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Classification of flexible $$M$$-curves of degree 8 up to isotopy. (English) Zbl 1105.14080
From the introduction: In the first part of his 16-th problem, Hilbert asked how the connected components of a plane real algebraic curve of degree $$m$$ can be arranged on $$\mathbb{R}\mathbb{P}^2$$ up to isotopy. At that time, the answer was known only for $$m\leq 5$$ solved this problem for $$m= 6$$ and Viro did it for $$m= 7$$. For $$m= 8$$, the complete answer is still unknown. We shall call the real scheme of a curve the arrangement of its connected components on $$\mathbb{R}\mathbb{P}^2$$ up to isotopy.
It is reasonable to start the classification with $$M$$-curves (a curve is called an $$M$$-curve if it has the maximal possible number $$(m-1)(m- 2)/2+ 1$$ of connected components). After the studies of Fiedler, Viro, Shustin, Korchagin, and Chevallier, there remained only 9 real schemes whose realizability was open. Here we exclude two of them:
Theorem 1.1. There do not exist real algebraic curves of degree 8 with real schemes
$\langle 1\sqcup 1(3)\langle 3\rangle\sqcup 1\langle 6\rangle\rangle, \quad \langle 1\sqcup 1\langle 6\rangle\sqcup 1\langle 13\rangle\rangle.$
This is an immediate consequence of Theorem 1.2 (a) below.
We say that a real scheme in $$\mathbb{R}\mathbb{P}^2$$ is realized by a real pseudo-holomorphic curve $$A$$ if it is isotopic in $$\mathbb{R}\mathbb{P}^2$$ to $$\mathbb{R} A= A\cap\mathbb{R}\mathbb{P}^2$$, the set of real points of $$A$$. Here we complete the classification of up to isotopy of the arrangements of 22 ovals on $$\mathbb{R}\mathbb{P}^2$$ realizable by real pseudo-holomorphic $$M$$-curves of degree 8:
Theorem 1.2. (a) The two real schemes in Theorem 1.1. are not realizable by real pseudo-holomorphic curves of degree 8. (b) The other seven real schemes (1) are realizable by real pseudoholomorphic curves of degree 8.

##### MSC:
 14P25 Topology of real algebraic varieties 57M25 Knots and links in the $$3$$-sphere (MSC2010)
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