Quadratic polynomial maps with Jacobian rank two

TL;DR

This paper classifies quadratic polynomial maps with Jacobian rank at most two over any field, revealing their structural forms, trace degree properties, and nilpotent Jacobian matrices' triangularity, extending previous results.

Contribution

It provides a comprehensive classification of quadratic polynomial maps with low-rank Jacobians, including their structural forms and nilpotent Jacobian matrices, generalizing prior work across different field characteristics.

Findings

Jacobian matrices with rank ≤ 2 have only two nonzero columns or three nonzero rows.

For quadratic maps with rank ≤ 2, the transcendence degree equals the Jacobian rank.

Nilpotent Jacobian matrices of quadratic maps are conjugate to triangular matrices with zero diagonals.

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 $N(0) = 0$.