Parameterizing and inverting analytic mappings with unit Jacobian

TL;DR

This paper characterizes matrices that produce mappings with constant Jacobian determinants when combined with analytic functions, and constructs polynomially invertible mappings with unit Jacobian using Hadamard products.

Contribution

It uniquely classifies matrices yielding constant Jacobian mappings and introduces a family of polynomially invertible mappings with unit Jacobian via Hadamard products.

Findings

Unique classification of matrices with constant Jacobian mappings.

Explicit recursive formula for inverses of constructed mappings.

Construction of polynomially invertible 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.