A Vanishing Conjecture on Differential Operators with Constant Coefficients

Abstract

In the recent progress [BE1], [Me] and [Z2], the well-known JC (Jacobian conjecture) ([BCW], [E]) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent). In this paper, we first show that the vanishing conjecture above, hence also the JC, is equivalent to a vanishing conjecture for all 2nd order homogeneous differential operators $\Lambda$ and $\Lambda$-nilpotent polynomials $P$ (the polynomials $P(z)$ satisfying ${\Lambda }^m{P}^m=0$ for all $m\ge 1$). We then transform some results in the literature on the JC, HN polynomials and the VC of the Laplace operators to certain results on $\Lambda$-nilpotent polynomials and the associated VC for 2nd order homogeneous differential operators $\Lambda$. This part of the paper can also be read as a short survey on HN polynomials and the associated VC in the more general setting. Finally, we discuss a still-to-be-understood connection of $\Lambda$-nilpotent polynomials in general with the classical orthogonal polynomials in one or more variables. This connection provides a conceptual understanding for the isotropic properties of homogeneous $\Lambda$-nilpotent polynomials for the 2nd order homogeneous full rank differential operators $\Lambda$ with constant coefficients.

Latex, 32 pages. Following the referee, some minor improvements have been done. Appeared in the proceeding of the joint international conference "Polynomial Automorphisms and related topics" (Hanoi, Vietnam, 2006) of the Institute of Mathematics, Hanoi, Vietnam, and the Adus Salam International Centreof Theoretical Physics (ICTP), Trieste, Italy

References in corpus (2)

• Two Results on Homogeneous Hessian Nilpotent Polynomials
• Some Properties of and Open Problems on Hessian Nilpotent Polynomials

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• Reconstruction of Planar Domains from Partial Integral Measurements
• Some Results on the Vanishing Conjecture of Differential Operators with Constant Coefficients
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• A Generalization of Mathieu Subspaces to Modules of Associative Algebras
• Analogue of the Duistermaat-van der Kallen Theorem for Group Algebras
• An introduction to Mathieu subspaces
• Mathieu Subspaces of Univariate Polynomial Algebras
• New Proofs for the Abhyankar-Gurjar Inversion Formula and the Equivalence of the Jacobian Conjecture and the Vanishing Conjecture
• A few remarks on the Generalized Vanishing Conjecture
• A Deformation of Commutative Polynomial Algebras in Even Numbers of Variables
• Remarks on Mathieu-Zhao Subspaces of Commutative Associative Algebras and Vertex Algebras