On the growth function of \(n\)-valued dynamics (Q6569615)

Let's introduce an \(n\)-valued product using the following rule over the field \N\[\Nx\ast y=\left[(\sqrt[n]{x}+\varepsilon^r\sqrt[n]{y})^n,r=1,\dots,n\right],\quad \varepsilon=e^{2\pi i/n}.\N\]\NConsider a function \(T:\mathbb{C}\longrightarrow\mathbb{C}^n\) that defines \(n\)-valued dynamics in the following way: \(T(y)=x\ast y\), for some \(x\in\mathbb{C}\). Then it is possible to define \(T^k(y)\) in a natural way as a result of application of the function \(T\) to all elements of \(T^{k-1}(y)\). E.g., \(T^2(y)=x\ast(x\ast y)\). This paper answers the question of \textit{V. M. Buchstaber} [Mosc. Math. J. 6, No. 1, 57--84 (2006; Zbl 1129.20045)] on the growth function in case of certain \(n\)-valued group. More precisely, the question is how many different elements are there in the set \(\{0\}\cup T(0)\cup\cdots\cup T^k(0)\)? This question is in close relation to specific discrete integrable systems. \N\NIn the present paper, the author finds a specific formula for the growth function in the case of prime \(n\). He also proves a polynomial asymptotic estimate of the growth function in the general case. At the end, he poses new conjectures and questions regarding growth functions. This paper is organized as follows. Section 1 is an introduction to the subject. Section 2 introduces necessary definitions to describe dynamics \(T(y)=x\ast y\). In Section 3, the author proves an explicit formula for the function of growth in the case of prime \(n\). In Section 4, some results of numerical calculations of growth polynomials are described. Finally, in Section 5 the author expresses some new conjectures and questions.