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placesbibalignatcardordanynum dblaccvarrefliterallists =cmsy8 -1000 On Newton’s Lemma

Robert F. Coleman Department of Mathematics University of California Berkeley, CA 94720 USA Harvey J. Stein Department of Mathematics Hebrew University Jerusalem Israel

On Newton’s Lemma

1in Robert F. Coleman Harvey J. Stein

Introduction ”This document was prepared using LAMS-TEX, and thus makes indirect use of . 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 E, Lemma 1 in general and give a new proof of Tougeron’s lemma A, Lemma 5.11 when the base is a complete local ring.

In section 1 we define some variations of the infinitesimal lifting property called the strong infinitesimal lifting property and the weak infinitesimal lifting property. We use them to deduce a multidimensional Newton’s lemma. In sections 2 and 3 we explore these variations in more detail. In section 4, we use our multidimentional Newton’s lemma to deduce Elkik’s lemma and a special case of Tougeron’s lemma. In section 6, we use our Newton’s lemma to deduce a special case of the Greenberg-Shappacher theorem.

We are grateful to H. W. Lenstra Jr. for provoking us to think about this topic. Notation If C is a ring and T is a C-module, we let _{C}T denote the annihilator of T in C. If S is a subset of C, (S) will denote the ideal generated by S in C.

Suppose m and n are sets and C is a ring. Then M_{m×n}(C) will denote the set of matrices over C whose rows are indexed by m and whose columns are indexed by n and are such that all but finitely many entries of any row are zero. Clearly if p is a set there is a natural multiplication from M_{m×n}(C) × M_{n×p}(C) to M_{m×p}(C). We call the elements of M_{m×n}(C) m × n matrices and the elements of M_{n×1}(C) n-column vectorsEDITORIAL NOTE - Check for consistent use of column vectorsnd sometimes n-vectors, and also denote them by M_{n}. If M is a matrix, (M) will denote the ideal of C generated by the entries of M. We let I_{m} denote the m × m identity matrix. In this context, if k is a non-negative integer we will also use k to denote the set {i1 ≤ i ≤ k}EDITORIAL NOTE - Maybe should use {i ∈ 0 ≤ i ≤ k − 1}, since this is the usual set theoretic construction of the integers. A_{n} to denote the ring of restricted power series in {X_{i}i ∈ n}.) Let X denote the

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EDITORIAL NOTE - Check for consistent use of column vectors a

EDITORIAL NOTE - Maybe should use {i ∈ 0 ≤ i ≤ k − 1}, since this is the usual set theoretic construction of the integers.

We call the ring of integers in a complete valued field with non-trivial non-archimedean absolute value ∥ ∥, a complete valued domain. If n is a set, let A_{n} denote the polynomial ring A[X_{i}] where i runs from 1 to n. (If A is a complete valued domain and n is finite we will also allowEDITORIAL NOTE - I’m not sure that the results hold in this case. They certainly don’t hold unless we work in the category of continuous homomorphisms, so we should at least mention this.You made a note here asking whether or not homomorphisms are automatically continuous. Well, since there are elements of A < x > which are transendental over A[x], we can map them independently of where x maps to, can’t we? For example, if A < x >→ A < x > is an A-algebra homomorphism mapping x