Hessian Nilpotent Formal Power Series and Their Deformed Inversion Pairs

Let P (z) be a formal power series in z = (z1, · · · , zn) with o(P (z)) ≥ 2 and t a formal parameter which commutes with z. We say P (z) is HN (Hessian nilpotent) if its Hessian matrix HesP (z) = ( ∂ 2 P ∂zi∂zj ) is nilpotent. The deformed inversion pair Qt(z) of P (z) by definition is the unique Qt(z) ∈ C[[z, t]] with o(Qt(z)) ≥ 2 such that the formal maps Gt(z) = z + t∇Q(z) and Ft(z) = z − t∇P (z) are inverse to each other. In this paper, for HNS (Hessian nilpotent power series) P (z), we first derive the PDE’s satisfied by Qt(z), {∆Qt |k, m ≥ 1} and exp(sQt(z)) (s ∈ C), where ∆ = ∑n i=1 ∂ 2 ∂z i is the Laplace operator. We then prove a uniform formula for Qkt (z) (k ≥ 1) and give a criterion for Hessian nilpotency of formal power series. By using a fundamental theorem on harmonic polynomials, we also give a criterion for Hessian nilpotency of homogeneous harmonic polynomials. Some identities, vanishing properties and isotropic properties of {∆kPm(z)|m, k ≥ 0} for HNS or HNP’s (Hessian nilpotent polynomials) P (z) are also proved. Finally, we reveal some close relationships of the deformed inversion pairs Qt(z) of HNS or HNP’s with the Heat equation and the Jacobian conjecture. In particular, by using the gradient reduction obtained by M. de Bondt, A. van den Essen [BE1] and G. Meng [M] for the Jacobian conjecture, we show that the Jacobian conjecture is equivalent to the following what we call vanishing conjecture: for any homogeneous HNP P (z) of degree d = 4, one should have ∆P(z) = 0 when m > 3 2 (3 − 1).