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Register a new userClassification of cubic homogeneous polynomial maps with Jacobian matrices of rank two
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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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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 $F: C^n rightarrow C^m$ 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.
In the paper, we first classify all polynomial maps $H$ of the following form: $H=big(H_1(x_1,x_2,ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2),ldots,H_n(x_1,x_2)big)$ with $JH$ nilpotent. After that, we generalize the structure of $H$ to $H=big(H_1(x_1,x_2,ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2,H_1),ldots,H_n(x_1,x_2,H_1)big)$.
We classify all quadratic homogeneous polynomial maps $H$ and Keller maps of the form $x + H$, for which $rk J H = 3$, over a field $K$ of arbitrary characteristic. In particular, we show that such a Keller map (up to a square part if $char K=2$) is a tame automorphism.
In this paper, which is work in progress, the results in [Singular Hessians, J. Algebra 282 (2004), no. 1, 195--204], for polynomial Hessians with determinant zero in small dimensions $r+1$, are generalized to similar results in arbitrary dimension, for polynomial Hessians with rank r. All of this is over a field $K$ of characteristic zero. The results in [Singular Hessians, J. Algebra 282 (2004), no. 1, 195--204] are also reproved in a different perspective. One of these results is the classification by Gordan and Noether of homogeneous polynomials in $5$ variables, for which the Hessians determinant is zero. This result is generalized to homogeneous polynomials in general, for which the Hessian rank is 4. Up to a linear transformation, such a polynomial is either contained in $K[x_1,x_2,x_3,x_4]$, or contained in $$ K[x_1,x_2,p_3(x_1,x_2)x_3+p_4(x_1,x_2)x_4+cdots+p_n(x_1,x_2)x_n] $$ for certain $p_3,p_4,ldots,p_n in K[x_1,x_2]$ which are homogeneous of the same degree. Furthermore, a new result which is similar to those in [Singular Hessians, J. Algebra 282 (2004), no. 1, 195--204], is added, namely about polynomials $h in K[x_1,x_2,x_3,x_4,x_5]$, for which the last four rows of the Hessian matrix of $t h$ are dependent. Here, $t$ is a variable, which is not one of those with respect to which the Hessian is taken. This result is generalized to arbitrary dimension as well: the Hessian rank of $t h$ is $4$ and the last row of the Hessian matrix of $t h$ is independent of the other rows.
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