Symmetric Jacobians

Bondt, Michiel de

doi:10.2478/s11533-013-0393-7

Cited by 2 publications

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“…We say that f K x y → K x y is an -morphism, if f = f . Note that the symmetry determined by is not any of the symmetries analyzed in [5] or [11].…”

Section: An Analogue Results For the Jacobian Conjecturementioning

confidence: 97%

The Starred Dixmier Conjecture forA₁

Moskowicz

Valqui²

2015

Communications in Algebra

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Let A 1 K = K X Y YX − XY = 1 be the first Weyl algebra over a characteristic zero field K, and let be the exchange involution on A 1 K given by X = Y and Y = X. The Dixmier conjecture of Dixmier (1968) asks the following question: Is every algebra endomorphism of the Weyl algebra A 1 K an automorphism? The aim of this paper is to prove that each -endomorphism of A 1 K is an automorphism. Here an -endomorphism of A 1 K is an endomorphism which preserves the involution . We also prove an analogue result for the Jacobian conjecture in dimension 2, called − JC 2 .

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“…We say that f K x y → K x y is an -morphism, if f = f . Note that the symmetry determined by is not any of the symmetries analyzed in [5] or [11].…”

Section: An Analogue Results For the Jacobian Conjecturementioning

confidence: 97%

The Starred Dixmier Conjecture forA₁

Moskowicz

Valqui²

2015

Communications in Algebra

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Polynomials with constant Hessian determinants in dimension three

Bondt

2015

Journal of Pure and Applied Algebra

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In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n ≤ 3 over a field K of characteristic zero. We do this by extending the following result for n ≤ 2 by F. Dillen to n ≤ 3: if f is a polynomial of degree larger than two in n ≤ 3 variables such that the Hessian determinant of f is constant, then after a suitable linear transformation (replacing f by f (T x) for some T ∈ GLn(K)), the Hessian matrix of f becomes zero below the anti-diagonal. The result does not hold for larger n.The proof of the case det Hf ∈ K * is based on the following result, which in turn is based on the already known case det Hf = 0: if f is a polynomial in n ≤ 3 variables such that det Hf = 0, then after a suitable linear transformation, there exists a positive weight function w on the variables such that the Hessian determinant of the w-leading part of f is nonzero. This result does not hold for larger n either (even if we replace 'positive' by 'nontrivial' above).In the last section, we show that the Jacobian conjecture holds for gradient maps over the reals whose linear part is the identity map, by proving that such gradient maps are translations (i.e. have degree 1) if they satisfy the Keller condition. We do this by showing that this problem is the polynomial case of the main result of [Pog]. For polynomials in dimension n ≤ 3, we generalize this result to arbitrary fields of characteristic zero.

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Symmetric Jacobians

Cited by 2 publications

References 10 publications

The Starred Dixmier Conjecture forA₁

The Starred Dixmier Conjecture forA₁

Polynomials with constant Hessian determinants in dimension three

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Symmetric Jacobians

Cited by 2 publications

References 10 publications

The Starred Dixmier Conjecture forA1

The Starred Dixmier Conjecture forA1

Polynomials with constant Hessian determinants in dimension three

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The Starred Dixmier Conjecture forA₁

The Starred Dixmier Conjecture forA₁