2015/01/31 by Michiel de Bondt
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Combinatorics #Dimension (graph theory) #Homogeneous #Mathematics #Pure mathematics #math.AG #msc:14R05 #msc:14R10 #msc:14R20
paper · pdf · doi:10.1007/s13366-017-0358-2
34 pages
openalex publication_date 2017/10/13 · arxiv created 2017/10/18 · arxiv updated 2017/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We give a proof in modern language of the following result by Paul Gordan and Max Nöther: a homogeneous quasi-translation in dimension 5 without linear invariants would be linearly conjugate to another such quasi-translation x + H , for which H5 is algebraically independent over \mathbb C of H1, H2, H3, H4 . Just like Gordan and Nöther, we apply this result to classify all homogeneous polynomials h in 5 indeterminates, for which the Hessian determinant is zero. Others claim to have reproved ‘the result of Gordan and Nöther in \mathbb P4 ’ as well, but their proofs have gaps, which can be fixed by using the above result about homogeneous quasi-translations. Furthermore, some of the proofs assume that h is irreducible, which Gordan and Nöther did not. We derive some other properties which H would have. One of them is that deg H ≥ 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 C5 which contains the fifth standard basis unit vector.