Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials

Abstract.

Any symmetric polynomial f∈R[X ₁, …, X _n] has a unique representation f = p(σ₁, …, σ_n) with p∈R[X ₁, …, X _n] in the elementary symmetric polynomials σ₁, …, σ_n. This paper investigates higher order symmetric polynomials; these are symmetric polynomials with a representation p, which is also symmetric. We present rewriting techniques for higher order symmetric polynomials and exact degree bounds for the generators of the corresponding invariant rings. Moreover, we point out how algorithms and degree bounds for these polynomials are related to Pascals triangle, Fibonacci numbers, Chebyshev polynomials, and cardinalities of finite distributive lattices of semi-ideals.