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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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


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:
  • Received by editor(s): December 22, 2010
  • Received by editor(s) in revised form: August 31, 2011
  • Published electronically: October 15, 2012
  • © Copyright 2012 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 1669-1685
  • MSC (2010): Primary 11F52; Secondary 11G09, 11J93, 11T55
  • DOI:
  • MathSciNet review: 3003278