论文标题
使用单位Jacobian参数化和反相分析映射
Parameterizing and inverting analytic mappings with unit Jacobian
论文作者
论文摘要
令$ x =（x_1，\ ldots，x_n）\ in {\ rm \ bf c}^n $是复杂变量的向量，用$ a =（a_ {jk}）$表示$ n \ geq 2，$和让$ n in \ in \ mathcal nondrimane nondrimane nondrimane nondrimant in nortion nortion nondirant in nortion nortion nortion nondirant in nortion nondirand in nortion nortion nondirantiand（a） $我们调查了映射$$ f =（f_1，\ ldots，f_n）：{\ rm \ bf c}^n \ rightArolow {\ rm \ rm \ rm \ rm \ rm \ bf c}^n，quad f [quad f [a quad f [a quad f [a，$ f _n）$ f =（f_1，\ ldots，f_n）： $$ f_j：x \ mapsto x_j +φ\ left（\ sum \ limits_ {k = 1}^n a_ {jk} x_k \ right），\ quad j = 1，\ ldots，n $ qul，n $ j jacobian的jacobian的jacobian在$ x $中相同等于$ x $ y的非零$ f _ $。 令$ u $为正方形矩阵，使得映射$ f [u，φ]（x）$的jacobian对于任何$ x $来说都是一个非零的常数，此外，对于任何分析函数$φ\ in \ nathcal {o}（O}（ω））。 $ m $阳性整数的总和以及$ m $元素上的排列。 对于任何$ d = 2,3，\ ldots $我们构建$ n $ - 份量的方形矩阵$ h（s），s \ in {\ rm \ bf c}^n $，以至于对任何矩阵$ u $ y $ as as apping $ x+\ weft（u \ odot h（s）x $ proffice themapping $ x+weft（u \ odot h（s）雅各布单元。我们证明任何此类映射是多项式可逆的，并为其反向提供了明确的递归公式。
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 $φ\in\mathcal{O}(Ω)$ be an analytic function defined in a nonempty domain $Ω\subset {\rm \bf C}.$ We investigate the family of mappings $$ f=(f_1,\ldots,f_n):{\rm \bf C}^n\rightarrow {\rm \bf C}^n, \quad f[A,φ](x):=x+φ(Ax) $$ with the coordinates $$ f_j : x \mapsto x_j + φ\left(\sum\limits_{k=1}^n a_{jk}x_k\right), \quad j=1,\ldots,n $$ 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 $f[U,φ](x)$ is a nonzero constant for any $x$ and moreover for any analytic function $φ\in\mathcal{O}(Ω).$ 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.
