Homogeneous quasi-translations in dimension 5

摘要

We give a proof in modern language of the following result by Paul Gordan and Max N?ther: a homogeneous quasi-translation in dimension 5 without linear invariants would be linearly conjugate to another such quasi-translation $$x + H$$ x + H , for which $$H_5$$ H 5 is algebraically independent over $${mathbb C}$$ C of $$H_1, H_2, H_3, H_4$$ H 1 , H 2 , H 3 , H 4 . Just like Gordan and N?ther, 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 N?ther in $${mathbb P}^4$$ P 4 ’ 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 N?ther did not. We derive some other properties which H would have. One of them is that $$deg H ge 15$$ 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 $${mathbb C}^5$$ C 5 which contains the fifth standard basis unit vector.