M-curves of degree 9 with three nests

TL;DR

This paper investigates the structure of degree 9 M-curves in real algebraic geometry, proving that among three nests, at least one contains an odd number of empty ovals, advancing understanding of Hilbert's sixteenth problem.

Contribution

It proves that for degree 9 M-curves with three nests, at least one nest has an odd number of empty ovals, supporting a conjecture about the distribution of ovals.

Findings

At least one of the three nests has an odd number of empty ovals.

Supports Korchagin's conjecture on the parity of ovals in nests.

Progress in classifying degree 9 M-curves in real algebraic geometry.

Abstract

The first part of Hilbert's sixteenth problem deals with the classification of the isotopy types realizable by real plane algebraic curves of a given degree $m$. For $m = 9$, the classification of the $M$-curves is still wide open. Let $C_9$ be an $M$-curve of degree 9 and $O$ be a non-empty oval of $C_9$. If $O$ contains in its interior $\alpha$ ovals that are all empty, we say that $O$ together with these $\alpha$ ovals forms a nest. The present paper deals with the $M$-curves with three nests. Let $\alpha_i, i = 1, 2, 3$ be the numbers of empty ovals in each nest. We prove that at least one of the $\alpha_i$ is odd. This is a step towards a conjecture of A. Korchagin, claiming that at least two of the $\alpha_i$ should be odd.