On curves with one place at infinity

Let $f$ be a plane curve. We give a procedure based on Abhyankar's approximate roots to detect if it has a single place at infinity, and if so construct its associated $\delta$-sequence, and consequently its value semigroup. Also for fixed genus (equivalently Frobenius number) we construct all $\delta$-sequences generating numerical semigroups with this given genus. For a $\delta$-sequence we present a procedure to construct all curves having this associated sequence. We also study the embeddings of such curves in the plane. In particular, we prove that polynomial curves might not have a unique embedding.Comment: 14 pages, 2 figure

Isolated factorizations and their applications in simplicial affine semigroups

We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. We also generalize $\alpha$-rectangular numerical semigroups to the context of simplicial affine semigroups and study their isolated factorizations. As a consequence of our results, we characterize those complete intersection simplicial affine semigroups with only one Betti minimal element in several ways. Moreover, we define Betti sorted and Betti divisible simplicial affine semigroups and characterize them in terms of gluings and their minimal presentations. Finally, we determine all the Betti divisible numerical semigroups, which turn out to be those numerical semigroups that are free for any arrangement of their minimal generators

Cyclotomic numerical semigroups

Given a numerical semigroup $S$, we let $\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s$ be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups $S$ such that $\mathrm P_S(x)$ has all its roots in the unit disc. We conjecture that $S$ is a cyclotomic numerical semigroup if and only if $S$ is a complete intersection numerical semigroup and present some evidence for it. Aside from the notion of cyclotomic numerical semigroup we introduce the notion of cyclotomic exponents and polynomially related numerical semigroups. We derive some properties and give some applications of these new concepts.Comment: 17 pages, accepted for publication in SIAM J. Discrete Mat