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arXiv · 0710.4662

Bounding the number of rational places using Weierstrass semigroups

Olav Geil·Ryutaroh Matsumoto
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Abstract

Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over GF(q) in one variable which possesses a rational place that has Lambda as its Weierstrass semigroup. We ask the question as to how many rational places such a function field can possibly have and we derive an upper bound in terms of the generators of Lambda and q. Our bound is an improvement to a bound by Lewittes which takes into account only the multiplicity of Lambda and q. From the new bound we derive significant improvements to Serre's upper bound in the cases q=2, 3 and 4. We finally show that Lewittes' bound has important implications to the theory of towers of function fields.

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Olav Geil, Ryutaroh Matsumoto. 2007-10-25. Bounding the number of rational places using Weierstrass semigroups. https://doi.org/10.1016/j.jpaa.2008.11.013

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