Weierstrass semigroups at totally ramified places of degree one on linearized function fields

A linearized function field $F$ can be viewed as a Galois extension of a rational function field $K(x)$. For a totally ramified place $Q$ of degree one in $F/K(x)$, we give a unified description of the set $G(Q)$ of gaps at $Q$. As a consequence, we explicitly provide a system of generators, the multiplicity, and the Frobenius number of the Weierstrass semigroup $H(Q)$. Moreover, we give a necessary and sufficient condition for $H(Q)$ to be symmetric. Then we investigate the minimal generating set of the Weierstrass semigroups at several totally ramified places of degree one. We not only explicitly describe the minimal generating set, but also provide functions whose coefficients of pole divisors lie in the minimal generating set. Finally, we investigate the linearized function field associated with the denominator of a separable polynomial and apply our results to present several examples.