M-curves of degree 9 with three nests

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.

37 pages, 24 figures, some corrections, more detailed proofs