  In 1876 in [8], the authors Paul Gordan and Max N\"other classify all
homogeneous polynomials h in at most five variables for which the Hessian
determinant vanishes. For that purpose, they study quasi-translations which are
associated with singular Hessians.
  We will explain what quasi-translations are and formulate some elementary
properties of them. Additionally, we classify all quasi-translations with
Jacobian rank one and all so-called irreducible homogeneous quasi-translations
with Jacobian rank two. The latter is an important result of [8]. Using these
results, we classify all quasi-translations in dimension at most three and all
homogeneous quasi-translations in dimension at most four.
  Furthermore, we describe the connection of quasi-translation with singular
Hessians, and as an application, we will classify all polynomials in dimension
two and all homogeneous polynomials in dimensions three and four whose Hessian
determinant vanishes. More precisely, we will show that up to linear terms,
these polynomials can be expressed in n-1 linear forms, where n is the
dimension, according to an invalid theorem of Hesse.
  In the last section, we formulate some known results and conjectures in
connection with quasi-translations and singular Hessians.

Algebraic Geometry