Subcoordinate representation of $p$-adic functions and generalization of Hensel's lemma

In this paper we describe a new representation of $p$-adic functions, the so-called subcoordinate representation. The main feature of the subcoordinate representation of a $p$-adic function is that the values of the function $f$ are given in the canonical form of the representation of $p$-adic numbers. The function $f$ itself is determined by a tuple of $p$-valued functions from the set $\{0,1,\dots,p-1\}$ into itself and by the order in which these functions are used to determine the values of $f$. We also give formulae that enable one to pass from the subcoordinate representation of a $1$-Lipschitz function to its van der Put series representation. The effective use of the subcoordinate representation of $p$-adic functions is illustrated by a study of the feasibility of generalizing Hensel's lemma.