Describing the closed class of polynomial functions modulo a power of a prime number by a relation

The closed class $\operatorname{Pol}_{p^m}$ in $p^m$-valued logic, where $p$ is a prime number, $1 \leqslant m \leqslant p$, is studied. This class consists of all functions that are polynomial modulo $p^m$. Criteria for polynomiality modulo $p^m$ of a function in $p^m$-valued logic are found. A relation describing the class $\operatorname{Pol}_{p^m}$ is obtained in an explicit form.