Ultrametric Smale's $\alpha$-theory

TL;DR

This paper extends Smale's $eta$-theory to ultrametric fields, providing a multivariate version of Hensel's lemma applicable to $p$-adic and similar fields.

Contribution

It introduces a novel ultrametric version of Smale's $eta$-theory, generalizing Hensel's lemma to multivariate cases over ultrametric fields.

Findings

Established a multivariate ultrametric $eta$-theory

Generalized Hensel's lemma for multivariate polynomials

Applicable to $p$-adic and other ultrametric fields

Abstract

We present a version of Smale's $\alpha$-theory for ultrametric fields, such as the $p$-adics and their extensions, which gives us a multivariate version of Hensel's lemma.