On Kummer extensions with one place at infinity

Let $K$ be the algebraic closure of ${\mathbb{F}}_q$. We provide an explicit description of the Weierstrass semigroup $H(Q_{\mathrm{\infty }})$ at the only place at infinity $Q_{\mathrm{\infty }}$ of the curve $\mathcal{X}$ defined by the Kummer extension with equation $y^m=f(x)$, where $f(x)\in K[x]$ is a polynomial satisfying $\gcd (m,\text{deg}f)=1$. As a consequence, we determine the Frobenius number and the multiplicity of $H(Q_{\mathrm{\infty }})$ in some cases, and we discuss sufficient conditions for the Weierstrass semigroup $H(Q_{\mathrm{\infty }})$ to be symmetric. Finally, we characterize certain maximal Castle curves of type $(\mathcal{X},Q_{\mathrm{\infty }})$.