Local invertibility of non-archimedean vector-valued functions
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 \(k=2\), \(n=2)\).
scientific article; zbMATH DE number 1187662