Towards the generalized Shapiro and Shapiro conjecture

TL;DR

This paper establishes a new asymptotic upper bound on the genus of curves that could challenge the generalized total reality conjecture, improving previous bounds and covering all known cases except the original conjecture.

Contribution

It provides a tighter asymptotic bound on the genus of potential counterexamples to the generalized total reality conjecture.

Findings

New bound g ≤ (1/4)d^2 + O(d) on genus

Covers all known cases except g=0

Advances understanding of the conjecture's limitations

Abstract

We find a new, asymptotically better, bound $g\le\frac14d^2+O(d)$ on the genus of a curve that may violate the generalized total reality conjecture. The bound covers all known cases except $g=0$ (the original conjecture).