Topological complexity of finding flex points on cubic plane curves

TL;DR

This paper establishes a lower bound on the topological complexity of locating flex points on cubic plane curves, using Schwarz genus bounds, and shows this bound is nearly optimal.

Contribution

It introduces a novel lower bound for the topological complexity of finding flex points on cubic curves, advancing understanding of the problem's inherent difficulty.

Findings

Lower bound for the topological complexity is established.

The bound is shown to be nearly optimal.

The approach involves bounding the Schwarz genus of a related cover.

Abstract

We prove a lower bound for the topological complexity, in the sense of Smale, of the problem of finding a flex point on a cubic plane curve. The key is to bound the Schwarz genus of a cover associated to this problem. We also show that our lower bound for the complexity is close to be the best possible.