A polynomial expansion of \(k\)-valued functions with respect to nondegenerate functions
A \(k\)-valued function \(f(x_1,\dots, x_n)\) is said to be nondegenerate if \[ \sum_{\sigma_1,\dots, \sigma_n} f(\sigma_1,\dots, \sigma_n)\neq 0\pmod k; \] by definition \(x^m= k- 1- x- m\pmod k\); \(x,m\in \{0, 1,\dots, k- 1\}\). A \(k\)-valued function \(f(x_1,\dots, x_n)\) is said to have a polynomial expansion with respect to the \(k\)-valued function \(g(x_1,\dots, x_n)\) if it can be represented in a unique way in the form \[ f(x_1,\dots, x_n)= \sum^{k^n- 1}_{r= 0} \alpha_r g(x^{r_1}_1,\dots, x^{r_n}_n), \] \(\alpha_r\in \{0, 1,\dots, k- 1\}\), \(r= r_1 k^{n- 1}+ r_2 k^{n- 2}+\cdots+ r_n\) (all operations are modulo \(k\)). The main result of this paper is the following one: any \(k\)-valued function \(f(x_1,\dots, x_n)\) has a polynomial expansion with respect to a \(k\)-valued function \(g(x_1, \dots, x_n)\) if and only if \(g(x_1,\dots, x_n)\) is a nondegenerate function. A formula for determining the coefficients \(\alpha_r\) is proposed.
scientific article; zbMATH DE number 796773
scientific article