Wenhua Zhao

Hessian nilpotent polynomials and the Jacobian conjecture

Let z = (z 1 ,···,z n ) and let A = Σ n i=1 ∂ 2 ∂z 2 i be the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to what we call the vanishing conjecture: for any homogeneous polynomial P(z) of degree d = 4, if Δ m P m (z) = 0 for all m ≥ 1, then Δ m P m+1 (z) = 0 when m > > 0, or equivalently, Δ m P m+1 (z) = 0 when m > 3 2(3 n-2 -1). It is also shown in this paper that the condition Δ m P m (z) = 0 (m ≥ 1) above is equivalent to the condition that P(z) is Hessian nilpotent, i.e. the Hessian matrix Hes P(z) = (∂ 2 P ∂z i ∂z j ) is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived.

D-log and formal flow for analytic isomorphisms of n-space

Given a formal map F = (F1 . . . , Fn) of the form z + higher order terms, we give tree expansion formulas and as- sociated algorithms for the D-Log of F and the formal flow Ft. The coefficients which appear in these formulas can be viewed as certain generalizations of the Bernoulli numbers and the Bernoulli polynomials. Moreover the coefficient polynomials in the formal flow formula coincide with the strict order polynomials in combina- torics for the partially ordered sets induced by trees. Applications of these formulas to the Jacobian Conjecture are discussed.

A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

Abstract In this paper, we construct explicitly a noncommutative symmetric (

Exponential formulas for the Jacobians and Jacobian matrices of analytic maps

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Some open problems on locally finite or locally nilpotent derivations and ℰ-derivations

CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the

The Gaussian Moments Conjecture and the Jacobian Conjecture

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SEMI-INFINITE FORMS AND TOPOLOGICAL VERTEX OPERATOR ALGEBRAS

CS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficients of the third generating function of the constructed

NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND THE INVERSION PROBLEM

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A generalization of Mathieu subspaces to modules of associative algebras

CS system with the order polynomials of rooted trees is also given and proved.

A family of invariants of rooted forests

Abstract Let F =( F 1 , F 2 ,…, F n ) be an n -tuple of formal power series in n variables of the form F ( z )= z + O (| z | 2 ). It is known that there exists a unique formal differential operator A= ∑ i=1 n a i (z)∂/∂z i such that F ( z )=exp( A ) z as formal series. In this article, we show the Jacobian J (F) and the Jacobian matrix J ( F ) of F can also be given by some exponential formulas. Namely, J (F)= exp (A+▽A)·1 , where ▽A(z)= ∑ i=1 n (∂a i /∂z i )(z) , and J ( F )=exp( A + R Ja )· I n × n , where I n × n is the identity matrix and R Ja is the multiplication operator by Ja for the right. As an immediate consequence, we get an elementary proof for the known result that J (F)≡1 if and only if ▽ A =0. Some consequences and applications of the exponential formulas as well as their relations with the well-known Jacobian Conjecture are also discussed.