An elementary symmetric polynomial 
,
for nonnegative integers 
and 
,
is the sum of all products of
distinct variables:
The convention is 
,
with 
for 
. For example, 
. The products 
form a vector basis for the symmetric
polynomials in these variables over any field,
where 
ranges over partitions whose parts are at most 
, including the empty partition
(Macdonald 1995, pp. 19-21).
For 
, the polynomial
map 
has Jacobian matrix 
, with determinant
where 
is the Vandermonde determinant (Bracciali
and Piñar 2023, p. 189). The polynomial
discriminant of the monic polynomial 
is 
. Consequently, 
is nonsingular iff
the roots 
are pairwise distinct.
See also
Chromatic Quasisymmetric Function,
Complete Homogeneous
Symmetric Polynomial,
Fundamental
Theorem of Symmetric Functions,
Polynomial
Discriminant,
Shareshian-Wachs Conjecture,
Symmetric Function,
Symmetric
Polynomial,
Vandermonde Determinant,
Vieta's Formulas
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References
Bracciali, C. F. and Piñar, M. A. "On Multivariate Orthogonal Polynomials and Elementary Symmetric Functions." Numer.
Algorithms 92, 183-206, 2023. https://doi.org/10.1007/s11075-022-01434-4.Macdonald,
I. G. Symmetric
Functions and Hall Polynomials, 2nd ed. Oxford, England: Oxford University
Press, 1995.Wu, X. "Recursive Algorithm Without Extra Function
Evaluations for the Jacobian Matrix of Viéta's Polynomial System with Applications."
Appl. Math. Comput. 162, 539-548, 2005. https://doi.org/10.1016/j.amc.2003.12.107.Referenced
on Wolfram|Alpha
Elementary Symmetric Polynomial
Cite this as:
Weisstein, Eric W. "Elementary Symmetric Polynomial."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ElementarySymmetricPolynomial.html
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