M-curves of degree 9 with three nests

Details M-curves of degree 9 with three nests M-curves of degree 9 with three nests

2008-06-27

arxiv.org/abs/0806.4446v2

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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