Singular Hessians

Bondt, Michiel de; Essen, Arno van den

doi:10.1016/j.jalgebra.2004.08.026

Cited by 16 publications

(31 citation statements)

References 9 publications

(21 reference statements)

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“…The main result of the paper is the following. Recently, I was delighted to learn that de Bondt and van den Essen [1,2] found some application of these results, which formed part of my Ph.D. thesis [9], in their study of Jacobian Conjecture.…”

Section: Introductionmentioning

confidence: 91%

“…So keeping in mind that is over k, we get (M) = (M) for every monomial M = X e 1 1 · · · X e n n ( ∈ k, e j ∈ N). Moreover, if f ∈ B\{0} is written as a sum of monomials, f = M j , then…”

Section: Gr D (H ) D (H ) For All Homogeneous Elements H Of Bmentioning

confidence: 99%

“…, X n ), resulting in X 1 ∈ p being a contradiction. • Given any monomial M = X e 1 1 · · · X e n n ( ∈ k, e j ∈ N), p ∩ k[M] = 0; in particular, X j − / ∈ p, for all j and all ∈ k.…”

Section: Gr D (H ) D (H ) For All Homogeneous Elements H Of Bmentioning

confidence: 99%

“…4. Let M be a collection of monomials X e 1 1 · · · X e n n (e j ∈ N), and let N ∈ M. Then the following are equivalent: Proof. We only prove the third statement here, because the rest are more or less well-known.…”

Section: Is Irreducible If and Only If D Is 2 D Has A Slice If Anmentioning

confidence: 99%

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Homogeneization of locally nilpotent derivations and an application to k[X,Y,Z]

Wang¹

2005

Journal of Pure and Applied Algebra

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Let k be a field of characteristic zero and let B be a graded k-algebra. We obtain information on a given derivation D : B → B by studying the behavior of the associated homogeneous derivation gr D.As an application, we give a complete classification of locally nilpotent derivations D :We study locally nilpotent derivations of polynomial rings over k, where k is a field of characteristic zero. Our special interest is in so called "nice" derivations over a polynomial ring k[X 1 , . . . , X n ], which satisfies D 2 X i = 0 for all i. The main result of the paper is the following.Y, Z] and let 0 = D : B → B be a locally nilpotent derivation. If D is irreducible and D 2 X = 0 = D 2 Y , then one of the following holds: 1. There exists a coordinate system (L 1 , L 2 , Z) of B, where L 1 and L 2 are linear forms in X, Y , such that D(L 1 ) = 0, D(L 2 ) ∈ k[L 1 ] and D(Z) ∈ k[L 1 , L 2 ] = k[X, Y ]. *

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Section: Introductionmentioning

confidence: 91%

Section: Gr D (H ) D (H ) For All Homogeneous Elements H Of Bmentioning

confidence: 99%

Section: Gr D (H ) D (H ) For All Homogeneous Elements H Of Bmentioning

confidence: 99%

Section: Is Irreducible If and Only If D Is 2 D Has A Slice If Anmentioning

confidence: 99%

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Homogeneization of locally nilpotent derivations and an application to k[X,Y,Z]

Wang¹

2005

Journal of Pure and Applied Algebra

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“…. , x n ], then x + H with H = ∇R • ∇h is a quasi-translation; see [3] or just the original paper [9]. It is remarkable that P. Gordan and M. Nöther already juggled with nilpotent derivations before derivations were invented.…”

Section: Final Remarksmentioning

confidence: 99%

Quasi-translations and counterexamples to the Homogeneous Dependence Problem

Bondt

2006

Proc. Amer. Math. Soc.

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Abstract. In this article, the author gives counterexamples to the Linear Dependence Problem for Homogeneous Nilpotent Jacobians for dimension 5 and up. This problem has been formulated as a conjecture/problem by several authors in connection to the Jacobian conjecture. In dimension 10 and up, cubic counterexamples are given.In the construction of these counterexamples, the author makes use of socalled quasi-translations, a special type of invertible polynomial maps. Quasitranslations can also be seen as a special type of locally nilpotent derivations.

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On the Theory of Gordan-Noether on Homogeneous Forms with Zero Hessian (Improved Version)

Watanabe¹,

Bondt²

2020

Springer Proceedings in Mathematics &Amp; Statistics

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We give a detailed proof for Gordan-Noether's results in "Ueber die algebraischen Formen, deren Hesse'sche Determinante identisch verschwindet." C. Lossen has written a paper in a similar direction as the present paper, but did not provide a proof for every result. In our paper, every result is proved. Furthermore, our paper is independent of Lossen's paper and includes a considerable number of new observations. An earlier version of this paper, namely [17], has been printed in version, a serious error has been corrected, namely [17, Lemma 5.2]. Lemma 5.2 has been replaced by a weaker statement, and Proposition 6.2 has been weakened along with that. Lemma 5.2 no longer suffices for the proof of Proposition 7.2. For that reason, a new section (Section 8) has been added to complete the proof of Proposition 7.2. Furthermore, several trivial errors have been corrected, and a section with new results has been added (Section 9).

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Singular Hessians

Abstract: Let k be a field of characteristic zero. For small n, we classify all f ∈ k [n] such that the Hessian of f is singular.

Cited by 16 publications

References 9 publications

Homogeneization of locally nilpotent derivations and an application to k[X,Y,Z]

Homogeneization of locally nilpotent derivations and an application to k[X,Y,Z]

Quasi-translations and counterexamples to the Homogeneous Dependence Problem

On the Theory of Gordan-Noether on Homogeneous Forms with Zero Hessian (Improved Version)

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