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On coefficient valuations of Eisenstein polynomials

Matthias KünzerEduard Wirsing — 2005

Journal de Théorie des Nombres de Bordeaux

Let p 3 be a prime, let n > m 1 . Let π n be the norm of ζ p n - 1 under C p - 1 , so that ( p ) [ π n ] | ( p ) is a purely ramified extension of discrete valuation rings of degree p n - 1 . The minimal polynomial of π n over ( π m ) is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at π m . The function field analogue, as introduced by Carlitz and Hayes, is studied as well.

Three two-dimensional Weyl steps in the circle problem I. The Hessian determinant

Ulrike M. A. VorhauerEduard Wirsing — 1999

Acta Arithmetica

1. Summary. In a sequence of three papers we study the circle problem and its generalization involving the logarithmic mean. Most of the deeper results in this area depend on estimates of exponential sums. For the circle problem itself Chen has carried out such estimates using three two-dimensional Weyl steps with complicated techniques. We make the same Weyl steps but our approach is simpler and clearer. Crucial is a good understanding of the Hessian determinant that appears and a simple...

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