An exact degree for multivariate special polynomials

We introduce certain special polynomials in an arbitrary number of indeterminates over a finite field. These polynomials generalize the special polynomials associated to the Goss zeta function and Goss-Dirichlet $L$-functions over the ring of polynomials in one indeterminate over a finite field and also capture the special values at non-positive integers of $L$-series associated to Drinfeld modules over Tate algebras defined over the same ring. We compute the exact degree in $t_0$ of these special polynomials and show that this degree is an invariant for a natural action of Goss' group of digit permutations. Finally, we characterize the vanishing of these multivariate special polynomials at $t_0=1$. This gives rise to a notion of trivial zeros for our polynomials generalizing that of the Goss zeta function mentioned above.