Weierstrass points on Kummer extensions

Abstract

Abstract For Kummer extensions given by ym = f(x), we discuss conditions for an integer to be a Weierstrass gap at a place P. In the case of fully ramified places, the conditions are necessary and sufficient. As a consequence, we extend independent results of several authors. Moreover, we show that if the Kummer extension is 𝔽 q 2 -maximal and f(x) ∈ 𝔽 q 2 [x] has at least two roots with the same multiplicity λ coprime to m, then m divides 2(q + 1). Under the extra condition that either m or the multiplicity of a third root of f(x) is odd, we conclude that m divides q + 1.