Math 2.0

Combinatorics Definition

Symmetric polynomial

Builds on Polynomials in several variables

Definition

Let RR be a commutative ring and let n≥1n \ge 1. A polynomial f∈R[x1,…,xn]f \in R[x_1, \dots, x_n] is symmetric if f(xσ(1),…,xσ(n))=f(x1,…,xn)f(x_{\sigma(1)}, \dots, x_{\sigma(n)}) = f(x_1, \dots, x_n) for every permutation σ\sigma of {1,…,n}\{1, \dots, n\}.

Remarks

Examples in two variables: x1+x2x_1 + x_2, x1x2x_1 x_2 and x12+x22x_1^2 + x_2^2. The polynomial x1−x2x_1 - x_2 is not symmetric. The symmetric polynomials form a subring of R[x1,…,xn]R[x_1, \dots, x_n], the ring of invariants of the symmetric group. The main examples are the elementary symmetric polynomials and the power sums. The coefficients of a monic polynomial are symmetric polynomials in its roots, by Vieta's formulas.

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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 polynomialdefinition
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Polynomials in several variables definition
  6. Symmetric polynomial definition

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9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Symmetric_polynomial, lead section
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