On the number of functions of $k$-valued logic which are polynomials modulo composite $k$

A function of $k$-valued logic is called polynomial if it may be represented by a polynomial modulo $k$. For any composite number $k$ we propose a uniquely defined canonical form of polynomials for polynomial functions of $k$-valued logic depending on an arbitrary number of variables. This canonical form is used to find, for any composite $k$, a formula for the number of $n$-place polynomial functions of $k$-valued logic. As a corollary, for any composite $k$ we find the asymptotic behaviour of the logarithm of the number of $n$-place polynomial functions of $k$-valued logic.