Polynomial Hessians with small rank

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 $p_3,p_4,\dots ,p_n\in K[x_1,x_2]$ 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 $th$ 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 $th$ is $4$ and the first row of the Hessian matrix of $th$ is independent of the other rows.

57 pages, introduction added, corrections made

References in corpus (1)

• Homogeneous quasi-translations in dimension 5

Cited by in corpus (1)

• On the theory of Gordan-Noether on homogeneous forms with zero Hessian (Improved version)