ON THE ZEROES OF GOSS POLYNOMIALS

Goss polynomials provide a substitute of trigonometric functions and their identities for the arithmetic of function fields. We study the Goss polynomials G(k)(X) for the lattice A = F-q[T] and obtain, in the case when q is prime, an explicit description of the Newton polygon N P(G(k) (X)) of the k-th Goss polynomial in terms of the q-adic expansion of k - 1. In the case of an arbitrary q, we have similar results on N P(G(k) (X)) for special classes of k, and we formulate a general conjecture about its shape. The proofs use rigid-analytic techniques and the arithmetic of power sums of elements of A.