Generalized Weierstrass semigroups and Riemann-Roch spaces for certain curves with separated variables

In this work we study the generalized Weierstrass semigroup $\widehat{H} ($\mathbf{P}$m)$atan$m$-tuple$$\mathbf{P}$m = (P{1}, \ldots , P{m})$ of rational points on certain curves admitting a plane model of the form $f(y) = g(x)$over$$\mathbb{F}$_{q}$,where${f(T),g(T)\in $\mathbb{F}$_q[T]}$. In particular, we compute the generating set $\widehat{\mathrm{\Gamma }}({\mathbf{P}}_m)$ of $\widehat{H}({\mathbf{P}}_m)$ and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in $\{P_1,\dots ,P_m\}$ on these curves, generalizing results of Maharaj, Matthews, and Pirsic.