Quadratic polynomial maps with Jacobian rank two

Abstract

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 $N(0) = 0$.