Rational curves with one place at infinity

TL;DR

This paper investigates polynomials with one place at infinity over algebraically closed fields, showing they are either equivalent to coordinates or have at most two rational members in their family, with singularity descriptions when two are rational.

Contribution

It establishes a classification for such polynomials, linking their structure to coordinate equivalence and rational elements, with detailed singularity analysis.

Findings

Polynomials with one place at infinity are either coordinate-equivalent or have at most two rational members.

When two rational elements exist, their singularities can be explicitly described.

The results extend understanding of polynomial behavior at infinity in algebraic geometry.

Abstract

Let K be an algebraically closed field of characteristic zero. Given a polynomial f(x,y) in K[x,y] with one place at infinity, we prove that either f is equivalent to a coordinate, or the family (f+c) has at most two rational elements. When (f+c) has two rational elements, we give a description of the singularities of these elements.