Integrality and Specialized Symmetric Functions

TL;DR

This paper investigates trace identities in algebraic structures using specialized symmetric functions, establishing new relations involving powers of elements and products of traces.

Contribution

It introduces novel trace identities involving powers and products of traces, utilizing specialized symmetric functions to prove these relations.

Findings

Established trace identities involving powers of elements.

Derived linear combinations reducing powers and traces.

Applied specialized symmetric functions to prove identities.

Abstract

Given $d_1,\ldots,d_k$ in the field $F$, there is a weighted trace function $F^k\rightarrow F$ given by $tr(x_1,\ldots,x_k)=\sum d_ix_i$. We prove that $F^k$ satisfies trace identities of the forms $\alpha(d_1,\ldots,d_k) x^N=$ a linear combination of terms with lower powers of $x$; and $tr(y_1)\cdots tr(y_n)=$ a linear combination of terms with fewer traces. The approach uses specialized symmetric functions.