Integrality and Specialized Symmetric Functions

Given $d_1,\dots ,d_k$ in the field $F$, there is a weighted trace function $F^k\to F$ given by $tr(x_1,\dots ,x_k)=\sum d_i{x}_i$. We prove that $F^k$ satisfies trace identities of the forms $\alpha (d_1,\dots ,d_k)x^N=$ 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.