arXiv subjects

Wenhua Zhao

Publications and source records attributed to Wenhua Zhao.

51 records · Page 3Linked to original sources

arXiv2007

A Noncommutative Symmetric System over the Grossman-Larson Hopf Algebra of Labeled Rooted Trees

In this paper, we construct explicitly a noncommutative symmetric (${\mathcal N}$CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the ${\mathcal N}$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 ${\mathcal N}$CS system with the order polynomials of rooted trees is also given and proved.

math.CO↗

arXiv2007

Noncommutative Symmetric Systems over Associative Algebras

This paper is the first of a sequence papers ([Z4]--[Z7]) on the {\it ${\mathcal N}$CS $(\text{noncommutative symmetric})$ systems} over differential operator algebras in commutative or noncommutative variables ([Z4]); the ${\mathcal N}$CS systems over the Grossman-Larson Hopf algebras ([GL],[F]) of labeled rooted trees ([Z6]); as well as their connections and applications to the inversion problem ([BCW],[E4]) and specializations of NCSFs ([Z5],[Z7]). In this paper, inspired by the seminal work [GKLLRT] on NCSFs (noncommutative symmetric functions), we first formulate the notion {\it ${\mathcal N}$CS systems} over associative $\mathbb Q$-algebras. We then prove some results for ${\mathcal N}$CS systems in general; the ${\mathcal N}$CS systems over bialgebras or Hopf algebras; and the universal ${\mathcal N}$CS system formed by the generating functions of certain NCSFs in [GKLLRT]. Finally, we review some of the main results that will be proved in the followed papers [Z4], [Z6] and [Z7] as some supporting examples for the general discussions given in this paper.

math.CO↗

arXiv2007

Two Results on Homogeneous Hessian Nilpotent Polynomials

Let $z=(z_1, ..., z_n)$ and $Δ=\sum_{i=1}^n \frac {\partial^2}{\partial z^2_i}$ the Laplace operator. A formal power series $P(z)$ is said to be {\it Hessian Nilpotent}(HN) if its Hessian matrix $\Hes P(z)=(\frac {\partial^2 P}{\partial z_i\partial z_j})$ is nilpotent. In recent developments in [BE1], [M] and [Z], the Jacobian conjecture has been reduced to the following so-called {\it vanishing conjecture}(VC) of HN polynomials: {\it for any homogeneous HN polynomial $P(z)$ $($of degree $d=4$$)$, we have $Δ^m P^{m+1}(z)=0$ for any $m>>0$.} In this paper, we first show that, the VC holds for any homogeneous HN polynomial $P(z)$ provided that the projective subvarieties ${\mathcal Z}_P$ and ${\mathcal Z}_{σ_2}$ of $\mathbb C P^{n-1}$ determined by the principal ideals generated by $P(z)$ and $σ_2(z):=\sum_{i=1}^n z_i^2$, respectively, intersect only at regular points of ${\mathcal Z}_P$. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps $F=z-\nabla P$ with $P(z)$ HN if $F$ has no non-zero fixed point $w\in \mathbb C^n$ with $\sum_{i=1}^n w_i^2=0$. Secondly, we show that the VC holds for a HN formal power series $P(z)$ if and only if, for any polynomial $f(z)$, $Δ^m (f(z)P(z)^m)=0$ when $m>>0$.

math.AG↗

arXiv2007

Deformations and Inversion Formulas For Formal Automorphisms in Noncommutative Variables

Let $z=(z_1, z_2, ..., z_n)$ be noncommutative free variables and $t$ a formal parameter which commutes with $z$. Let $k$ be any unital integral domain of any characteristic and $F_t(z)=z-H_t(z)$ with $H_t(z)\in {k[[t]]< < z >>}^{\times n}$ and the order $o(H_t(z))\geq 2$. Note that $F_t(z)$ can be viewed as a deformation of the formal map $F(z):=z-H_{t=1}(z)$ when it makes sense (for example, when $H_t(z)\in {k[t]< < z >>}^{\times n}$). The inverse map $G_t(z)$ of $F_t(z)$ can always be written as $G_t(z)=z+M_t(z)$ with $M_t(z)\in {k[[t]]< < z >>}^{\times n}$ and $o(M_t(z))\geq 2$. In this paper, we first derive the PDE's satisfied by $M_t(z)$ and $u(F_t), u(G_t)\in {k[[t]]< < z >>}$ with $u(z)\in {k< < z >>}$ in the general case as well as in the special case when $H_t(z)=tH(z)$ for some $H(z)\in {k< < z >>}^{\times n}$. We also show that the formal power series above are actually characterized by certain Cauchy problems of these PDE's. Secondly, we apply the derived PDE's to prove a recurrent inversion formula for formal maps in noncommutative variables. Finally, for the case char. $k=0$, we derive an expansion inversion formula by the planar binary rooted trees.

math.GM↗

arXiv2005

Noncommutative Symmetric Functions and the Inversion Problem

Let $K$ be any unital commutative $\bQ$-algebra and $z=(z_1, z_2, ..., z_n)$ commutative or noncommutative variables. Let $t$ be a formal central parameter and $\kttzz$ the formal power series algebra of $z$ over $K[[t]]$. In \cite{GTS-II}, for each automorphism $F_t(z)=z-H_t(z)$ of $\kttzz$ with $H_{t=0}(z)=0$ and $o(H(z))\geq 1$, a \cNcs (noncommutative symmetric) system (\cite{GTS-I}) $\Oft$ has been constructed. Consequently, we get a Hopf algebra homomorphism $\cSft: \cNsf \to \cDzz$ from the Hopf algebra $\cNsf$ (\cite{G-T}) of NCSF's (noncommutative symmetric functions). In this paper, we first give a list for the identities between any two sequences of differential operators in the \cNcs system $\Oft$ by using some identities of NCSF's derived in \cite{G-T} and the homomorphism $\cSft$. Secondly, we apply these identities to derive some formulas in terms of differential operator in the system $\Oft$ for the Taylor series expansions of $u(F_t)$ and $u(F_t^{-1})$ $(u(z)\in \kttzz)$; the D-Log and the formal flow of $F_t$ and inversion formulas for the inverse map of $F_t$. Finally, we discuss a connection of the well-known Jacobian conjecture with NCSF's.

math.CV↗

arXiv2005

NCS Systems over Differential Operator Algebras and the Grossman-Larson Hopf Algebras of Labeled Rooted Trees

Let $K$ be any unital commutative $\bQ$-algebra and $W$ any non-empty subset of $\bN^+$. Let $z=(z_1, ..., z_n)$ be commutative or noncommutative free variables and $t$ a formal central parameter. % Denote uniformly by $\kzz$ and $\kttzz$ the formal power series algebras % of $z$ over $K$ and $K[[t]]$, respectively. Let $\cDazz$ $(α\geq 1)$ be the unital algebra generated by the differential operators of $\kzz$ which increase the degree in $z$ by at least $α-1$ and $ \ataz $ the group of automorphisms $F_t(z)=z-H_t(z)$ of $\kttzz$ with $o(H_t(z))\geq α$ and $H_{t=0}(z)=0$. First, we study a connection of the \cNcs systems $Ω_{F_t}$ $(F_t\in \ataz)$ (\cite{GTS-I}, \cite{GTS-II}) over the differential operators algebra $\cDazz$ and the \cNcs system $Ω_\bT^W$ (\cite{GTS-IV}) over the Grossman-Larson Hopf algebra $\cH_{GL}^W$ (\cite{GL}, \cite{F1}, \cite{F2}) of $W$-labeled rooted trees. We construct a Hopf algebra homomorphism $\mathcal A_{F_t}: \cH_{GL}^W \to \cDazz$ $(F_t\in \ataz)$ such that $\mathcal A_{F_t}^{\times 5}(Ω_\bT^W) =Ω_{F_t}$. Secondly, we generalize the tree expansion formulas for the inverse map (\cite{BCW}, \cite{Wr3}), the D-Log and the formal flow (\cite{WZ}) of $F_t$ in the commutative case to the noncommutative case. Thirdly, we prove the injectivity of the specialization $\cT:{\mathcal N}Sym \to \cH_{GL}^{\bN^+}$ (\cite{GTS-IV}) of NCSF's (noncommutative symmetric functions) (\cite{G-T}). Finally, we show the family of the specializations $\cS_{F_t}$ of NCSF's with all $n\geq 1$ and the polynomial automorphisms $F_t=z-H_t(z)$ with $H_t(z)$ homogeneous and the Jacobian matrix $JH_t$ strictly lower triangular can distinguish any two different NCSF's. The graded dualized versions of the main results above are also discussed.

math.CO↗

arXiv2004

Hessian Nilpotent Polynomials and the Jacobian Conjecture

Let $z=(z_1, ..., z_n)$ and $Δ=\sum_{i=1}^n \fr {\p^2}{\p z^2_i}$ 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 the following what we call {\it vanishing conjecture}: for any homogeneous polynomial $P(z)$ of degree $d=4$, if $Δ^m P^m(z)=0$ for all $m \geq 1$, then $Δ^m P^{m+1}(z)=0$ when $m>>0$, or equivalently, $Δ^m P^{m+1}(z)=0$ when $m> \fr 32 (3^{n-2}-1)$. It is also shown in this paper that the condition $Δ^m P^m(z)=0$ ($m \geq 1$) above is equivalent to the condition that $P(z)$ is Hessian nilpotent, i.e. the Hessian matrix $\Hes P(z)=(\fr {\p^2 P}{\p z_i\p z_j})$ is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen \cite{BE1} 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.

math.CV↗

arXiv2004

Inversion Problem, Legendre Transform and Inviscid Burgers' Equations

Let $F(z)=z-H(z)$ with order $o(H(z))\geq 1$ be a formal map from $\bC^n$ to $\bC^n$ and $G(z)$ the formal inverse map of $F(z)$. We first study the deformation $F_t(z)=z-tH(z)$ of $F(z)$ and its formal inverse $G_t(z)=z+tN_t(z)$. (Note that $G_{t=1}(z)=G(z)$ when $o(H(z))\geq 2$.) We show that $N_t(z)$ is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for $G_t(z)$. Secondly, motivated by the gradient reduction obtained by M. de Bondt, A. van den Essen \cite{BE1} and G. Meng \cite{M} for the Jacobian conjecture, we consider the formal maps $F(z)=z-H(z)$ satisfying the gradient condition, i.e. $H(z)=\nabla P(z)$ for some $P(z)\in \bC[[z]]$ of order $o(P(z))\geq 2$. We show that, under the gradient condition, $N_t(z)=\nabla Q_t(z)$ for some $Q_t(z)\in \bC[[z, t]]$ and the PDE satisfied by $N_t(z)$ becomes the $n$-dimensional inviscid Burgers' equation, from which a recurrent formula for $Q_t(z)$ also follows. Furthermore, we clarify some close relationships among the inversion problem, Legendre transform and the inviscid Burgers' equations. In particular the Jacobian conjecture is reduced to a problem on the inviscid Burgers' equations. Finally, under the gradient condition, we derive a binary rooted tree expansion inversion formula for $Q_t(z)$. The recurrent inversion formula and the binary rooted tree expansion inversion formula derived in this paper can also be used as computational algorithms for solutions of certain Cauchy problems of the inviscid Burgers' equations and Legendre transforms of the power series $f(z)$ of $o(f(z))\geq 2$.

math.CV↗

arXiv2004

Recurrent Inversion Formulas

Let $F(z)=z-H(z)$ with $o(H(z))\geq 2$ be a formal map from $\bC^n$ to $\bC^n$ and $G(z)$ the formal inverse of $F(z)$. In this paper, we fist study the deformation $F_t(z)=z-tH(z)$ and its formal inverse map $G_t(z)$. We then derive two recurrent formulas for the formal inverse $G(z)$. The first formula in certain situations provides a more efficient method for the calculation of $G(z)$ than other well known inversion formulas. The second one is differential free but only works when $H(z)$ is homogeneous of degree $d\geq 2$. Finally, we reveal a close relationship of the inversion problem with a Cauchy problem of a PDE. When the Jacobian matrix $JF(z)$ is symmetric, the PDE coincides with the $n$-dimensional inviscid Burgers' equation in Diffusion theory.

math.CV↗

arXiv2003

A New Approach to Order Polynomials of Labeled Posets and Their Generalizations

In this paper, we first give formulas for the order polynomial $Ω(\Pw; t)$ and the Eulerian polynomial $e(\Pw; λ)$ of a finite labeled poset $(P, ω)$ using the adjacency matrix of what we call the $ω$-graph of $(P, ω)$. We then derive various recursion formulas for $Ω(\Pw; t)$ and $e(\Pw; λ)$ and discuss some applications of these formulas to Bernoulli numbers and Bernoulli polynomials. Finally, we give a recursive algorithm using a single linear operator on a vector space. This algorithm provides a uniform method to construct a family of new invariants for labeled posets $(\Pw)$, which includes the order polynomial $Ω(\Pw; t)$ and the invariant $\tilde e(\Pw; λ) =\frac {e(\Pw; λ)}{(1-λ)^{|P|+1}}$. The well-known quasi-symmetric function invariant of labeled posets and a further generalization of our construction are also discussed.

math.CO↗

arXiv2003

A Family of Invariants of Rooted Forests

Let $A$ be a commutative $k$-algebra over a field of $k$ and $Ξ$ a linear operator defined on $A$. We define a family of $A$-valued invariants $Ψ$ for finite rooted forests by a recurrent algorithm using the operator $Ξ$ and show that the invariant $Ψ$ distinguishes rooted forests if (and only if) it distinguishes rooted trees $T$, and if (and only if) it is {\it finer} than the quantity $α(T)=|\text{Aut}(T)|$ of rooted trees $T$. We also consider the generating function $U(q)=\sum_{n=1}^\infty U_n q^n$ with $U_n =\sum_{T\in \bT_n} \frac 1{α(T)} Ψ(T)$, where $\bT_n$ is the set of rooted trees with $n$ vertices. We show that the generating function $U(q)$ satisfies the equation $Ξ\exp U(q)= q^{-1} U(q)$. Consequently, we get a recurrent formula for $U_n$ $(n\geq 1)$, namely, $U_1=Ξ(1)$ and $U_n =ΞS_{n-1}(U_1, U_2, >..., U_{n-1})$ for any $n\geq 2$, where $S_n(x_1, x_2, ...)$ $(n\in \bN)$ are the elementary Schur polynomials. We also show that the (strict) order polynomials and two well known quasi-symmetric function invariants of rooted forests are in the family of invariants $Ψ$ and derive some consequences about these well-known invariants from our general results on $Ψ$. Finally, we generalize the invariant $Ψ$ to labeled planar forests and discuss its certain relations with the Hopf algebra $\mathcal H_{P, R}^D$ in \cite{F} spanned by labeled planar forests.

math.CO↗

arXiv2002

Exponential Formulas for the Jacobians and Jacobian Matrices of Analytic Maps

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 $A=\sum_{i=1}^n a_i(z)\frac {\p}{\p z_i}$ such that $F(z)=exp (A)z$ as formal series. In this article, we show the Jacobian ${\cal J}(F)$ and the Jacobian matrix $J(F)$ of $F$ can also be given by some exponential formulas. Namely, ${\cal J}(F)=\exp (A+\triangledown A)\cdot 1$, where $\triangledown A(z)= \sum_{i=1}^n \frac {\p a_i}{\p z_i}(z)$, and $J(F)=\exp(A+R_{Ja})\cdot I_{n\times n}$, where $I_{n\times n}$ 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 ${\cal J}(F)\equiv 1$ if and only if $\triangledown A=0$. Some consequences and applications of the exponential formulas as well as their relations with the well known Jacobian Conjecture are also discussed.

math.CV↗

arXiv2002

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

Given a formal map $F=(F_1...,F_n)$ of the form $z+\text{higher}$ order terms, we give tree expansion formulas and associated algorithms for the D-Log of F and the formal flow F_t. 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 combinatorics for the partially ordered sets induced by trees. Applications of these formulas to the Jacobian Conjecture are discussed.

math.CV↗

arXiv2002

Some Reductions on Jacobian Problem in Two Variables

Let $f=(f_1, f_2)$ be a regular sequence of affine curves in $\bC^2$. Under some reduction conditions achieved by composing with some polynomial automorphisms of $\bC^2$, we show that the intersection number of curves $(f_i)$ in $\bC^2$ equals to the coefficient of the leading term $x^{n-1}$ in $g_2$, where $n=°f_i$ $(i=1, 2)$ and $(g_1, g_2)$ is the unique solution of the equation $y{\mathcal J}(f)=g_1f_1+g_2f_2$ with $°g_i\leq n-1$. So the well-known Jacobian problem is reduced to solving the equation above. Furthermore, by using the result above, we show that the Jacobian problem can also be reduced to a special family of polynomial maps.

math.AG↗

arXiv1999

Semi-infinite forms and topological vertex operator algebras

Semi-infinite forms on the moduli spaces of genus-zero Riemann surfaces with punctures and local coordinates are introduced. A partial operad for semi-infinite forms is constructed. Using semi-infinite forms and motivated by a partial suboperad of the partial operad for semi-infinite forms, topological vertex partial operads of type $k<0$ and strong topological vertex partial operads of type $k<0$ are constructed. It is proved that the category of (locally-)grading-restricted (strong) topological vertex operator algebras of type $k<0$ and the category of (weakly) meromorphic ${\Bbb Z}\times {\Bbb Z}$-graded algebras over the (strong) topological vertex partial operad of type k are isomorphic. As an application of this isomorphism theorem, the following conjecture of Lian-Zuckerman and Kimura-Voronov-Zuckerman is proved: A strong topological vertex operator algebra gives a homotopy Gerstenhaber algebra. These results hold in particular for the tensor product of the moonshine module vertex operator algebra, the vertex algebra constructed {from} a rank 2 Lorentz lattice and the ghost vertex operator algebra, studied in detail first by Lian and Zuckerman.

math.QA↗