Polynomial Hessians with small rank

TL;DR

This paper generalizes previous results on polynomial Hessians with zero determinant to arbitrary dimensions, classifies polynomials with specific Hessian ranks, and introduces new related results over fields of characteristic zero.

Contribution

It extends known classifications of polynomials with small Hessian rank to higher dimensions and provides new insights into the structure of such polynomials and their Hessian matrices.

Findings

Classification of polynomials with Hessian rank 4 in arbitrary dimensions

Generalization of Gordan-Noether's result to homogeneous polynomials

New results on polynomials with dependent Hessian rows involving an extra variable

Abstract

In this paper, 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 first row of the Hessian matrix of $t h$ is independent of the other rows.