Local invertibility of non-archimedean vector-valued functions
type: Item
instance of: scholarly article
title: Local invertibility of non-archimedean vector-valued functions
zbMATH Open document ID: 0922.46065
author: Stany De Smedt
published in: Annales Mathématiques Blaise Pascal
publication date: 1998-10-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/79194
review text: Let \(E\) be an \(n\)-dimensional non-Archimedean vector space, \(F\) an \(m\)-dimensional non-Archimedean vector space and \(X\) a nonempty subset of \(E\) without isolated points. A function \(f: X\to F\) is said to be continuously differentiable at \(a\), i.e., \(f\) is \(C^1\) at \(a\), if there is a function \(g\) and a linear function \(L\) (depending on \(a\)) such that for all \(x\neq y\in X\): \[ f(x)- f(y)= L(x- y)+ g(x- y)\text{ with } \lim_{(x,y)\to (a,a)} {| g(y- x)|\over| y-x|}= 0. \] The linear function \(L\) is then denoted by \(df_a\) and is called the differential of \(f\) at \(a\). Its matrix \(f'(a)\) is called the derivative of \(f\) at \(a\). Thus \(f'(a)\) is the unique \(m\times n\) matrix such that \(df_a(x)= f'(a)x\) for all \(x\in X\). The function \(f\) is \(C^2\) at \(a\) if there is a linear function \(L_1\) and a bilinear function \(L_2\) (both depending on \(a\)) such that for all \(x\neq y\in X\): \[ f(x)- f(y)= L_1(x- y)+ L_2(x- y)(x- y)+ g(x- y)\text{ with }\lim_{(x,y)\to (a,a)} {| g(y- x)|\over| y-x|^2}= 0. \] A similar definition can be given for \(C^k\)-functions. The following two theorems proved in this paper deal with the local invertibility of non-archimedean vector-valued \(C^1\) and \(C^k\)-function: Theorem 1. Let \(f:E\to E\) be \(C^1\) at \(a\) and suppose \(\det f'(a)\neq 0\), then \(f\) is locally in vertible. The local inverse \(g\) of \(f\) is \(C^1\) at \(f(a)\) and \(g'(f(a))=(f'(a))^{-1}\). Theorem 2. Let \(f:E\to E\) be \(C^k\) at \(a\) and suppose \(\det f'(a)\neq 0\), then \(f\) is locally invertible. The local inverse \(g\) of \(f\) is \(C^k\) at \(f(a)\). (The proof of Theorem 2 is given for the case \(k=2\), \(n=2)\).
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 1187662
zbMATH Keywords: non-Archimedean vector space
zbMATH Keywords: differential
zbMATH Keywords: derivative
reviewed by: T. D. Narang
MaRDI profile type: Publication
OpenAlex ID: W2322342514
cites work: The van der Put base for \(C^ n\)-functions
cites work: \(p\)-adic continuously differentiable functions of several variables
cites work: Q3913621
cites work: Q3346515
DOI: 10.5802/AMBP.102
description: scientific article; zbMATH DE number 1187662