Number of singular points of a genus $g$ curve with one point at infinity

TL;DR

This paper establishes bounds on the maximum number of singular points for plane algebraic curves with a single point at infinity, confirming a conjecture relating singularities to the curve's topological complexity.

Contribution

It provides new bounds on singular points for such curves and confirms the Zaidenberg and Lin conjecture for the case of one point at infinity.

Findings

Bound N < (17/11) b_1(C) asymptotically for large b_1

Confirmed Zaidenberg and Lin conjecture N ≤ 2b_1 + 1

Established a relationship between singular points and Betti number

Abstract

We bound the maximal number N of singular points of a plane algebraic curve C that has precisely one place at infinity with one branch in terms of its first Betti number $b_1(C)$. Asymptotically we prove that $N<\sim{17/11}b_1(C)$ for large $b_1$. In particular, in the case of curves with one place at infinity, we confirm the Zaidenberg and Lin conjecture stating that $N\le 2b_1+1$.