On the zeroes of Goss polynomials

Gekeler, Ernst-Ulrich

doi:10.1090/S0002-9947-2012-05699-6

On the zeroes of Goss polynomials
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Trans. Amer. Math. Soc. 365 (2013), 1669-1685 Request permission

Abstract:

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 $A=\mathbb {F}_q[T]$ and obtain, in the case when $q$ is prime, an explicit description of the Newton polygon $NP(G_k(X))$ 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 $NP(G_k(X))$ 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 $A$.

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