205: 1195: 1190: 1180: 1167: 1162: 1152: 1142: 1137: 1124: 1119: 1109: 1099: 1094: 1081: 1076: 1066: 1061: 1051: 1038: 1028: 1023: 1013: 1008: 1002: 975: 970: 960: 950: 945: 932: 927: 917: 912: 902: 889: 879: 874: 864: 859: 853: 829: 816: 810: 789: 783: 762: 756: 751:is not symmetric, since if one exchanges 732: 719: 713: 675: 665: 652: 621: 615: 588: 578: 562: 552: 536: 526: 510: 500: 490: 484: 436: 426: 413: 397: 392: 382: 369: 359: 354: 341: 336: 326: 321: 312: 284: 279: 266: 261: 255: 7252:, second ed. Oxford: Clarendon Press. 7250:Symmetric Functions and Hall Polynomials 7240:Symmetric Functions and Hall Polynomials 1344:of the roots, originally in the form of 7157:and a symmetric polynomial, and form a 7138:Analogous to symmetric polynomials are 7104:Symmetric polynomials are important to 2700:, no products at all can be formed, so 1544:, the roots will exist in the field of 7142:: polynomials that, rather than being 2626:, the elementary symmetric polynomial 2578:Special kinds of symmetric polynomials 5796:, for instance for the given example 3094:that this is always possible see the 299:{\displaystyle X_{1}^{3}+X_{2}^{3}-7} 7: 2443:These are in fact just instances of 1333:is given by a symmetric polynomial. 4264:power sum polynomials; for example 1552:roots is expressed by the relation 1318:implies that a polynomial function 3806:, in other words for which α is a 2451:: although for a given polynomial 25: 5348:containing the rational numbers. 4260:can be so expressed in the first 3954:where α is the partition of 2665: = 0 there is only the 805:one gets a different polynomial, 7100:Symmetric polynomials in algebra 5317:. But rational coefficients are 5298:to zero), but since it involves 2609:Elementary symmetric polynomials 2523:elementary symmetric polynomials 1338:elementary symmetric polynomials 180:elementary symmetric polynomials 4155:coefficients. More precisely, 3975:Power-sum symmetric polynomials 2737:following more detailed facts: 2615:Elementary symmetric polynomial 1360:Consider a monic polynomial in 845:. Similarly in three variables 7153:These are all products of the 6968: 6936: 6917: 6885: 6866: 6856: 6775: 6749: 6736: 6710: 6685: 6658: 6344: 6311: 6305: 6137: 6131: 6077: 6064: 6025: 6020: 6002: 5991: 5952: 5947: 5935: 5924: 5885: 5880: 5874: 5859: 5820: 5477: 5438: 5268: 5229: 5213: 5174: 5161: 5122: 4927: 4888: 4883: 4871: 4832: 4806: 4793: 4767: 4735: 4708: 4681: 4655: 4650: 4638: 4551: 4525: 4520: 4508: 4455: 4428: 4402: 4376: 4363: 4337: 4310: 4284: 4068: 4036: 3981:Power sum symmetric polynomial 3941: 3909: 3893: 3861: 3800: ≥ ... ≥ α 3494: 3455: 3450: 3432: 3291: 3252: 3247: 3229: 3102:Monomial symmetric polynomials 3068: 3042: 3029: 3003: 2978: 2951: 2496: 2486: 2381: 2371: 2287: 2277: 2270: 2088: 2073: 2063: 1740: 1721: 1715: 1696: 1693: 1674: 672: 645: 433: 406: 1: 7207:Graduate Texts in Mathematics 7116:. They are also important in 5391:) is the sum of all distinct 5362:For each nonnegative integer 3814:: every symmetric polynomial 3172:monomial symmetric polynomial 3166:) this can be abbreviated to 2541:. A basic fact, known as the 217:, are of great importance in 4159:Any symmetric polynomial in 7122:ring of symmetric functions 6480:) with 1 ≤ 2692:) = 1, while for 838:{\displaystyle X_{2}-X_{1}} 741:{\displaystyle X_{1}-X_{2}} 215:ring of symmetric functions 32:ring of symmetric functions 7312: 7181:Stanley symmetric function 7131: 7085: 5355: 3978: 3962:parts 1 (followed by 2612: 2514:{\displaystyle (-1)^{n-i}} 1281: 29: 7268:Enumerative Combinatorics 7238:Macdonald, I.G. (1979), 6435:Any symmetric polynomial 2741:any symmetric polynomial 3214:). For instance one has 3200:where β ranges over all 2551:symmetric polynomial in 174:in one variable and its 7148:sign of the permutation 7140:alternating polynomials 7134:Alternating polynomials 7128:Alternating polynomials 3808:partition of an integer 7155:Vandermonde polynomial 6981: 6855: 6788: 6358: 5742: 5282: 4843: 4614: 4472: 4138: 3948: 3762: 3412: 3081: 2605:that are fundamental. 2515: 2434: 2322: 1750: 1490: 1263: 985: 839: 799: 772: 742: 685: 598: 446: 300: 7186:Muirhead's inequality 7110:representation theory 7094:representation theory 7070:) with 1 ≤ 6982: 6835: 6789: 6359: 5743: 5283: 4844: 4615: 4473: 4139: 3949: 3763: 3413: 3146:where the exponents α 3082: 2790:) with 1 ≤ 2763:polynomial expression 2619:For each nonnegative 2516: 2449:polynomial expression 2435: 2302: 1751: 1491: 1348:, later developed in 1264: 986: 840: 800: 798:{\displaystyle X_{2}} 773: 771:{\displaystyle X_{1}} 743: 686: 599: 447: 301: 223:representation theory 191:polynomial expression 172:roots of a polynomial 18:Symmetric polynomials 6832: 6597: 5803: 5425: 4859: 4630: 4500: 4271: 4023: 3848: 3818:can be written as a 3424: 3221: 2896: 2761:can be written as a 2483: 1766: 1559: 1374: 1001: 852: 809: 782: 755: 712: 614: 483: 311: 254: 203:complete homogeneous 92:symmetric polynomial 43:symmetric polynomial 7296:Symmetric functions 7176:Newton's identities 7159:quadratic extension 6632: 6614: 6484: ≤ 6304: 6266: 6248: 6220: 6182: 6154: 6130: 6112: 6094: 5734: 5716: 5678: 5660: 5642: 5581: 5543: 5515: 5497: 5101: 5063: 5045: 5007: 4989: 4951: 4623:has the expression 4609: 4571: 4130: 4106: 4088: 3826:. In particular if 3754: 3739: 3711: 3696: 3668: 3643: 3615: 3600: 3582: 3557: 3529: 3514: 3407: 3359: 3311: 2931: 2913: 2765:in the polynomials 2572:Newton's identities 2521:, are known as the 1508:in some field 1346:Lagrange resolvents 1248: 1233: 1215: 1200: 1172: 1147: 1129: 1104: 1086: 1071: 1033: 1018: 980: 955: 937: 922: 884: 869: 402: 364: 346: 331: 289: 271: 7264:Richard P. Stanley 7260:(paperback, 1998). 7171:Symmetric function 6977: 6784: 6618: 6600: 6354: 6352: 6290: 6252: 6234: 6206: 6168: 6140: 6116: 6098: 6080: 5738: 5720: 5702: 5664: 5646: 5628: 5567: 5529: 5501: 5483: 5278: 5276: 5087: 5049: 5031: 4993: 4975: 4937: 4839: 4838: 4610: 4595: 4557: 4468: 4467: 4134: 4116: 4092: 4074: 3944: 3820:linear combination 3758: 3740: 3725: 3697: 3682: 3654: 3629: 3601: 3586: 3568: 3543: 3515: 3500: 3408: 3393: 3345: 3297: 3204:permutations of (α 3130:can be written as 3077: 2917: 2899: 2511: 2430: 2428: 2349: 2338: 2027: 2006: 1746: 1499:with coefficients 1486: 1259: 1234: 1219: 1201: 1186: 1158: 1133: 1115: 1090: 1072: 1057: 1019: 1004: 981: 966: 941: 923: 908: 870: 855: 835: 795: 768: 738: 681: 644: 594: 442: 388: 350: 332: 317: 296: 275: 257: 193:in the roots of a 101:of the subscripts 7216:978-0-387-95385-4 7082:Schur polynomials 6499:For example, for 5402:in the variables 5330:Newton identities 4755: 4696: 4416: 4325: 3985:For each integer 2816:For example, for 2323: 2037: 1979: 1342:permutation group 1306:roots in a given 617: 211:Schur polynomials 16:(Redirected from 7303: 7235: 7088:Schur polynomial 6986: 6984: 6983: 6978: 6967: 6966: 6948: 6947: 6935: 6934: 6916: 6915: 6897: 6896: 6884: 6883: 6874: 6873: 6854: 6849: 6793: 6791: 6790: 6785: 6774: 6773: 6761: 6760: 6748: 6747: 6735: 6734: 6722: 6721: 6709: 6708: 6693: 6692: 6683: 6682: 6670: 6669: 6657: 6656: 6631: 6626: 6613: 6608: 6589: 6539: 6363: 6361: 6360: 6355: 6353: 6343: 6342: 6333: 6332: 6323: 6322: 6303: 6298: 6289: 6288: 6276: 6275: 6265: 6260: 6247: 6242: 6233: 6232: 6219: 6214: 6205: 6204: 6192: 6191: 6181: 6176: 6164: 6163: 6153: 6148: 6129: 6124: 6111: 6106: 6093: 6088: 6070: 6063: 6062: 6050: 6049: 6037: 6036: 6024: 6023: 5990: 5989: 5977: 5976: 5964: 5963: 5951: 5950: 5923: 5922: 5910: 5909: 5897: 5896: 5884: 5883: 5858: 5857: 5845: 5844: 5832: 5831: 5819: 5818: 5747: 5745: 5744: 5739: 5733: 5728: 5715: 5710: 5701: 5700: 5688: 5687: 5677: 5672: 5659: 5654: 5641: 5636: 5627: 5626: 5614: 5613: 5604: 5603: 5594: 5593: 5580: 5575: 5566: 5565: 5553: 5552: 5542: 5537: 5525: 5524: 5514: 5509: 5496: 5491: 5476: 5475: 5463: 5462: 5450: 5449: 5437: 5436: 5287: 5285: 5284: 5279: 5277: 5267: 5266: 5254: 5253: 5241: 5240: 5228: 5227: 5212: 5211: 5199: 5198: 5186: 5185: 5173: 5172: 5160: 5159: 5147: 5146: 5134: 5133: 5121: 5120: 5105: 5100: 5095: 5086: 5085: 5073: 5072: 5062: 5057: 5044: 5039: 5030: 5029: 5017: 5016: 5006: 5001: 4988: 4983: 4974: 4973: 4961: 4960: 4950: 4945: 4926: 4925: 4913: 4912: 4900: 4899: 4887: 4886: 4848: 4846: 4845: 4840: 4831: 4830: 4818: 4817: 4805: 4804: 4792: 4791: 4779: 4778: 4766: 4765: 4756: 4748: 4743: 4742: 4733: 4732: 4720: 4719: 4707: 4706: 4697: 4689: 4680: 4679: 4667: 4666: 4654: 4653: 4619: 4617: 4616: 4611: 4608: 4603: 4594: 4593: 4581: 4580: 4570: 4565: 4550: 4549: 4537: 4536: 4524: 4523: 4477: 4475: 4474: 4469: 4463: 4462: 4453: 4452: 4440: 4439: 4427: 4426: 4417: 4409: 4401: 4400: 4388: 4387: 4375: 4374: 4362: 4361: 4349: 4348: 4336: 4335: 4326: 4318: 4309: 4308: 4296: 4295: 4283: 4282: 4256: > 4143: 4141: 4140: 4135: 4129: 4124: 4105: 4100: 4087: 4082: 4067: 4066: 4048: 4047: 4035: 4034: 3953: 3951: 3950: 3945: 3940: 3939: 3921: 3920: 3908: 3907: 3892: 3891: 3873: 3872: 3860: 3859: 3767: 3765: 3764: 3759: 3753: 3748: 3738: 3733: 3724: 3723: 3710: 3705: 3695: 3690: 3681: 3680: 3667: 3662: 3653: 3652: 3642: 3637: 3625: 3624: 3614: 3609: 3599: 3594: 3581: 3576: 3567: 3566: 3556: 3551: 3539: 3538: 3528: 3523: 3513: 3508: 3493: 3492: 3480: 3479: 3467: 3466: 3454: 3453: 3417: 3415: 3414: 3409: 3406: 3401: 3392: 3391: 3382: 3381: 3369: 3368: 3358: 3353: 3344: 3343: 3331: 3330: 3321: 3320: 3310: 3305: 3290: 3289: 3277: 3276: 3264: 3263: 3251: 3250: 3086: 3084: 3083: 3078: 3067: 3066: 3054: 3053: 3041: 3040: 3028: 3027: 3015: 3014: 3002: 3001: 2986: 2985: 2976: 2975: 2963: 2962: 2950: 2949: 2930: 2925: 2912: 2907: 2696: > 2520: 2518: 2517: 2512: 2510: 2509: 2445:Vieta's formulas 2439: 2437: 2436: 2431: 2429: 2422: 2421: 2409: 2408: 2399: 2398: 2389: 2388: 2363: 2362: 2348: 2347: 2337: 2321: 2316: 2301: 2300: 2269: 2268: 2250: 2249: 2240: 2239: 2227: 2226: 2217: 2216: 2198: 2197: 2188: 2187: 2169: 2168: 2156: 2155: 2146: 2145: 2136: 2135: 2123: 2122: 2110: 2109: 2100: 2099: 2087: 2086: 2055: 2054: 2035: 2034: 2031: 2026: 2025: 2016: 2015: 2005: 1974: 1973: 1964: 1963: 1939: 1938: 1929: 1928: 1910: 1909: 1900: 1899: 1887: 1886: 1877: 1876: 1860: 1859: 1840: 1839: 1821: 1820: 1808: 1807: 1788: 1787: 1755: 1753: 1752: 1747: 1739: 1738: 1714: 1713: 1692: 1691: 1670: 1669: 1654: 1653: 1641: 1640: 1631: 1630: 1612: 1611: 1596: 1595: 1577: 1576: 1540:is the field of 1495: 1493: 1492: 1487: 1485: 1484: 1469: 1468: 1456: 1455: 1446: 1445: 1427: 1426: 1411: 1410: 1392: 1391: 1268: 1266: 1265: 1260: 1258: 1257: 1247: 1242: 1232: 1227: 1214: 1209: 1199: 1194: 1185: 1184: 1171: 1166: 1157: 1156: 1146: 1141: 1128: 1123: 1114: 1113: 1103: 1098: 1085: 1080: 1070: 1065: 1056: 1055: 1043: 1042: 1032: 1027: 1017: 1012: 990: 988: 987: 982: 979: 974: 965: 964: 954: 949: 936: 931: 921: 916: 907: 906: 894: 893: 883: 878: 868: 863: 844: 842: 841: 836: 834: 833: 821: 820: 804: 802: 801: 796: 794: 793: 777: 775: 774: 769: 767: 766: 747: 745: 744: 739: 737: 736: 724: 723: 690: 688: 687: 682: 680: 679: 670: 669: 657: 656: 643: 603: 601: 600: 595: 593: 592: 583: 582: 567: 566: 557: 556: 541: 540: 531: 530: 515: 514: 505: 504: 495: 494: 451: 449: 448: 443: 441: 440: 431: 430: 418: 417: 401: 396: 387: 386: 374: 373: 363: 358: 345: 340: 330: 325: 305: 303: 302: 297: 288: 283: 270: 265: 195:monic polynomial 166: 107: 100: 89: 83: 77: 21: 7311: 7310: 7306: 7305: 7304: 7302: 7301: 7300: 7281: 7280: 7217: 7197: 7194: 7167: 7136: 7130: 7102: 7090: 7084: 7069: 7060: 7053: 7044: 7035: 7028: 7021: 7012: 7005: 6958: 6939: 6920: 6907: 6888: 6875: 6865: 6830: 6829: 6821: 6812: 6805: 6765: 6752: 6739: 6726: 6713: 6700: 6684: 6674: 6661: 6648: 6595: 6594: 6588: 6581: 6575: 6568: 6561: 6554: 6547: 6541: 6538: 6531: 6524: 6517: 6510: 6504: 6479: 6470: 6463: 6454: 6445: 6430: 6421: 6414: 6405: 6396: 6389: 6382: 6373: 6351: 6350: 6334: 6324: 6314: 6280: 6267: 6224: 6196: 6183: 6155: 6068: 6067: 6054: 6041: 6028: 5997: 5981: 5968: 5955: 5930: 5914: 5901: 5888: 5869: 5862: 5849: 5836: 5823: 5810: 5801: 5800: 5795: 5786: 5775: 5766: 5759: 5751:The polynomial 5692: 5679: 5618: 5605: 5595: 5585: 5557: 5544: 5516: 5467: 5454: 5441: 5428: 5423: 5422: 5418:. For instance 5417: 5408: 5390: 5381: 5374: 5360: 5354: 5327: 5304: 5297: 5275: 5274: 5258: 5245: 5232: 5219: 5203: 5190: 5177: 5164: 5151: 5138: 5125: 5112: 5103: 5102: 5077: 5064: 5021: 5008: 4965: 4952: 4930: 4917: 4904: 4891: 4866: 4857: 4856: 4822: 4809: 4796: 4783: 4770: 4757: 4734: 4724: 4711: 4698: 4671: 4658: 4633: 4628: 4627: 4585: 4572: 4541: 4528: 4503: 4498: 4497: 4485:variables with 4454: 4444: 4431: 4418: 4392: 4379: 4366: 4353: 4340: 4327: 4300: 4287: 4274: 4269: 4268: 4251: 4242: 4235: 4222: 4213: 4206: 4197: 4188: 4181: 4174: 4165: 4058: 4039: 4026: 4021: 4020: 4015: 4006: 3999: 3983: 3977: 3931: 3912: 3899: 3883: 3864: 3851: 3846: 3845: 3805: 3799: 3795: 3791: 3784: 3777: 3715: 3672: 3644: 3616: 3558: 3530: 3484: 3471: 3458: 3427: 3422: 3421: 3383: 3373: 3360: 3335: 3322: 3312: 3281: 3268: 3255: 3224: 3219: 3218: 3213: 3207: 3195: 3186: 3179: 3165: 3159: 3154:natural numbers 3151: 3145: 3136: 3129: 3120: 3104: 3058: 3045: 3032: 3019: 3006: 2993: 2977: 2967: 2954: 2941: 2894: 2893: 2888: 2882: 2875: 2868: 2861: 2854: 2847: 2840: 2833: 2826: 2789: 2780: 2773: 2760: 2751: 2731: 2722: 2715: 2708: 2691: 2682: 2675: 2660: 2650: 2641: 2634: 2617: 2611: 2604: 2595: 2588: 2580: 2540: 2531: 2495: 2481: 2480: 2478: 2427: 2426: 2413: 2400: 2390: 2380: 2364: 2354: 2351: 2350: 2339: 2286: 2254: 2241: 2231: 2218: 2202: 2189: 2179: 2160: 2147: 2137: 2127: 2114: 2101: 2091: 2072: 2056: 2046: 2043: 2042: 2029: 2028: 2017: 2007: 1965: 1949: 1930: 1920: 1901: 1891: 1878: 1868: 1861: 1845: 1842: 1841: 1831: 1812: 1799: 1789: 1773: 1764: 1763: 1730: 1705: 1683: 1661: 1645: 1632: 1622: 1597: 1581: 1568: 1557: 1556: 1546:complex numbers 1531: 1522: 1507: 1476: 1460: 1447: 1437: 1412: 1396: 1383: 1372: 1371: 1358: 1295:polynomials of 1286: 1280: 1275: 1249: 1176: 1148: 1105: 1047: 1034: 999: 998: 956: 898: 885: 850: 849: 825: 812: 807: 806: 785: 780: 779: 758: 753: 752: 728: 715: 710: 709: 671: 661: 648: 612: 611: 584: 574: 558: 548: 532: 522: 506: 496: 486: 481: 480: 475: 468: 461: 432: 422: 409: 378: 365: 309: 308: 252: 251: 247:are symmetric: 246: 239: 231: 164: 155: 148: 137: 126: 119: 109: 102: 98: 85: 79: 75: 66: 59: 49: 35: 28: 23: 22: 15: 12: 11: 5: 7309: 7307: 7299: 7298: 7293: 7283: 7282: 7279: 7278: 7261: 7246:I.G. Macdonald 7243: 7236: 7215: 7193: 7190: 7189: 7188: 7183: 7178: 7173: 7166: 7163: 7132:Main article: 7129: 7126: 7106:linear algebra 7101: 7098: 7086:Main article: 7083: 7080: 7065: 7058: 7049: 7040: 7033: 7026: 7017: 7010: 7003: 6997: 6996: 6976: 6973: 6970: 6965: 6961: 6957: 6954: 6951: 6946: 6942: 6938: 6933: 6930: 6927: 6923: 6919: 6914: 6910: 6906: 6903: 6900: 6895: 6891: 6887: 6882: 6878: 6872: 6868: 6864: 6861: 6858: 6853: 6848: 6845: 6842: 6838: 6817: 6810: 6801: 6795: 6794: 6783: 6780: 6777: 6772: 6768: 6764: 6759: 6755: 6751: 6746: 6742: 6738: 6733: 6729: 6725: 6720: 6716: 6712: 6707: 6703: 6699: 6696: 6691: 6687: 6681: 6677: 6673: 6668: 6664: 6660: 6655: 6651: 6647: 6644: 6641: 6638: 6635: 6630: 6625: 6621: 6617: 6612: 6607: 6603: 6586: 6579: 6573: 6566: 6559: 6552: 6545: 6536: 6529: 6522: 6515: 6508: 6497: 6496: 6489: 6475: 6468: 6459: 6450: 6443: 6426: 6419: 6410: 6401: 6394: 6387: 6378: 6371: 6365: 6364: 6349: 6346: 6341: 6337: 6331: 6327: 6321: 6317: 6313: 6310: 6307: 6302: 6297: 6293: 6287: 6283: 6279: 6274: 6270: 6264: 6259: 6255: 6251: 6246: 6241: 6237: 6231: 6227: 6223: 6218: 6213: 6209: 6203: 6199: 6195: 6190: 6186: 6180: 6175: 6171: 6167: 6162: 6158: 6152: 6147: 6143: 6139: 6136: 6133: 6128: 6123: 6119: 6115: 6110: 6105: 6101: 6097: 6092: 6087: 6083: 6079: 6076: 6073: 6071: 6069: 6066: 6061: 6057: 6053: 6048: 6044: 6040: 6035: 6031: 6027: 6022: 6019: 6016: 6013: 6010: 6007: 6004: 6000: 5996: 5993: 5988: 5984: 5980: 5975: 5971: 5967: 5962: 5958: 5954: 5949: 5946: 5943: 5940: 5937: 5933: 5929: 5926: 5921: 5917: 5913: 5908: 5904: 5900: 5895: 5891: 5887: 5882: 5879: 5876: 5872: 5868: 5865: 5863: 5861: 5856: 5852: 5848: 5843: 5839: 5835: 5830: 5826: 5822: 5817: 5813: 5809: 5808: 5791: 5784: 5771: 5764: 5755: 5749: 5748: 5737: 5732: 5727: 5723: 5719: 5714: 5709: 5705: 5699: 5695: 5691: 5686: 5682: 5676: 5671: 5667: 5663: 5658: 5653: 5649: 5645: 5640: 5635: 5631: 5625: 5621: 5617: 5612: 5608: 5602: 5598: 5592: 5588: 5584: 5579: 5574: 5570: 5564: 5560: 5556: 5551: 5547: 5541: 5536: 5532: 5528: 5523: 5519: 5513: 5508: 5504: 5500: 5495: 5490: 5486: 5482: 5479: 5474: 5470: 5466: 5461: 5457: 5453: 5448: 5444: 5440: 5435: 5431: 5413: 5406: 5386: 5379: 5370: 5356:Main article: 5353: 5350: 5342:characteristic 5325: 5302: 5295: 5289: 5288: 5273: 5270: 5265: 5261: 5257: 5252: 5248: 5244: 5239: 5235: 5231: 5226: 5222: 5218: 5215: 5210: 5206: 5202: 5197: 5193: 5189: 5184: 5180: 5176: 5171: 5167: 5163: 5158: 5154: 5150: 5145: 5141: 5137: 5132: 5128: 5124: 5119: 5115: 5111: 5108: 5106: 5104: 5099: 5094: 5090: 5084: 5080: 5076: 5071: 5067: 5061: 5056: 5052: 5048: 5043: 5038: 5034: 5028: 5024: 5020: 5015: 5011: 5005: 5000: 4996: 4992: 4987: 4982: 4978: 4972: 4968: 4964: 4959: 4955: 4949: 4944: 4940: 4936: 4933: 4931: 4929: 4924: 4920: 4916: 4911: 4907: 4903: 4898: 4894: 4890: 4885: 4882: 4879: 4876: 4873: 4869: 4865: 4864: 4850: 4849: 4837: 4834: 4829: 4825: 4821: 4816: 4812: 4808: 4803: 4799: 4795: 4790: 4786: 4782: 4777: 4773: 4769: 4764: 4760: 4754: 4751: 4746: 4741: 4737: 4731: 4727: 4723: 4718: 4714: 4710: 4705: 4701: 4695: 4692: 4686: 4683: 4678: 4674: 4670: 4665: 4661: 4657: 4652: 4649: 4646: 4643: 4640: 4636: 4621: 4620: 4607: 4602: 4598: 4592: 4588: 4584: 4579: 4575: 4569: 4564: 4560: 4556: 4553: 4548: 4544: 4540: 4535: 4531: 4527: 4522: 4519: 4516: 4513: 4510: 4506: 4479: 4478: 4466: 4461: 4457: 4451: 4447: 4443: 4438: 4434: 4430: 4425: 4421: 4415: 4412: 4407: 4404: 4399: 4395: 4391: 4386: 4382: 4378: 4373: 4369: 4365: 4360: 4356: 4352: 4347: 4343: 4339: 4334: 4330: 4324: 4321: 4315: 4312: 4307: 4303: 4299: 4294: 4290: 4286: 4281: 4277: 4247: 4240: 4231: 4225: 4224: 4218: 4211: 4202: 4193: 4186: 4179: 4170: 4163: 4145: 4144: 4133: 4128: 4123: 4119: 4115: 4112: 4109: 4104: 4099: 4095: 4091: 4086: 4081: 4077: 4073: 4070: 4065: 4061: 4057: 4054: 4051: 4046: 4042: 4038: 4033: 4029: 4011: 4004: 3993: 3979:Main article: 3976: 3973: 3972: 3971: 3943: 3938: 3934: 3930: 3927: 3924: 3919: 3915: 3911: 3906: 3902: 3898: 3895: 3890: 3886: 3882: 3879: 3876: 3871: 3867: 3863: 3858: 3854: 3801: 3797: 3796: ≥ α 3793: 3789: 3782: 3775: 3769: 3768: 3757: 3752: 3747: 3743: 3737: 3732: 3728: 3722: 3718: 3714: 3709: 3704: 3700: 3694: 3689: 3685: 3679: 3675: 3671: 3666: 3661: 3657: 3651: 3647: 3641: 3636: 3632: 3628: 3623: 3619: 3613: 3608: 3604: 3598: 3593: 3589: 3585: 3580: 3575: 3571: 3565: 3561: 3555: 3550: 3546: 3542: 3537: 3533: 3527: 3522: 3518: 3512: 3507: 3503: 3499: 3496: 3491: 3487: 3483: 3478: 3474: 3470: 3465: 3461: 3457: 3452: 3449: 3446: 3443: 3440: 3437: 3434: 3430: 3419: 3405: 3400: 3396: 3390: 3386: 3380: 3376: 3372: 3367: 3363: 3357: 3352: 3348: 3342: 3338: 3334: 3329: 3325: 3319: 3315: 3309: 3304: 3300: 3296: 3293: 3288: 3284: 3280: 3275: 3271: 3267: 3262: 3258: 3254: 3249: 3246: 3243: 3240: 3237: 3234: 3231: 3227: 3209: 3205: 3191: 3184: 3177: 3161: 3157: 3147: 3141: 3134: 3125: 3118: 3103: 3100: 3088: 3087: 3076: 3073: 3070: 3065: 3061: 3057: 3052: 3048: 3044: 3039: 3035: 3031: 3026: 3022: 3018: 3013: 3009: 3005: 3000: 2996: 2992: 2989: 2984: 2980: 2974: 2970: 2966: 2961: 2957: 2953: 2948: 2944: 2940: 2937: 2934: 2929: 2924: 2920: 2916: 2911: 2906: 2902: 2886: 2880: 2873: 2866: 2859: 2852: 2845: 2838: 2831: 2824: 2814: 2813: 2802: 2799: 2785: 2778: 2769: 2756: 2749: 2727: 2720: 2713: 2704: 2687: 2680: 2673: 2661:instead.) For 2656: 2646: 2639: 2630: 2613:Main article: 2610: 2607: 2600: 2593: 2586: 2579: 2576: 2547:, states that 2536: 2529: 2508: 2505: 2502: 2498: 2494: 2491: 2488: 2474: 2459:or not, being 2441: 2440: 2425: 2420: 2416: 2412: 2407: 2403: 2397: 2393: 2387: 2383: 2379: 2376: 2373: 2370: 2367: 2365: 2361: 2357: 2353: 2352: 2346: 2342: 2336: 2333: 2330: 2326: 2320: 2315: 2312: 2309: 2305: 2299: 2296: 2293: 2289: 2285: 2282: 2279: 2275: 2272: 2267: 2264: 2261: 2257: 2253: 2248: 2244: 2238: 2234: 2230: 2225: 2221: 2215: 2212: 2209: 2205: 2201: 2196: 2192: 2186: 2182: 2178: 2175: 2172: 2167: 2163: 2159: 2154: 2150: 2144: 2140: 2134: 2130: 2126: 2121: 2117: 2113: 2108: 2104: 2098: 2094: 2090: 2085: 2082: 2079: 2075: 2071: 2068: 2065: 2062: 2059: 2057: 2053: 2049: 2045: 2044: 2041: 2032: 2030: 2024: 2020: 2014: 2010: 2004: 2001: 1998: 1995: 1992: 1989: 1986: 1982: 1977: 1972: 1968: 1962: 1959: 1956: 1952: 1948: 1945: 1942: 1937: 1933: 1927: 1923: 1919: 1916: 1913: 1908: 1904: 1898: 1894: 1890: 1885: 1881: 1875: 1871: 1867: 1864: 1862: 1858: 1855: 1852: 1848: 1844: 1843: 1838: 1834: 1830: 1827: 1824: 1819: 1815: 1811: 1806: 1802: 1798: 1795: 1792: 1790: 1786: 1783: 1780: 1776: 1772: 1771: 1757: 1756: 1745: 1742: 1737: 1733: 1729: 1726: 1723: 1720: 1717: 1712: 1708: 1704: 1701: 1698: 1695: 1690: 1686: 1682: 1679: 1676: 1673: 1668: 1664: 1660: 1657: 1652: 1648: 1644: 1639: 1635: 1629: 1625: 1621: 1618: 1615: 1610: 1607: 1604: 1600: 1594: 1591: 1588: 1584: 1580: 1575: 1571: 1567: 1564: 1527: 1520: 1512:. There exist 1503: 1497: 1496: 1483: 1479: 1475: 1472: 1467: 1463: 1459: 1454: 1450: 1444: 1440: 1436: 1433: 1430: 1425: 1422: 1419: 1415: 1409: 1406: 1403: 1399: 1395: 1390: 1386: 1382: 1379: 1357: 1354: 1328:if and only if 1282:Main article: 1279: 1276: 1274: 1271: 1270: 1269: 1256: 1252: 1246: 1241: 1237: 1231: 1226: 1222: 1218: 1213: 1208: 1204: 1198: 1193: 1189: 1183: 1179: 1175: 1170: 1165: 1161: 1155: 1151: 1145: 1140: 1136: 1132: 1127: 1122: 1118: 1112: 1108: 1102: 1097: 1093: 1089: 1084: 1079: 1075: 1069: 1064: 1060: 1054: 1050: 1046: 1041: 1037: 1031: 1026: 1022: 1016: 1011: 1007: 992: 991: 978: 973: 969: 963: 959: 953: 948: 944: 940: 935: 930: 926: 920: 915: 911: 905: 901: 897: 892: 888: 882: 877: 873: 867: 862: 858: 832: 828: 824: 819: 815: 792: 788: 765: 761: 749: 748: 735: 731: 727: 722: 718: 692: 691: 678: 674: 668: 664: 660: 655: 651: 647: 642: 639: 636: 633: 630: 627: 624: 620: 605: 604: 591: 587: 581: 577: 573: 570: 565: 561: 555: 551: 547: 544: 539: 535: 529: 525: 521: 518: 513: 509: 503: 499: 493: 489: 473: 466: 459: 453: 452: 439: 435: 429: 425: 421: 416: 412: 408: 405: 400: 395: 391: 385: 381: 377: 372: 368: 362: 357: 353: 349: 344: 339: 335: 329: 324: 320: 316: 306: 295: 292: 287: 282: 278: 274: 269: 264: 260: 244: 237: 230: 227: 160: 153: 146: 138:) = 131: 124: 117: 71: 64: 57: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 7308: 7297: 7294: 7292: 7289: 7288: 7286: 7277: 7276:0-521-56069-1 7273: 7269: 7265: 7262: 7259: 7258:0-19-850450-0 7255: 7251: 7247: 7244: 7241: 7237: 7234: 7230: 7226: 7222: 7218: 7212: 7208: 7204: 7200: 7196: 7195: 7191: 7187: 7184: 7182: 7179: 7177: 7174: 7172: 7169: 7168: 7164: 7162: 7160: 7156: 7151: 7149: 7145: 7141: 7135: 7127: 7125: 7123: 7119: 7118:combinatorics 7115: 7114:Galois theory 7111: 7107: 7099: 7097: 7095: 7089: 7081: 7079: 7077: 7074: ≤ 7073: 7068: 7064: 7057: 7052: 7048: 7043: 7039: 7032: 7025: 7020: 7016: 7009: 7002: 6994: 6990: 6974: 6971: 6963: 6959: 6955: 6952: 6949: 6944: 6940: 6931: 6928: 6925: 6921: 6912: 6908: 6904: 6901: 6898: 6893: 6889: 6880: 6876: 6870: 6862: 6859: 6851: 6846: 6843: 6840: 6836: 6828: 6827: 6826: 6823: 6820: 6816: 6809: 6804: 6800: 6781: 6778: 6770: 6766: 6762: 6757: 6753: 6744: 6740: 6731: 6727: 6723: 6718: 6714: 6705: 6701: 6697: 6694: 6689: 6679: 6675: 6671: 6666: 6662: 6653: 6649: 6645: 6642: 6639: 6636: 6633: 6628: 6623: 6619: 6615: 6610: 6605: 6601: 6593: 6592: 6591: 6585: 6578: 6572: 6565: 6558: 6551: 6544: 6535: 6528: 6521: 6514: 6507: 6502: 6494: 6490: 6487: 6483: 6478: 6474: 6467: 6462: 6458: 6453: 6449: 6442: 6438: 6434: 6433: 6432: 6429: 6425: 6418: 6413: 6409: 6404: 6400: 6393: 6386: 6381: 6377: 6370: 6347: 6339: 6335: 6329: 6325: 6319: 6315: 6308: 6300: 6295: 6291: 6285: 6281: 6277: 6272: 6268: 6262: 6257: 6253: 6249: 6244: 6239: 6235: 6229: 6225: 6221: 6216: 6211: 6207: 6201: 6197: 6193: 6188: 6184: 6178: 6173: 6169: 6165: 6160: 6156: 6150: 6145: 6141: 6134: 6126: 6121: 6117: 6113: 6108: 6103: 6099: 6095: 6090: 6085: 6081: 6074: 6072: 6059: 6055: 6051: 6046: 6042: 6038: 6033: 6029: 6017: 6014: 6011: 6008: 6005: 5998: 5994: 5986: 5982: 5978: 5973: 5969: 5965: 5960: 5956: 5944: 5941: 5938: 5931: 5927: 5919: 5915: 5911: 5906: 5902: 5898: 5893: 5889: 5877: 5870: 5866: 5864: 5854: 5850: 5846: 5841: 5837: 5833: 5828: 5824: 5815: 5811: 5799: 5798: 5797: 5794: 5790: 5783: 5779: 5774: 5770: 5763: 5758: 5754: 5735: 5730: 5725: 5721: 5717: 5712: 5707: 5703: 5697: 5693: 5689: 5684: 5680: 5674: 5669: 5665: 5661: 5656: 5651: 5647: 5643: 5638: 5633: 5629: 5623: 5619: 5615: 5610: 5606: 5600: 5596: 5590: 5586: 5582: 5577: 5572: 5568: 5562: 5558: 5554: 5549: 5545: 5539: 5534: 5530: 5526: 5521: 5517: 5511: 5506: 5502: 5498: 5493: 5488: 5484: 5480: 5472: 5468: 5464: 5459: 5455: 5451: 5446: 5442: 5433: 5429: 5421: 5420: 5419: 5416: 5412: 5405: 5401: 5398: 5394: 5389: 5385: 5378: 5373: 5369: 5365: 5359: 5351: 5349: 5347: 5343: 5339: 5335: 5331: 5324: 5320: 5316: 5312: 5308: 5301: 5294: 5271: 5263: 5259: 5255: 5250: 5246: 5242: 5237: 5233: 5224: 5220: 5216: 5208: 5204: 5200: 5195: 5191: 5187: 5182: 5178: 5169: 5165: 5156: 5152: 5148: 5143: 5139: 5135: 5130: 5126: 5117: 5113: 5109: 5107: 5097: 5092: 5088: 5082: 5078: 5074: 5069: 5065: 5059: 5054: 5050: 5046: 5041: 5036: 5032: 5026: 5022: 5018: 5013: 5009: 5003: 4998: 4994: 4990: 4985: 4980: 4976: 4970: 4966: 4962: 4957: 4953: 4947: 4942: 4938: 4934: 4932: 4922: 4918: 4914: 4909: 4905: 4901: 4896: 4892: 4880: 4877: 4874: 4867: 4855: 4854: 4853: 4835: 4827: 4823: 4819: 4814: 4810: 4801: 4797: 4788: 4784: 4780: 4775: 4771: 4762: 4758: 4752: 4749: 4744: 4739: 4729: 4725: 4721: 4716: 4712: 4703: 4699: 4693: 4690: 4684: 4676: 4672: 4668: 4663: 4659: 4647: 4644: 4641: 4634: 4626: 4625: 4624: 4605: 4600: 4596: 4590: 4586: 4582: 4577: 4573: 4567: 4562: 4558: 4554: 4546: 4542: 4538: 4533: 4529: 4517: 4514: 4511: 4504: 4496: 4495: 4494: 4492: 4488: 4484: 4464: 4459: 4449: 4445: 4441: 4436: 4432: 4423: 4419: 4413: 4410: 4405: 4397: 4393: 4389: 4384: 4380: 4371: 4367: 4358: 4354: 4350: 4345: 4341: 4332: 4328: 4322: 4319: 4313: 4305: 4301: 4297: 4292: 4288: 4279: 4275: 4267: 4266: 4265: 4263: 4259: 4255: 4250: 4246: 4239: 4234: 4230: 4221: 4217: 4210: 4205: 4201: 4196: 4192: 4185: 4178: 4173: 4169: 4162: 4158: 4157: 4156: 4154: 4150: 4131: 4126: 4121: 4117: 4113: 4110: 4107: 4102: 4097: 4093: 4089: 4084: 4079: 4075: 4071: 4063: 4059: 4055: 4052: 4049: 4044: 4040: 4031: 4027: 4019: 4018: 4017: 4014: 4010: 4003: 3997: 3992: 3988: 3982: 3974: 3969: 3966: − 3965: 3961: 3957: 3936: 3932: 3928: 3925: 3922: 3917: 3913: 3904: 3900: 3896: 3888: 3884: 3880: 3877: 3874: 3869: 3865: 3856: 3852: 3844: 3843: 3842: 3840: 3837: ≤ 3836: 3831: 3829: 3825: 3821: 3817: 3813: 3809: 3804: 3788: 3781: 3778: = 3774: 3755: 3750: 3745: 3741: 3735: 3730: 3726: 3720: 3716: 3712: 3707: 3702: 3698: 3692: 3687: 3683: 3677: 3673: 3669: 3664: 3659: 3655: 3649: 3645: 3639: 3634: 3630: 3626: 3621: 3617: 3611: 3606: 3602: 3596: 3591: 3587: 3583: 3578: 3573: 3569: 3563: 3559: 3553: 3548: 3544: 3540: 3535: 3531: 3525: 3520: 3516: 3510: 3505: 3501: 3497: 3489: 3485: 3481: 3476: 3472: 3468: 3463: 3459: 3447: 3444: 3441: 3438: 3435: 3428: 3420: 3403: 3398: 3394: 3388: 3384: 3378: 3374: 3370: 3365: 3361: 3355: 3350: 3346: 3340: 3336: 3332: 3327: 3323: 3317: 3313: 3307: 3302: 3298: 3294: 3286: 3282: 3278: 3273: 3269: 3265: 3260: 3256: 3244: 3241: 3238: 3235: 3232: 3225: 3217: 3216: 3215: 3212: 3203: 3199: 3194: 3190: 3183: 3176: 3173: 3169: 3164: 3155: 3150: 3144: 3140: 3133: 3128: 3124: 3117: 3113: 3109: 3101: 3099: 3097: 3093: 3074: 3071: 3063: 3059: 3055: 3050: 3046: 3037: 3033: 3024: 3020: 3016: 3011: 3007: 2998: 2994: 2990: 2987: 2982: 2972: 2968: 2964: 2959: 2955: 2946: 2942: 2938: 2935: 2932: 2927: 2922: 2918: 2914: 2909: 2904: 2900: 2892: 2891: 2890: 2885: 2879: 2872: 2865: 2858: 2851: 2844: 2837: 2830: 2823: 2819: 2811: 2807: 2803: 2800: 2797: 2794: ≤ 2793: 2788: 2784: 2777: 2772: 2768: 2764: 2759: 2755: 2748: 2744: 2740: 2739: 2738: 2735: 2730: 2726: 2719: 2712: 2707: 2703: 2699: 2695: 2690: 2686: 2679: 2672: 2668: 2667:empty product 2664: 2659: 2654: 2649: 2645: 2638: 2633: 2629: 2625: 2622: 2616: 2608: 2606: 2603: 2599: 2592: 2585: 2577: 2575: 2573: 2568: 2563: 2559: 2554: 2550: 2546: 2545: 2539: 2535: 2528: 2524: 2506: 2503: 2500: 2492: 2489: 2477: 2473: 2469: 2464: 2462: 2458: 2454: 2450: 2446: 2423: 2418: 2414: 2410: 2405: 2401: 2395: 2391: 2385: 2377: 2374: 2368: 2366: 2359: 2355: 2344: 2340: 2334: 2331: 2328: 2324: 2318: 2313: 2310: 2307: 2303: 2297: 2294: 2291: 2283: 2280: 2273: 2265: 2262: 2259: 2255: 2251: 2246: 2242: 2236: 2232: 2228: 2223: 2219: 2213: 2210: 2207: 2203: 2199: 2194: 2190: 2184: 2180: 2176: 2173: 2170: 2165: 2161: 2157: 2152: 2148: 2142: 2138: 2132: 2128: 2124: 2119: 2115: 2111: 2106: 2102: 2096: 2092: 2083: 2080: 2077: 2069: 2066: 2060: 2058: 2051: 2047: 2039: 2022: 2018: 2012: 2008: 2002: 1999: 1996: 1993: 1990: 1987: 1984: 1980: 1975: 1970: 1966: 1960: 1957: 1954: 1950: 1946: 1943: 1940: 1935: 1931: 1925: 1921: 1917: 1914: 1911: 1906: 1902: 1896: 1892: 1888: 1883: 1879: 1873: 1869: 1865: 1863: 1856: 1853: 1850: 1846: 1836: 1832: 1828: 1825: 1822: 1817: 1813: 1809: 1804: 1800: 1796: 1793: 1791: 1784: 1781: 1778: 1774: 1762: 1761: 1760: 1743: 1735: 1731: 1727: 1724: 1718: 1710: 1706: 1702: 1699: 1688: 1684: 1680: 1677: 1671: 1666: 1662: 1658: 1655: 1650: 1646: 1642: 1637: 1633: 1627: 1623: 1619: 1616: 1613: 1608: 1605: 1602: 1598: 1592: 1589: 1586: 1582: 1578: 1573: 1569: 1565: 1562: 1555: 1554: 1553: 1551: 1547: 1543: 1539: 1535: 1530: 1526: 1519: 1515: 1511: 1506: 1502: 1481: 1477: 1473: 1470: 1465: 1461: 1457: 1452: 1448: 1442: 1438: 1434: 1431: 1428: 1423: 1420: 1417: 1413: 1407: 1404: 1401: 1397: 1393: 1388: 1384: 1380: 1377: 1370: 1369: 1368: 1367: 1363: 1355: 1353: 1351: 1350:Galois theory 1347: 1343: 1339: 1334: 1332: 1329: 1325: 1321: 1317: 1313: 1309: 1305: 1301: 1298: 1294: 1291: 1285: 1284:Galois theory 1278:Galois theory 1277: 1272: 1254: 1250: 1244: 1239: 1235: 1229: 1224: 1220: 1216: 1211: 1206: 1202: 1196: 1191: 1187: 1181: 1177: 1173: 1168: 1163: 1159: 1153: 1149: 1143: 1138: 1134: 1130: 1125: 1120: 1116: 1110: 1106: 1100: 1095: 1091: 1087: 1082: 1077: 1073: 1067: 1062: 1058: 1052: 1048: 1044: 1039: 1035: 1029: 1024: 1020: 1014: 1009: 1005: 997: 996: 995: 976: 971: 967: 961: 957: 951: 946: 942: 938: 933: 928: 924: 918: 913: 909: 903: 899: 895: 890: 886: 880: 875: 871: 865: 860: 856: 848: 847: 846: 830: 826: 822: 817: 813: 790: 786: 763: 759: 733: 729: 725: 720: 716: 708: 707: 706: 703: 701: 697: 676: 666: 662: 658: 653: 649: 640: 637: 634: 631: 628: 625: 622: 618: 610: 609: 608: 589: 585: 579: 575: 571: 568: 563: 559: 553: 549: 545: 542: 537: 533: 527: 523: 519: 516: 511: 507: 501: 497: 491: 487: 479: 478: 477: 472: 465: 458: 437: 427: 423: 419: 414: 410: 403: 398: 393: 389: 383: 379: 375: 370: 366: 360: 355: 351: 347: 342: 337: 333: 327: 322: 318: 314: 307: 293: 290: 285: 280: 276: 272: 267: 262: 258: 250: 249: 248: 243: 236: 228: 226: 224: 220: 219:combinatorics 216: 212: 208: 204: 198: 196: 192: 189: 185: 181: 177: 173: 168: 163: 159: 152: 145: 141: 135: 130: 123: 116: 112: 106: 97: 93: 88: 82: 74: 70: 63: 56: 52: 48: 44: 40: 33: 19: 7267: 7249: 7239: 7202: 7152: 7143: 7137: 7103: 7091: 7075: 7071: 7066: 7062: 7055: 7050: 7046: 7041: 7037: 7030: 7023: 7018: 7014: 7007: 7000: 6998: 6992: 6988: 6824: 6818: 6814: 6807: 6802: 6798: 6796: 6583: 6576: 6570: 6563: 6556: 6549: 6542: 6533: 6526: 6519: 6512: 6505: 6500: 6498: 6492: 6485: 6481: 6476: 6472: 6465: 6460: 6456: 6451: 6447: 6440: 6436: 6427: 6423: 6416: 6411: 6407: 6402: 6398: 6391: 6384: 6379: 6375: 6368: 6366: 5792: 5788: 5781: 5777: 5772: 5768: 5761: 5756: 5752: 5750: 5414: 5410: 5403: 5399: 5387: 5383: 5376: 5371: 5367: 5363: 5361: 5333: 5322: 5318: 5314: 5310: 5306: 5299: 5292: 5290: 4851: 4622: 4490: 4486: 4482: 4480: 4261: 4257: 4253: 4248: 4244: 4237: 4232: 4228: 4226: 4219: 4215: 4208: 4203: 4199: 4194: 4190: 4183: 4176: 4171: 4167: 4160: 4148: 4146: 4012: 4008: 4001: 3995: 3990: 3986: 3984: 3967: 3963: 3959: 3955: 3838: 3834: 3832: 3827: 3823: 3815: 3802: 3786: 3779: 3772: 3770: 3210: 3201: 3197: 3192: 3188: 3181: 3174: 3171: 3167: 3162: 3148: 3142: 3138: 3131: 3126: 3122: 3115: 3107: 3105: 3089: 2883: 2877: 2870: 2863: 2856: 2849: 2842: 2835: 2828: 2821: 2817: 2815: 2805: 2795: 2791: 2786: 2782: 2775: 2770: 2766: 2757: 2753: 2746: 2742: 2733: 2728: 2724: 2717: 2710: 2705: 2701: 2697: 2693: 2688: 2684: 2677: 2670: 2662: 2657: 2652: 2647: 2643: 2636: 2631: 2627: 2623: 2618: 2601: 2597: 2590: 2583: 2581: 2566: 2561: 2558:coefficients 2557: 2552: 2548: 2542: 2537: 2533: 2526: 2475: 2471: 2467: 2465: 2456: 2452: 2442: 1758: 1549: 1542:real numbers 1537: 1533: 1528: 1524: 1517: 1513: 1509: 1504: 1500: 1498: 1365: 1361: 1359: 1335: 1330: 1323: 1319: 1311: 1303: 1299: 1287: 1273:Applications 993: 750: 704: 700:discriminant 693: 606: 470: 463: 456: 454: 241: 234: 232: 199: 187: 176:coefficients 169: 161: 157: 150: 143: 139: 133: 128: 121: 114: 110: 104: 91: 86: 80: 72: 68: 61: 54: 50: 42: 36: 7291:Polynomials 7199:Lang, Serge 3792:for which α 103:1, 2, ..., 96:permutation 94:if for any 39:mathematics 7285:Categories 7233:0984.00001 7192:References 6987:, for all 5340:of finite 1364:of degree 1293:univariate 47:polynomial 7248:(1995), 7144:invariant 6953:… 6929:− 6902:… 6860:− 6837:∑ 6779:− 6643:− 6634:− 5393:monomials 5217:− 4745:− 4406:− 4111:⋯ 4053:… 3998:,0,...,0) 3926:… 3905:α 3878:… 3072:− 2988:− 2933:− 2504:− 2490:− 2411:⋯ 2375:− 2332:≠ 2325:∏ 2304:∑ 2295:− 2281:− 2263:− 2252:⋯ 2211:− 2200:⋯ 2174:⋯ 2158:⋯ 2112:⋯ 2081:− 2067:− 2040:⋮ 2000:≤ 1988:≤ 1981:∑ 1958:− 1944:⋯ 1915:⋯ 1854:− 1829:− 1826:⋯ 1823:− 1810:− 1797:− 1782:− 1728:− 1719:⋯ 1703:− 1681:− 1617:⋯ 1606:− 1590:− 1432:⋯ 1421:− 1405:− 823:− 726:− 659:− 638:≤ 626:≤ 619:∏ 569:− 543:− 517:− 291:− 207:power sum 188:symmetric 7266:(1999), 7201:(2002), 7165:See also 6406:), ..., 4487:integral 4198:), ..., 4153:rational 3841:one has 3771:Clearly 3202:distinct 3112:monomial 3108:additive 2810:integral 1310:. These 229:Examples 108:one has 7225:1878556 7203:Algebra 7061:, ..., 7036:, ..., 7013:, ..., 6813:, ..., 6471:, ..., 6446:, ..., 6422:, ..., 6397:, ..., 6374:, ..., 5787:, ..., 5767:, ..., 5409:, ..., 5382:, ..., 4243:, ..., 4214:, ..., 4189:, ..., 4166:, ..., 4007:, ..., 3970:zeros). 3187:, ..., 3121:, ..., 3090:(for a 2781:, ..., 2752:, ..., 2723:, ..., 2683:, ..., 2642:, ..., 2621:integer 2596:, ..., 2532:, ..., 1322:of the 1302:having 221:and in 156:, ..., 127:, ..., 67:, ..., 7274: 7256: 7231: 7223: 7213: 7112:, and 7022:) and 6999:Since 5397:degree 5319:always 4252:) for 3208:,...,α 3170:. The 3160:,...,α 2855:, and 2461:simple 2036: 1516:roots 1297:degree 696:square 209:, and 5338:field 3958:into 3812:basis 3092:proof 1523:,..., 1308:field 1290:monic 90:is a 45:is a 7272:ISBN 7254:ISBN 7211:ISBN 6562:) = 6540:and 6525:) = 5346:ring 3152:are 2876:) = 2841:) = 2808:has 1994:< 778:and 632:< 240:and 125:σ(2) 118:σ(1) 41:, a 7229:Zbl 6491:If 6439:in 5780:in 5395:of 3137:... 3098:). 2804:if 2745:in 2669:so 2549:any 2525:in 1550:all 1532:of 702:). 78:in 37:In 7287:: 7227:, 7221:MR 7219:, 7205:, 7150:. 7108:, 6782:7. 6582:+ 6569:+ 6555:, 6532:+ 6518:, 4223:). 2869:, 2848:+ 2834:, 2716:, 2589:, 1352:. 476:: 469:, 462:, 225:. 205:, 167:. 149:, 132:σ( 120:, 60:, 7076:n 7072:k 7067:n 7063:X 7059:1 7056:X 7054:( 7051:k 7047:h 7042:n 7038:X 7034:1 7031:X 7029:( 7027:0 7024:h 7019:n 7015:X 7011:1 7008:X 7006:( 7004:0 7001:e 6995:. 6993:n 6989:k 6975:0 6972:= 6969:) 6964:n 6960:X 6956:, 6950:, 6945:1 6941:X 6937:( 6932:i 6926:k 6922:h 6918:) 6913:n 6909:X 6905:, 6899:, 6894:1 6890:X 6886:( 6881:i 6877:e 6871:i 6867:) 6863:1 6857:( 6852:k 6847:0 6844:= 6841:i 6819:n 6815:X 6811:1 6808:X 6806:( 6803:n 6799:h 6776:) 6771:2 6767:X 6763:, 6758:1 6754:X 6750:( 6745:2 6741:h 6737:) 6732:2 6728:X 6724:, 6719:1 6715:X 6711:( 6706:1 6702:h 6698:3 6695:+ 6690:3 6686:) 6680:2 6676:X 6672:, 6667:1 6663:X 6659:( 6654:1 6650:h 6646:2 6640:= 6637:7 6629:3 6624:2 6620:X 6616:+ 6611:3 6606:1 6602:X 6587:2 6584:X 6580:2 6577:X 6574:1 6571:X 6567:1 6564:X 6560:2 6557:X 6553:1 6550:X 6548:( 6546:2 6543:h 6537:2 6534:X 6530:1 6527:X 6523:2 6520:X 6516:1 6513:X 6511:( 6509:1 6506:h 6501:n 6493:P 6488:. 6486:n 6482:k 6477:n 6473:X 6469:1 6466:X 6464:( 6461:k 6457:h 6452:n 6448:X 6444:1 6441:X 6437:P 6428:n 6424:X 6420:1 6417:X 6415:( 6412:n 6408:h 6403:n 6399:X 6395:1 6392:X 6390:( 6388:1 6385:h 6380:n 6376:X 6372:1 6369:X 6348:. 6345:) 6340:3 6336:X 6330:2 6326:X 6320:1 6316:X 6312:( 6309:+ 6306:) 6301:2 6296:3 6292:X 6286:2 6282:X 6278:+ 6273:3 6269:X 6263:2 6258:2 6254:X 6250:+ 6245:2 6240:3 6236:X 6230:1 6226:X 6222:+ 6217:2 6212:2 6208:X 6202:1 6198:X 6194:+ 6189:3 6185:X 6179:2 6174:1 6170:X 6166:+ 6161:2 6157:X 6151:2 6146:1 6142:X 6138:( 6135:+ 6132:) 6127:3 6122:3 6118:X 6114:+ 6109:3 6104:2 6100:X 6096:+ 6091:3 6086:1 6082:X 6078:( 6075:= 6065:) 6060:3 6056:X 6052:, 6047:2 6043:X 6039:, 6034:1 6030:X 6026:( 6021:) 6018:1 6015:, 6012:1 6009:, 6006:1 6003:( 5999:m 5995:+ 5992:) 5987:3 5983:X 5979:, 5974:2 5970:X 5966:, 5961:1 5957:X 5953:( 5948:) 5945:1 5942:, 5939:2 5936:( 5932:m 5928:+ 5925:) 5920:3 5916:X 5912:, 5907:2 5903:X 5899:, 5894:1 5890:X 5886:( 5881:) 5878:3 5875:( 5871:m 5867:= 5860:) 5855:3 5851:X 5847:, 5842:2 5838:X 5834:, 5829:1 5825:X 5821:( 5816:3 5812:h 5793:n 5789:X 5785:1 5782:X 5778:k 5773:n 5769:X 5765:1 5762:X 5760:( 5757:k 5753:h 5736:. 5731:3 5726:3 5722:X 5718:+ 5713:2 5708:3 5704:X 5698:2 5694:X 5690:+ 5685:3 5681:X 5675:2 5670:2 5666:X 5662:+ 5657:3 5652:2 5648:X 5644:+ 5639:2 5634:3 5630:X 5624:1 5620:X 5616:+ 5611:3 5607:X 5601:2 5597:X 5591:1 5587:X 5583:+ 5578:2 5573:2 5569:X 5563:1 5559:X 5555:+ 5550:3 5546:X 5540:2 5535:1 5531:X 5527:+ 5522:2 5518:X 5512:2 5507:1 5503:X 5499:+ 5494:3 5489:1 5485:X 5481:= 5478:) 5473:3 5469:X 5465:, 5460:2 5456:X 5452:, 5447:1 5443:X 5439:( 5434:3 5430:h 5415:n 5411:X 5407:1 5404:X 5400:k 5388:n 5384:X 5380:1 5377:X 5375:( 5372:k 5368:h 5364:k 5334:n 5326:1 5323:e 5315:n 5311:n 5307:n 5303:3 5300:p 5296:3 5293:X 5272:. 5269:) 5264:3 5260:X 5256:, 5251:2 5247:X 5243:, 5238:1 5234:X 5230:( 5225:3 5221:p 5214:) 5209:3 5205:X 5201:, 5196:2 5192:X 5188:, 5183:1 5179:X 5175:( 5170:2 5166:p 5162:) 5157:3 5153:X 5149:, 5144:2 5140:X 5136:, 5131:1 5127:X 5123:( 5118:1 5114:p 5110:= 5098:2 5093:3 5089:X 5083:2 5079:X 5075:+ 5070:3 5066:X 5060:2 5055:2 5051:X 5047:+ 5042:2 5037:3 5033:X 5027:1 5023:X 5019:+ 5014:3 5010:X 5004:2 4999:1 4995:X 4991:+ 4986:2 4981:2 4977:X 4971:1 4967:X 4963:+ 4958:2 4954:X 4948:2 4943:1 4939:X 4935:= 4928:) 4923:3 4919:X 4915:, 4910:2 4906:X 4902:, 4897:1 4893:X 4889:( 4884:) 4881:1 4878:, 4875:2 4872:( 4868:m 4836:. 4833:) 4828:2 4824:X 4820:, 4815:1 4811:X 4807:( 4802:1 4798:p 4794:) 4789:2 4785:X 4781:, 4776:1 4772:X 4768:( 4763:2 4759:p 4753:2 4750:1 4740:3 4736:) 4730:2 4726:X 4722:, 4717:1 4713:X 4709:( 4704:1 4700:p 4694:2 4691:1 4685:= 4682:) 4677:2 4673:X 4669:, 4664:1 4660:X 4656:( 4651:) 4648:1 4645:, 4642:2 4639:( 4635:m 4606:2 4601:2 4597:X 4591:1 4587:X 4583:+ 4578:2 4574:X 4568:2 4563:1 4559:X 4555:= 4552:) 4547:2 4543:X 4539:, 4534:1 4530:X 4526:( 4521:) 4518:1 4515:, 4512:2 4509:( 4505:m 4491:n 4483:n 4465:. 4460:3 4456:) 4450:2 4446:X 4442:, 4437:1 4433:X 4429:( 4424:1 4420:p 4414:2 4411:1 4403:) 4398:2 4394:X 4390:, 4385:1 4381:X 4377:( 4372:1 4368:p 4364:) 4359:2 4355:X 4351:, 4346:1 4342:X 4338:( 4333:2 4329:p 4323:2 4320:3 4314:= 4311:) 4306:2 4302:X 4298:, 4293:1 4289:X 4285:( 4280:3 4276:p 4262:n 4258:n 4254:k 4249:n 4245:X 4241:1 4238:X 4236:( 4233:k 4229:p 4220:n 4216:X 4212:1 4209:X 4207:( 4204:n 4200:p 4195:n 4191:X 4187:1 4184:X 4182:( 4180:1 4177:p 4172:n 4168:X 4164:1 4161:X 4149:n 4132:. 4127:k 4122:n 4118:X 4114:+ 4108:+ 4103:k 4098:2 4094:X 4090:+ 4085:k 4080:1 4076:X 4072:= 4069:) 4064:n 4060:X 4056:, 4050:, 4045:1 4041:X 4037:( 4032:k 4028:p 4013:n 4009:X 4005:1 4002:X 4000:( 3996:k 3994:( 3991:m 3987:k 3968:k 3964:n 3960:k 3956:k 3942:) 3937:n 3933:X 3929:, 3923:, 3918:1 3914:X 3910:( 3901:m 3897:= 3894:) 3889:n 3885:X 3881:, 3875:, 3870:1 3866:X 3862:( 3857:k 3853:e 3839:n 3835:k 3828:P 3824:P 3816:P 3803:n 3798:2 3794:1 3790:α 3787:m 3783:β 3780:m 3776:α 3773:m 3756:. 3751:3 3746:3 3742:X 3736:2 3731:2 3727:X 3721:1 3717:X 3713:+ 3708:2 3703:3 3699:X 3693:3 3688:2 3684:X 3678:1 3674:X 3670:+ 3665:3 3660:3 3656:X 3650:2 3646:X 3640:2 3635:1 3631:X 3627:+ 3622:3 3618:X 3612:3 3607:2 3603:X 3597:2 3592:1 3588:X 3584:+ 3579:2 3574:3 3570:X 3564:2 3560:X 3554:3 3549:1 3545:X 3541:+ 3536:3 3532:X 3526:2 3521:2 3517:X 3511:3 3506:1 3502:X 3498:= 3495:) 3490:3 3486:X 3482:, 3477:2 3473:X 3469:, 3464:1 3460:X 3456:( 3451:) 3448:1 3445:, 3442:2 3439:, 3436:3 3433:( 3429:m 3418:, 3404:3 3399:3 3395:X 3389:2 3385:X 3379:1 3375:X 3371:+ 3366:3 3362:X 3356:3 3351:2 3347:X 3341:1 3337:X 3333:+ 3328:3 3324:X 3318:2 3314:X 3308:3 3303:1 3299:X 3295:= 3292:) 3287:3 3283:X 3279:, 3274:2 3270:X 3266:, 3261:1 3257:X 3253:( 3248:) 3245:1 3242:, 3239:1 3236:, 3233:3 3230:( 3226:m 3211:n 3206:1 3198:x 3193:n 3189:X 3185:1 3182:X 3180:( 3178:α 3175:m 3168:X 3163:n 3158:1 3149:i 3143:n 3139:X 3135:1 3132:X 3127:n 3123:X 3119:1 3116:X 3075:7 3069:) 3064:2 3060:X 3056:, 3051:1 3047:X 3043:( 3038:1 3034:e 3030:) 3025:2 3021:X 3017:, 3012:1 3008:X 3004:( 2999:2 2995:e 2991:3 2983:3 2979:) 2973:2 2969:X 2965:, 2960:1 2956:X 2952:( 2947:1 2943:e 2939:= 2936:7 2928:3 2923:2 2919:X 2915:+ 2910:3 2905:1 2901:X 2887:2 2884:X 2881:1 2878:X 2874:2 2871:X 2867:1 2864:X 2862:( 2860:2 2857:e 2853:2 2850:X 2846:1 2843:X 2839:2 2836:X 2832:1 2829:X 2827:( 2825:1 2822:e 2818:n 2806:P 2798:; 2796:n 2792:k 2787:n 2783:X 2779:1 2776:X 2774:( 2771:k 2767:e 2758:n 2754:X 2750:1 2747:X 2743:P 2734:n 2729:n 2725:X 2721:2 2718:X 2714:1 2711:X 2709:( 2706:k 2702:e 2698:n 2694:k 2689:n 2685:X 2681:1 2678:X 2676:( 2674:0 2671:e 2663:k 2658:k 2653:k 2648:n 2644:X 2640:1 2637:X 2635:( 2632:k 2628:e 2624:k 2602:n 2598:X 2594:2 2591:X 2587:1 2584:X 2567:K 2562:K 2553:n 2538:n 2534:x 2530:1 2527:x 2507:i 2501:n 2497:) 2493:1 2487:( 2476:i 2472:a 2468:P 2457:K 2453:P 2424:. 2419:n 2415:x 2406:2 2402:x 2396:1 2392:x 2386:n 2382:) 2378:1 2372:( 2369:= 2360:0 2356:a 2345:j 2341:x 2335:i 2329:j 2319:n 2314:1 2311:= 2308:i 2298:1 2292:n 2288:) 2284:1 2278:( 2274:= 2271:) 2266:1 2260:n 2256:x 2247:2 2243:x 2237:1 2233:x 2229:+ 2224:n 2220:x 2214:2 2208:n 2204:x 2195:2 2191:x 2185:1 2181:x 2177:+ 2171:+ 2166:n 2162:x 2153:4 2149:x 2143:3 2139:x 2133:1 2129:x 2125:+ 2120:n 2116:x 2107:3 2103:x 2097:2 2093:x 2089:( 2084:1 2078:n 2074:) 2070:1 2064:( 2061:= 2052:1 2048:a 2023:j 2019:x 2013:i 2009:x 2003:n 1997:j 1991:i 1985:1 1976:= 1971:n 1967:x 1961:1 1955:n 1951:x 1947:+ 1941:+ 1936:3 1932:x 1926:2 1922:x 1918:+ 1912:+ 1907:3 1903:x 1897:1 1893:x 1889:+ 1884:2 1880:x 1874:1 1870:x 1866:= 1857:2 1851:n 1847:a 1837:n 1833:x 1818:2 1814:x 1805:1 1801:x 1794:= 1785:1 1779:n 1775:a 1744:. 1741:) 1736:n 1732:x 1725:t 1722:( 1716:) 1711:2 1707:x 1700:t 1697:( 1694:) 1689:1 1685:x 1678:t 1675:( 1672:= 1667:0 1663:a 1659:+ 1656:t 1651:1 1647:a 1643:+ 1638:2 1634:t 1628:2 1624:a 1620:+ 1614:+ 1609:1 1603:n 1599:t 1593:1 1587:n 1583:a 1579:+ 1574:n 1570:t 1566:= 1563:P 1538:K 1534:P 1529:n 1525:x 1521:1 1518:x 1514:n 1510:K 1505:i 1501:a 1482:0 1478:a 1474:+ 1471:t 1466:1 1462:a 1458:+ 1453:2 1449:t 1443:2 1439:a 1435:+ 1429:+ 1424:1 1418:n 1414:t 1408:1 1402:n 1398:a 1394:+ 1389:n 1385:t 1381:= 1378:P 1366:n 1362:t 1331:f 1324:n 1320:f 1312:n 1304:n 1300:n 1255:3 1251:X 1245:4 1240:2 1236:X 1230:2 1225:1 1221:X 1217:+ 1212:4 1207:3 1203:X 1197:2 1192:2 1188:X 1182:1 1178:X 1174:+ 1169:2 1164:3 1160:X 1154:2 1150:X 1144:4 1139:1 1135:X 1131:+ 1126:4 1121:3 1117:X 1111:2 1107:X 1101:2 1096:1 1092:X 1088:+ 1083:2 1078:3 1074:X 1068:4 1063:2 1059:X 1053:1 1049:X 1045:+ 1040:3 1036:X 1030:2 1025:2 1021:X 1015:4 1010:1 1006:X 977:4 972:3 968:X 962:2 958:X 952:2 947:1 943:X 939:+ 934:2 929:3 925:X 919:4 914:2 910:X 904:1 900:X 896:+ 891:3 887:X 881:2 876:2 872:X 866:4 861:1 857:X 831:1 827:X 818:2 814:X 791:2 787:X 764:1 760:X 734:2 730:X 721:1 717:X 677:2 673:) 667:j 663:X 654:i 650:X 646:( 641:n 635:j 629:i 623:1 590:3 586:X 580:2 576:X 572:2 564:3 560:X 554:1 550:X 546:2 538:2 534:X 528:1 524:X 520:2 512:3 508:X 502:2 498:X 492:1 488:X 474:3 471:X 467:2 464:X 460:1 457:X 438:4 434:) 428:2 424:X 420:+ 415:1 411:X 407:( 404:+ 399:3 394:2 390:X 384:1 380:X 376:+ 371:2 367:X 361:3 356:1 352:X 348:+ 343:2 338:2 334:X 328:2 323:1 319:X 315:4 294:7 286:3 281:2 277:X 273:+ 268:3 263:1 259:X 245:2 242:X 238:1 235:X 165:) 162:n 158:X 154:2 151:X 147:1 144:X 142:( 140:P 136:) 134:n 129:X 122:X 115:X 113:( 111:P 105:n 99:σ 87:P 81:n 76:) 73:n 69:X 65:2 62:X 58:1 55:X 53:( 51:P 34:. 20:)

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Knowledge

Symmetric polynomial

Index