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 t0t0 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 t0=1t0=1. This gives rise to a notion of trivial zeros for our polynomials generalizing that of the Goss zeta function mentioned above.