A Vanishing Conjecture on Differential Operators with Constant Coefficients

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.