Generalization of Hensel's lemma: Finding the roots of p-adic Lipschitz functions

• We present generalization of Hensel lemma for nonpolynomial and nonsmooth p-adic functions.
• The novel method of finding of roots of p-adic functions is applicable in the class of Lipschitz function.
• An analog of the p-adic Newton method is used to find algorithmically approximate solutions.

In this paper we consider the problem of finding the roots of p-adic functions. In the case, where the function is defined by a polynomial with integer p-adic coefficients, using Hensel's lifting lemma helps us find the roots of the p-adic function.We generalize Hensel's lifting lemma for a wider class of p  -adic functions, namely, the functions which satisfy the Lipschitz condition with constant pα,α≥0, in particular, the functions of this class may be non-differentiable. The paper also presents an iterative procedure for finding approximate (in p  -adic metric) values of the root of pαpα-Lipschitz functions, thus generalizing the p-adic analogue of Newton's method for such a class of functions.