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Register a new userPolynomial maps with invertible sums of Jacobian matrices and of directional Derivatives
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Let $F: C^n rightarrow C^m$ be a polynomial map with $degF=d geq 2$. We prove that $F$ is invertible if $m = n$ and $sum^{d-1}_{i=1} JF(alpha_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines $L = {beta + mu gamma | mu in C} subseteq C^n$ ($gamma e 0$), $F|_L$ is linearly rectifiable, if and only if $sum^{d-1}_{i=1} JF(alpha_i) cdot gamma e 0$ for all $alpha_i in L$. This appears to be the case for all affine lines $L$ when $F$ is injective and $d le 3$. We also prove that if $m = n$ and $sum^{n}_{i=1} JF(alpha_i)$ is invertible for all $alpha_i in C^n$, then $F$ is a composition of an invertible linear map and an invertible polynomial map $X+H$ with linear part $X$, such that the subspace generated by ${JH(alpha) | alpha in C^n}$ consists of nilpotent matrices.
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Let $K$ be any field and $x = (x_1,x_2,ldots,x_n)$. We classify all matrices $M in {rm Mat}_{m,n}(K[x])$ whose entries are polynomials of degree at most 1, for which ${rm rk} M le 2$. As a special case, we describe all such matrices $M$, which are the Jacobian matrix $J H$ (the matrix of partial derivatives) of a polynomial map $H$ from $K^n$ to $K^m$. Among other things, we show that up to composition with linear maps over $K$, $M = J H$ has only two nonzero columns or only three nonzero rows in this case. In addition, we show that ${rm trdeg}_K K(H) = {rm rk} J H$ for quadratic polynomial maps $H$ over $K$ such that $frac12 in K$ and ${rm rk} J H le 2$. Furthermore, we prove that up to conjugation with linear maps over $K$, nilpotent Jacobian matrices $N$ of quadratic polynomial maps, for which ${rm rk} N le 2$, are triangular (with zeroes on the diagonal), regardless of the characteristic of $K$. This generalizes several results by others. In addition, we prove the same result for Jacobian matrices $N$ of quadratic polynomial maps, for which $N^2 = 0$. This generalizes a result by others, namely the case where $frac12 in K$ and $N(0) = 0$.
Let $K$ be any field with $textup{char}K eq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ with $textup{rk} JHleq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map then $F$ is invertible, and furthermore $F$ is tame if the dimension $n eq 4$.
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We compute by hand all quadratic homogeneous polynomial maps $H$ and all Keller maps of the form $x + H$, for which ${rm rk} J H = 3$, over a field of arbitrary characteristic. Furthermore, we use computer support to compute Keller maps of the form $x + H$ with ${rm rk} J H = 4$, namely: $bullet$ all such maps in dimension $5$ over fields with $frac12$; $bullet$ all such maps in dimension $6$ over fields without $frac12$. We use these results to prove the following over fields of arbitrary characteristic: for Keller maps $x + H$ for which ${rm rk} J H le 4$, the rows of $J H$ are dependent over the base field.
We formulate explicitly the necessary and sufficient conditions for the local invertibility of a field transformation involving derivative terms. Our approach is to apply the method of characteristics of differential equations, by treating such a transformation as differential equations that give new variables in terms of original ones. The obtained results generalise the well-known and widely used inverse function theorem. Taking into account that field transformations are ubiquitous in modern physics and mathematics, our criteria for invertibility will find many useful applications.
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