Quasi-translations and singular Hessians. (English) Zbl 1396.14054
At the beginning of the paper, the author introduces quasi-transitions, singular Hessians and Jacobians concretely and proves that Hesse’s theorem is true if \(n\leq 4\) and for non-homogeneous cases, it is true for \(n\leq 2\). Thus, the author classifies all polynomials of dimension 2 and all homogeneous polynomials of dimension 3 and 4 with singular Hessian. Then the author introduces quasi-degree and gives a method to construct new quasi-translation from already existing ones. In Section 5, the author gives a way to construct a homogeneous quasi-translation in dimension \(n+1\) that related to a quasi-translation in dimension \(n\). Then he classifies all quasi-translations with Jacobian rank \(\leq 1\) and the irreducible homogeneous quasi-translations with Jacobian rank \(\leq 2\). As a consequence, the author classifies all quasi-translations with dimension \(\leq 3\) and all homogeneous quasi-transitions with dimension \(\leq 4\). Finally, the author poses three problems that related to quasi-translations and singular Hessians.
Reviewer: Yan Dan (Changsha)
MSC:
| 14R05 | Classification of affine varieties |
| 14R10 | Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) |
| 14R20 | Group actions on affine varieties |
| 13N15 | Derivations and commutative rings |
Keywords:
quasi-translation; Hessian; determinant zero; homogeneous; locally nilpotent derivation; algebraic dependence; linear dependenceReferences:
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