turkmath.org

We are going to introduce the concept of Weierstrass semigroup at one point in an algebraic curve and its relevance for the study of properties of the curve. For Kummer extensions ym = f(x); f(x) a polynomial, we discuss conditions for an integer be a Weierstrass gap at a point P. For the totally ramified points, the condition will be necessary and suffcient. As a consequence we present an application to maximal curves: we present a class of polynomials f(x) in Fq2 [x] for which the Fq2 -maximality of ym = f(x) implies that m is a divisor of q +1.