A Viro-Zvonilov-type inequality for Q-flexible curves of odd degree

TL;DR

This paper introduces a new inequality for odd degree flexible curves, extending classical bounds on ovals, by defining an analogue of the Arnold surface and applying it to Q-flexible embeddings.

Contribution

It develops a Viro--Zvonilov-type inequality for odd degree flexible curves using a novel Arnold surface analogue and explores its application to flexible curves in quadrics and non-orientable cases.

Findings

Derived an upper bound on the number of ovals for odd degree flexible curves.

Extended the inequality to flexible curves in quadrics.

Proposed a potential definition for non-orientable flexible curves.

Abstract

We define an analogue of the Arnold surface for odd degree flexible curves, and we use it to double branch cover $Q$-flexible embeddings, where $Q$-flexible is a condition to be added to the classical notion of a flexible curve. This allows us to obtain a Viro--Zvonilov-type inequality: an upper bound on the number of non-empty ovals of a curve of odd degree. We investigate our method for flexible curves in a quadric to derive a similar bound in two cases. We also digress around a possible definition of non-orientable flexible curves, for which our method still works and a similar inequality holds.