О взаимных расположениях двух М-кривых степени 4

Итоги науки и техники. Современная математика и ее приложения. Тематические обзоры. 2023. Т. 222. С. 69–82.

Пучкова Н. Д.

We consider the problem of topological classification of mutual dispositions in the real projective plane of two M-curves of degree 4. We studi  arrangements which are subject to the maximality condition (the oval of one of these curves has 16 pairwise different common points with the oval of the other of them) and  some combinatorial condition to select a special type of  such arrangements.  Pairwise different topological models of arrangements of this type are listed, which satisfy the known facts about the topology of nonsingular curves and the topological consequences of Bezout's theorem. There are more than 2000 such models. Examples of curves of degree 8 realizing some of these models are given, and it is proved that 1734 models cannot be realized by curves of degree 8. Proofs of non-realizability are carried out by Orevkov's method based on the theory of braids and links.

Keywords: Orevkov's method, метод Оревкова, plane real algebraic curves, decomposable curves, quasi-positive braids, Murasugi-Tristram inequality, Fox-Milnor condition, Плоские вещественные алгебраические кривые, распадающиеся кривые, квазиположительные косы, неравенство Мурасуги-Тристрама, условие Фокса-Милнора