Parameterizing and Inverting Analytic Mappings with Unit Jacobian

Abstract

Let $x=(x_1,\ldots,x_n)\in {\rm \bf C}^n$ be a vector of complex variables, denote by $A=(a_{jk})$ a square matrix of size $n\geq 2,$ and let $\varphi\in\mathcal{O}(\Omega)$ be an analytic function defined in a nonempty domain $\Omega\subset {\rm \bf C}.$ We investigate the family of mappings $$ f=(f_1,\ldots,f_n):{\rm \bf C}^n\rightarrow {\rm \bf C}^n, \quad f[A,\varphi](x):=x+\varphi(Ax) $$ with the coordinates $$ f_j : x \mapsto x_j + \varphi\left(\sum\limits_{k=1}^n a_{jk}x_k\right), \quad j=1,\ldots,n $$ whose Jacobian is identically equal to a nonzero constant for any $x$ such that all of $f_j$ are well-defined. Let $U$ be a square matrix such that the Jacobian of the mapping $f[U,\varphi](x)$ is a nonzero constant for any $x$ and moreover for any analytic function $\varphi\in\mathcal{O}(\Omega).$ We show that any such matrix $U$ is uniquely defined, up to a suitable permutation similarity of matrices, by a partition of the dimension $n$ into a sum of $m$ positive integers together with a permutation on $m$ elements. For any $d=2,3,\ldots$ we construct $n$-parametric family of square matrices $H(s), s\in {\rm \bf C}^n$ such that for any matrix $U$ as above the mapping $x+\left((U\odot H(s))x\right)^d$ defined by the Hadamard product $U\odot H(s)$ has unit Jacobian. We prove any such mapping to be polynomially invertible and provide an explicit recursive formula for its inverse.