Conductors of Abhyankar-Moh semigroups of even degrees

TL;DR

This paper characterizes all possible conductor values of Abhyankar-Moh semigroups of even degrees, providing explicit constructions for each, advancing understanding of their algebraic structure.

Contribution

It proves that every conductor value can be realized by an Abhyankar-Moh semigroup of even degree with a constructive method.

Findings

All conductor values are achievable for even degree semigroups.

Explicit families of semigroups are constructed for each conductor value.

The proof is constructive, providing concrete examples.

Abstract

In their paper on the embeddings of the line in the plane, Abhyankar and Moh proved an important inequality, now known as the Abhyankar-Moh inequality, which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. Barrolleta, Garc\'ia Barroso and P\loski studied the semigroups of integers satisfying the Abhyankar-Moh inequality and call them Abhyankar-Moh semigroups. They described such semigroups with the maximum conductor. In this paper we prove that all possible conductor values are achieved for the Abhyankar-Moh semigroups of even degree. Our proof is constructive, explicitly describing families that achieve a given value as its conductor.