On curves with one place at infinity

TL;DR

This paper presents a method using Abhyankar's approximate roots to identify plane curves with a single point at infinity, constructs their associated semigroups, and explores their embeddings, including non-uniqueness of polynomial curve embeddings.

Contribution

It introduces a procedure to detect single-place infinity curves, construct their delta-sequences, and analyze their embeddings, including non-uniqueness phenomena.

Findings

Procedure to detect single-place at infinity curves using approximate roots.

Construction of delta-sequences and value semigroups for such curves.

Proof that polynomial curves can have multiple embeddings in the plane.

Abstract

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.