Homogeneous quasi-translations in dimension 5

TL;DR

This paper revisits a classical result on homogeneous quasi-translations in five dimensions, providing modern proofs and applying it to classify polynomials with zero Hessian determinant, while clarifying previous assumptions and extending properties.

Contribution

It offers a modern proof of Gordan and N"other's result, clarifies assumptions in prior classifications, and derives new properties of quasi-translations in dimension five.

Findings

Homogeneous quasi-translations in dimension 5 are linearly conjugate to a form with algebraically independent components.

The degree of the polynomial component H is at least 15.

The Zariski closure of H's image is an irreducible component of V(H), with other components being 3-dimensional linear subspaces.

Abstract

We give a proof in modern language of the following result by Paul Gordan and Max N\"other: a homogeneous quasi-translation in dimension $5$ without linear invariants would be linearly conjugate to another such quasi-translation $x + H$, for which $H_5$ is algebraically independent over $\mathbb C$ of $H_1, H_2, H_3, H_4$. Just like Gordan and N\"other, we apply this result to classify all homogeneous polynomials $h$ in $5$ indeterminates from which the Hessian determinant is zero. Others claim to have reproved 'the result of Gordan and N\"other in $\mathbb P^4$' as well, but some of them assume that $h$ is irreducible, which Gordan and N\"other did not. Furthermore, they do not use the above result about homogeneous quasi-translations in dimension $5$ for their classifications. (There is however one paper which could use this result very well, to fix a gap caused by an error.) We derive some other properties which $H$ would have. One of them is that ${\rm deg}\, H \ge 15$, for which we give a proof which is less computational than another proof of it by Dayan Liu. Furthermore, we show that the Zariski closure of the image of $H$ would be an irreducible component of $V(H)$, and prove that every other irreducible component of $V(H)$ would be a $3$-dimensional linear subspace of $\mathbb C^5$ which contains the fifth standard basis unit vector.