Multivalued groups and Newton polyhedron

Abstract

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 an right triangle.

7 pages