Integrality and specialized symmetric functions

Abstract

Given \(d_1,\ldots,d_k\) in the field $F$, there is a weighted trace function \(F^k\rightarrow F\) given by \({\rm tr}(x_1,\ldots,x_k)=\sum d_ix_i\). 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 \({\rm tr}(y_1)\cdots {\rm tr}(y_n)=\) a linear combination of terms with fewer traces. The approach uses specialized symmetric functions.