Polars of Artin–schreier Curves
نویسنده
چکیده
1. Introduction. In this paper we study some properties of the special family of Artin–Schreier curves related to the theory of polar curves we developed in [2]. Our goal is to show how this theory can be carried out in concrete situations and also show how it can be used to determine new upper bounds for the number of rational points of projective plane curves over finite fields. The method we will employ to get such bounds is a generalization in some direction of the one introduced by Stöhr and Voloch in [4]. In general, our method gives better bounds than Weil's, essentially for curves of large degree with big genus (with respect to the degree). This is the case for example when the curve is smooth of large degree, so we could apply our method successfully to Fermat curves in [1]. However, an Artin–Schreier curve tends to have low genus with respect to its degree, moreover, it possesses a natural trivial bound, which is frequently better than Weil's and ours. Nevertheless , for these curves we still can improve in some cases all known upper bounds. Throughout this paper we will use the notation of [2]. Let K be an algebraically closed field and let Z : F = 0 be a projective irreducible plane curve in P
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