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

In this work we study the generalized Weierstrass semigroup $widehat{H} (mathbf{P}_m)$ at an $m$-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 $widehat{Gamma}(mathbf{P}_m)$ of $widehat{H} (mathbf{P}_m)$ and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in ${P_{1}, ldots , P_{m}}$ on these curves, generalizing results of Maharaj, Matthews, and Pirsic.