论文标题
关于身份的适当性和线性力量的一些观察
Some observations on the properness of Identity plus linear powers
论文作者
论文摘要
For $2$ vectors $x,y\in \mathbb{R}^m$, we use the notation $x * y =(x_1y_1,\ldots ,x_my_m)$, and if $x=y$ we also use the notation $x^2=x*x$ and define by induction $x^k=x*(x^{k-1})$.我们在$ \ mathbb {r}^m $上使用$ <，> $。对于$ a $ a $ m \ times m $矩阵，带有$ \ mathbb {r} $系数的系数，我们可以分配一个地图$ f_a（x）= x+（ax）^3：〜\ mathbb {r}^m \ rightArrow \ rightArrow \ rightbb \ mathbb {r}^m $。一个矩阵$ a $是druzkowski iff $ det（jf_a（x））= 1 $ in \ in \ mathbb {r}^m $ in \ mathbb {r}^m $。 最近，江刘发布了关于arxiv的预印本，通过在$ f_a（x）$的适当性中通过实际数字中的某些不平等现象显示出$ f_a（x）$的证明。在证据中，刘的确断言$ f_a（x）$在$ a $上的$ f_a（x）$，请参阅本文的主体，以获取更多详细信息。 受此预印本的启发，我们在本文中研究了以上映射$ f_a（x）$的问题（即使对于不是Druzkowski的矩阵$ a $），这是正确的。我们获得了适当性和非专业性质的各种必要条件和足够条件。在获得$ 3 $的多项式方程系统的非零解决方案的存在方面，如果获得了$ 3 $的情况，则适用于适当性的完整表征，如果获得了$ a $具有corank $ 1 $的情况。扩展这一点，我们提出了一个新的猜想，并将某些应用程序与（实际的）雅各布猜想讨论。我们还考虑了更通用的地图$ x \ pm（ax）^k $或$ x \ pm a（x^k）$的适当性。由于Druzkowski的结果，我们的结果可以应用于$ \ Mathbb {C}^M $或$ \ Mathbb {r}^m $的所有多项式自我映射。
For $2$ vectors $x,y\in \mathbb{R}^m$, we use the notation $x * y =(x_1y_1,\ldots ,x_my_m)$, and if $x=y$ we also use the notation $x^2=x*x$ and define by induction $x^k=x*(x^{k-1})$. We use $<,>$ for the usual inner product on $\mathbb{R}^m$. For $A$ an $m\times m$ matrix with coefficients in $\mathbb{R}$, we can assign a map $F_A(x)=x+(Ax)^3:~\mathbb{R}^m\rightarrow \mathbb{R}^m$. A matrix $A$ is Druzkowski iff $det(JF_A(x))=1$ for all $x\in \mathbb{R}^m$. Recently, Jiang Liu posted a preprint on arXiv asserting a proof of the Jacobian conjecture, by showing the properness of $F_A(x)$ when $A$ is Druzkowski, via some inequalities in the real numbers. In the proof, indeed Liu asserted the properness of $F_A(x)$ under more general conditions on $A$, see the main body of this paper for more detail. Inspired by this preprint, we research in this paper on the question of to what extend the above maps $F_A(x)$ (even for matrices $A$ which are not Druzkowski) can be proper. We obtain various necessary conditions and sufficient conditions for both properness and non-properness properties. A complete characterisation of the properness, in terms of the existence of non-zero solutions to a system of polynomial equations of degree at most $3$, in the case where $A$ has corank $1$, is obtained. Extending this, we propose a new conjecture, and discuss some applications to the (real) Jacobian conjecture. We also consider the properness of more general maps $x\pm (Ax)^k$ or $x\pm A(x^k)$. By a result of Druzkowski, our results can be applied to all polynomial self-mappings of $\mathbb{C}^m$ or $\mathbb{R}^m$.
