Geodesic
On mutual arrangements of two $M$-curves of degree~$4$
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 3, Tome 222 (2023), pp. 69-82

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the problem of topological classification of mutual arrangements in the real projective plane of two $M$-curves of degree $4$. We study arrangements under the maximality condition (the oval of one of these curves has $16$ pairwise distinct common points with the oval of the other curve) and some combinatorial condition to select a special type of such arrangements. We list pairwise different topological models of arrangements of this type that satisfy 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; we prove that 1728 models cannot be realized by curves of degree $8$. Proofs of the nonrealizability are performed out by Orevkov's method based on the theory of braids and links.
Keywords: plane real algebraic curve, quasi-positive braid, Orevkov's method, Murasugi–Tristram inequality, Fox–Milnor condition.
Mots-clés : decomposable curve 
@article{INTO_2023_222_a6,
     author = {N. D. Puchkova},
     title = {On mutual arrangements of two $M$-curves of degree~$4$},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {69--82},
     publisher = {mathdoc},
     volume = {222},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2023_222_a6/}
}
TY  - JOUR
AU  - N. D. Puchkova
TI  - On mutual arrangements of two $M$-curves of degree~$4$
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2023
SP  - 69
EP  - 82
VL  - 222
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2023_222_a6/
LA  - ru
ID  - INTO_2023_222_a6
ER  - 
%0 Journal Article
%A N. D. Puchkova
%T On mutual arrangements of two $M$-curves of degree~$4$
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2023
%P 69-82
%V 222
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2023_222_a6/
%G ru
%F INTO_2023_222_a6
N. D. Puchkova. On mutual arrangements of two $M$-curves of degree~$4$. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 3, Tome 222 (2023), pp. 69-82. http://geodesic.mathdoc.fr/item/INTO_2023_222_a6/