On Newton's lemma

The non-Archimedean Newton's lemma on obtaining precise solutions of systems of equations from approximate ones may be described heuristically as the assertion that over a suitable base, an approximate solution to a system of equations which is sufficiently far from the singular locus of the system is close to an actual solution of that system. Versions of this lemma have been proven by several authors, notably, Hensel, Tougeron, Artin and Elkik. In this article, we attempt to introduce some coherence into this topic by explaining the notion of “sufficiently far” by tying it up with a generalization of the infinitesimal lifting property. In particular, we strengthen Elkik's lemma Elkik (1973) [E, Lemma 1], in general and give a new proof of Tougeron's lemma Artin (1969) [A, Lemma 5.11] when the base is a complete local ring.