Math 2.0

Combinatorics Definition

Elementary symmetric polynomials

Builds on Symmetric polynomialPolynomials in several variables

Definition

Let RR be a commutative ring and let n≥1n \ge 1. For 0≤k≤n0 \le k \le n, the kk-th elementary symmetric polynomial in x1,…,xnx_1, \dots, x_n is ek=∑1≤i1<i2<⋯<ik≤nxi1xi2⋯xik∈R[x1,…,xn].e_k = \sum_{1 \le i_1 < i_2 < \dots < i_k \le n} x_{i_1} x_{i_2} \cdots x_{i_k} \in R[x_1, \dots, x_n].

Remarks

So e0=1e_0 = 1, e1=x1+⋯+xne_1 = x_1 + \dots + x_n and en=x1x2⋯xne_n = x_1 x_2 \cdots x_n. One sets ek=0e_k = 0 for k>nk > n. In three variables, e2=x1x2+x1x3+x2x3e_2 = x_1 x_2 + x_1 x_3 + x_2 x_3. The generating identity is ∏i=1n(t+xi)=∑k=0nektn−k\prod_{i=1}^n (t + x_i) = \sum_{k=0}^n e_k t^{n-k}. Each eke_k is symmetric and homogeneous of degree kk.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionPolynomials in several variables, definition, no evidence levelPolynomials in severa…definitionSymmetric polynomial, definition, no evidence levelSymmetric polynomialdefinitionElementary symmetric polynomials, definition, no evidence levelElementary symmetric …definition
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  1. Group concept
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  5. Polynomials in several variables definition
  6. Symmetric polynomial definition
  7. Elementary symmetric polynomials definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial, Definition
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WritingW1 stated
MSC05E05
DefinesElementary symmetric polynomial
UsesSymmetric polynomial, Polynomials in several variables

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