Slope Estimates of Artin–Schreier Curves

2003/07/01 by Jasper Scholten, Hui June Zhu · 3 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Polynomial and algebraic computation

paper · pdf · doi:10.1023/a:1024116216156

Abstract

Let X/\open F_p be an Artin–Schreier curve defined by the affine equation y p − y = f ( x ) where f ( x ) ∈ \open F_p [ x ] is monic of degree d . In this paper we develop a method for estimating the first slope of the Newton polygon of X . Denote this first slope by NP 1 ( X/\open F_p ). We use our method to prove that if p > d ≥ 2 then NP 1 ( X/\open F_p ) ≥ [lceil ]( p −1)/ d [rceil ]/( p − 1). If p > 2 d ≥ 4, we give a sufficient condition for the equality to hold.

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