Symmetric Polynomial
A symmetric polynomial on n variables x_1, ..., x_n (also called a totally symmetric polynomial) is a function that is unchanged by any permutation of its variables. In other words, the symmetric polynomials satisfy
| f(y_1,y_2,...,y_n)=f(x_1,x_2,...,x_n), |
(1)
|
where y_i=x_(pi(i)) and pi being an arbitrary permutation of the indices 1, 2, ..., n.
For fixed n, the set of all symmetric polynomials in n variables forms an algebra of dimension n. The coefficients of a univariate polynomial f(x) of degree n are algebraically independent symmetric polynomials in the roots of f, and thus form a basis for the set of all such symmetric polynomials.
There are four common homogeneous bases for the symmetric polynomials, each of which is indexed by a partition lambda (Dumitriu et al. 2004). Letting l be the length of lambda, the elementary functions e_lambda, complete homogeneous functions h_lambda, and power-sum functions p_lambda are defined for l=1 by
and for l>1 by
where s is one of e, h or p. In addition, the monomial functions m_lambda are defined as
where S_lambda is the set of permutations giving distinct terms in the sum and lambda is considered to be infinite.
As several different abbreviations and conventions are in common use, care must be taken when determining which symmetric polynomial is in use.
The elementary symmetric polynomials Pi_k(x_1,...,x_n) (sometimes denoted sigma_k or e_lambda) on n variables {x_1,...,x_n} are defined by
The kth elementary symmetric polynomial is implemented in the Wolfram Language as SymmetricPolynomial [k, {x1, ..., xn}]. SymmetricReduction [f, {x1, ..., xn}] gives a pair of polynomials {p,q} in x_1, ..., x_n where p is the symmetric part and q is the remainder.
Alternatively, Pi_j(x_1,...,x_n) can be defined as the coefficient of x^(n-j) in the generating function
| [画像: product_(1<=i<=n)(x+x_i). ] |
(13)
|
For example, on four variables x_1, ..., x_4, the elementary symmetric polynomials are
The power sum S_p(x_1,...,x_n) is defined by
The relationship between S_p and Pi_1, ..., Pi_p is given by the so-called Newton-Girard formulas. The related function s_p(Pi_1,...,Pi_n) with arguments given by the elementary symmetric polynomials (not x_n) is defined by
It turns out that s_p(Pi_1,...,Pi_n) is given by the coefficients of the generating function
| ln(1+Pi_1t+Pi_2t^2+Pi_3t^3+...)=sum_(k=1)^infty(s_k)/kt^k =Pi_1t+1/2(-Pi_1^2+2Pi_2)t^2+1/3(Pi_1^3-3Pi_1Pi_2+3Pi_3)t^3+..., |
(21)
|
so the first few values are
In general, s_p can be computed from the determinant
(Littlewood 1958, Cadogan 1971). In particular,
(Schroeppel 1972), as can be verified by plugging in and multiplying through.
See also
Fundamental Theorem of Symmetric Functions, Jack Polynomial, Zonal Polynomial, Newton-Girard Formulas, Orthogonal Polynomials, Power Sum, Symmetric Function, Vieta's FormulasPortions of this entry contributed by David Terr
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References
Borwein, P. and Erdélyi, T. Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 5, 1995.Cadogan, C. C. "The Möbius Function and Connected Graphs." J. Combin. Th. B 11, 193-200, 1971.Dumitriu, I.; Edelman, A.; and Shuman, G. "MOPS: Multivariate Orthogonal Polynomials (Symbolically)." J. Symbolic Comput. 42, 587-620, 2007. https://doi.org/10.1016/j.jsc.200701005.Littlewood, J. E. A University Algebra, 2nd ed. London, England: Heinemann, 1958.Schroeppel, R. Item 6 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 4, Feb. 1972. https://www.inwap.com/pdp10/hbaker/hakmem/geometry.html#item6.Séroul, R. "Newton-Girard Formulas." §10.12 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 278-279, 2000.Referenced on Wolfram|Alpha
Symmetric PolynomialCite this as:
Terr, David and Weisstein, Eric W. "Symmetric Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SymmetricPolynomial.html