  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.
