Geodesic
Multivalued groups and Newton polyhedron
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1590-1596.

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On the set of complex number $\mathbb{C}$ it is possible to define $n$-valued group for any positive integer $n$. The $n$-multiplication defines a symmetric polynomial $p_n = p_n (x, y, z)$ with integer coefficients. By the theorem on symmetric polynomials, one can present $p_n$ as polynomial in elementary symmetric polynomials $e_1$, $e_2$, $e_3$. V. M. Buchstaber formulated a question on description coefficients of this polynomial. Also, he formulated the next question: How to describe the Newton polyhedron of $p_n$? In the present paper we find all coefficients of $p_n$ under monomials of the form $e_1^i e_2^j$ and prove that the Newton polyhedron of $p_n$ is a right triangle.
Keywords: multi-set, multivalued group, symmetric polynomial, Newton polyhedron. 
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V. G. Bardakov; T. A. Kozlovskaya. Multivalued groups and Newton polyhedron. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1590-1596. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a23/

[1] V.M. Buchstaber, “$n$-valued groups: theory and applications”, Mosc. Math. J., 6:1 (2006), 57–84 | DOI | MR | Zbl

[2] V.M. Buchstaber, S.P. Novikov, “Formal groups, power systems and Adams operators”, Mat. Sb., N.Ser., 84(126) (1971), 81–118 | MR | Zbl