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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the zeroes of Goss polynomials


Author: Ernst-Ulrich Gekeler
Journal: Trans. Amer. Math. Soc. 365 (2013), 1669-1685
MSC (2010): Primary 11F52; Secondary 11G09, 11J93, 11T55
Published electronically: October 15, 2012
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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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Additional Information

Ernst-Ulrich Gekeler
Affiliation: FR 6.1 Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken, Germany
Email: gekeler@math.uni-sb.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05699-6
PII: S 0002-9947(2012)05699-6
Received by editor(s): December 22, 2010
Received by editor(s) in revised form: August 31, 2011
Published electronically: October 15, 2012
Article copyright: © Copyright 2012 American Mathematical Society