Math 2.0

Combinatorics Definition

Power sum symmetric polynomials

Builds on Symmetric polynomial

Definition

Let RR be a commutative ring and let n≥1n \ge 1. For k≥1k \ge 1, the kk-th power sum in x1,…,xnx_1, \dots, x_n is pk=x1k+x2k+⋯+xnk∈R[x1,…,xn].p_k = x_1^k + x_2^k + \dots + x_n^k \in R[x_1, \dots, x_n].

Remarks

Each pkp_k is symmetric and homogeneous of degree kk. Newton's identities relate the power sums to the elementary symmetric polynomials. If RR contains ℚ\mathbb{Q}, then every symmetric polynomial in R[x1,…,xn]R[x_1, \dots, x_n] is a polynomial in p1,…,pnp_1, \dots, p_n. This fails over ℤ\mathbb{Z}: e2=12(p12−p2)e_2 = \tfrac{1}{2}(p_1^2 - p_2) needs the coefficient 12\tfrac{1}{2}.

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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 polynomialdefinitionPower sum symmetric polynomials, definition, no evidence levelPower sum symmetric p…definition
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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
  7. Power sum symmetric polynomials definition

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2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Power_sum_symmetric_polynomial, Definition
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