arXiv subjects

Wenhua Zhao

Publications and source records attributed to Wenhua Zhao.

At least 37 records · Page 2Linked to original sources

arXiv2017

The LNED and LFED Conjectures for Algebraic Algebras

Let $K$ be a field of characteristic zero and $\mathcal A$ a $K$-algebra such that all the $K$-subalgebras generated by finitely many elements of $\mathcal A$ are finite dimensional over $K$. A $K$-$\mathcal E$-derivation of $\mathcal A$ is a $K$-linear map of the form $\operatorname{I}-ϕ$ for some $K$-algebra endomorphism $ϕ$ of $\mathcal A$, where $\operatorname{I}$ denotes the identity map of $\mathcal A$. In this paper we first show that for all locally finite $K$-derivations $D$ and locally finite $K$-algebra automorphisms $ϕ$ of $\mathcal A$, the images of $D$ and $\operatorname{I}-ϕ$ do not contain any nonzero idempotent of $\mathcal A$. We then use this result to show some cases of the LFED and LNED conjectures proposed in [Z4]. More precisely, We show the LNED conjecture for $\mathcal A$, and the LFED conjecture for all locally finite $K$-derivations of $\mathcal A$ and all locally finite $K$-$\mathcal E$-derivations of the form $δ=\operatorname{I}-ϕ$ with $ϕ$ being surjective. In particular, both conjectures are proved for all finite dimensional $K$-algebras. Furthermore, some finite extensions of derivations and automorphism to inner derivations and inner automorphisms, respectively, have also been established. This result is not only crucial in the proofs of the results above, but also interesting on its own right.

math.RA↗

arXiv2017

Idempotents in Intersection of the Kernel and the Image of Locally Finite Derivations and $\mathcal E$-derivations

Let $K$ be a field of characteristic zero, $\mathcal A$ a $K$-algebra and $δ$ a $K$-derivation of $\mathcal A$ or $K$-$\mathcal E$-derivation of $\mathcal A$ (i.e., $δ=\operatorname{Id}_A-ϕ$ for some $K$-algebra endomorphism $ϕ$ of $\mathcal A$). Motivated by the Idempotent conjecture proposed in [Z4], we first show that for every idempotent $e$ lying in both the kernel ${\mathcal A}^δ$ and the image $\operatorname{Im}δ\!:=δ({\mathcal A})$ of $δ$, the principal ideal $(e)\subseteq \operatorname{Im} δ$ if $δ$ is a locally finite $K$-derivation or a locally nilpotent $K$-$\mathcal E$-derivation of $\mathcal A$; and $e{\mathcal A}, {\mathcal A}e \subseteq \operatorname{Im} δ$ if $δ$ is a locally finite $K$-$\mathcal E$-derivation of $\mathcal A$. Consequently, the Idempotent conjecture holds for all locally finite $K$-derivations and all locally nilpotent $K$-$\mathcal E$-derivations of $\mathcal A$. We then show that $1_{\mathcal A} \in \operatorname{Im} δ$, (if and) only if $δ$ is surjective, which generalizes the same result [GN, W] for locally nilpotent $K$-derivations of commutative $K$-algebras to locally finite $K$-derivations and $K$-$\mathcal E$-derivations $δ$ of all $K$-algebras $\mathcal A$.

math.RA↗

arXiv2017

The LFED and LNED Conjectures for Laurent Polynomial Algebras

Let $R$ be an integral domain of characteristic zero, $x=(x_1, x_2, ..., x_n)$ $n$ commutative free variables, and ${\mathcal A}_n:=R[x^{-1}, x]$, i.e., the Laurent polynomial algebra in $x$ over $R$. In this paper we first classify all locally finite or locally nilpotent $R$-derivations and $R$-$\mathcal E$-derivations of ${\mathcal A}_n$, where by an $R$-$\mathcal E$-derivation of ${\mathcal A}_n$ we mean an $R$-linear map of the form $\operatorname{Id}_{{\mathcal A}_n}-ϕ$ for some $R$-algebra endomorphism $ϕ$ of ${\mathcal A}_n$. In particular, we show that ${\mathcal A}_n$ has no nonzero locally nilpotent $R$-derivations or $R$-$\mathcal E$-derivations. Consequently, the LNED conjecture proposed in [Z4] for ${\mathcal A}_n$ follows. We then show some cases of the LFED conjecture proposed in [Z4] for ${\mathcal A}_n$. In particular, we show that both the LFED and LNED conjectures hold for the Laurent polynomial algebras in one or two commutative free variables over a field of characteristic zero.

math.AC↗

arXiv2015

The Gaussian Moments Conjecture and the Jacobian Conjecture

We first propose what we call the Gaussian Moments Conjecture. We then show that the Jacobian Conjecture follows from the Gaussian Moments Conjecture. We also give a counter-example to a more general statement known as the Moments Vanishing Conjecture .

math.AC↗

arXiv2011

Some Results on the Vanishing Conjecture of Differential Operators with Constant Coefficients

In this paper we prove four cases of the vanishing conjecture of differential operators with constant coefficients and also a conjecture on the Laurent polynomials with no holomorphic parts, which were proposed in [Zh3] by the third named author. We also give two examples to show that the generalizations of both the vanishing conjecture and the Duistermaat-van der Kallen theorem [DK] to Laurent formal power series do not hold in general.

math.AC↗

arXiv2010

Images of Locally Finite Derivations of Polynomial Algebras in Two Variables

In this paper we show that the image of any locally finite $k$-derivation of the polynomial algebra $k[x, y]$ in two variables over a field $k$ of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image $Im D$ of every $k$-derivation $D$ of $k[x, y]$ such that $1\in Im D$ and $div D=0$ is a Mathieu subspace of $k[x, y]$.

math.AC↗

arXiv2010

Mathieu Subspaces of Univariate Polynomial Algebras

We first give a characterization for Mathieu subspaces of univariate polynomial algebras over fields in terms of their radicals. We then deduce that for some classes of classical univariate orthogonal polynomials the Image Conjecture is true. We also prove two special cases of the one-dimensional Image Conjecture for univariate polynomial algebras $A[t]$ over commutative $\Bbb Q$-algebras $A$.

math.AC↗

arXiv2010

New Proofs for the Abhyankar-Gurjar Inversion Formula and the Equivalence of the Jacobian Conjecture and the Vanishing Conjecture

We first give a new proof and also a new formulation for the Abhyankar-Gurjar inversion formula for formal maps of affine spaces. We then use the reformulated Abhyankar-Gurjar formula to give a more straightforward proof for the equivalence of the Jacobian Conjecture with a special case of the Vanishing Conjecture of (homogeneous) quadratic differential operators with constant coefficients.

math.AG↗

arXiv2010

Analogue of the Duistermaat-van der Kallen Theorem for Group Algebras

Let $G$ be a group, $R$ an integral domain, and $V_G$ the subspace of the group algebra $R[G]$ consisting of all the elements of $R[G]$ whose coefficient of the identity element $1_G$ of $G$ is equal to zero. Motivated by the Mathieu conjecture [M], the Duistermaat-van der Kallen theorem [DK], and also by recent studies on the notion of Mathieu subspaces introduced in [Z4] and [Z6], we show that for finite groups $G$, $V_G$ under certain conditions also forms a Mathieu subspace of the group algebra $R[G]$. We also show that for the free abelian groups $G=\Bbb Z^n$ $(n\ge 1)$ and any integral domain $R$ of positive characteristic, $V_G$ fails to be a Mathieu subspace of $R[G]$, which is equivalent to saying that the Duistermaat-van der Kallen theorem [DK] cannot be generalized to any field or integral domain of positive characteristic.

math.RA↗

arXiv2010

On the Image Conjecture

The Image Conjecture was formulated by the third author, who showed that it implied his Vanishing Conjecture, which is equivalent to the famous Jacobian Conjecture. We prove various cases of the Image Conjecture and show how it leads to another fascinating and elusive assertion that we here dub the Factorial Conjecture. Various cases of the Factorial Conjecture are proved.

math.RA↗

arXiv2010

Mathieu Subspaces of Associative Algebras

Motivated by the Mathieu conjecture [Ma], the image conjecture [Z3] and the well-known Jacobian conjecture [K] (see also [BCW] and [E]), the notion of Mathieu subspaces as a natural generalization of the notion of ideals has been introduced recently in [Z4] for associative algebras. In this paper, we first study algebraic elements in the radicals of Mathieu subspaces of associative algebras over fields and prove some properties and characterizations of Mathieu subspaces with algebraic radicals. We then give some characterizations or classifications for strongly simple algebras (the algebras with no non-trivial Mathieu subspaces) over arbitrary commutative rings, and for quasi-stable algebras (the algebras all of whose subspaces that do not contain the identity element of the algebra are Mathieu spaces) over arbitrary fields. Furthermore, co-dimension one Mathieu subspaces and the minimal non-trivial Mathieu subspaces of the matrix algebras over fields are also completely determined.

math.RA↗

arXiv2010

A Generalization of Mathieu Subspaces to Modules of Associative Algebras

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $\mathcal A$, which was introduced recently in [Z4] and [Z6], to $\mathcal A$-modules $\mathcal M$. The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets $σ(N)$ and $τ(N)$ of stable elements and quasi-stable elements, respectively, for all $R$-subspaces $N$ of $\mathcal A$-modules $\mathcal M$, where $R$ is the base ring of $\mathcal A$. We then prove some general properties of the sets $σ(N)$ and $τ(N)$. Furthermore, examples from certain modules of the quasi-stable algebras [Z6], matrix algebras over fields and polynomial algebras are also studied.

math.RT↗

arXiv2010

Images of Commuting Differential Operators of Order One with Constant Leading Coefficients

We first study some properties of images of commuting differential operators of polynomial algebras of order one with constant leading coefficients. We then propose what we call the image conjecture on these differential operators and show that the Jacobian conjecture [BCW], [E], [Bo] (hence also the Dixmier conjecture [D]) and the vanishing conjecture [Z3] of differential operators with constant coefficients are actually equivalent to certain special cases of the image conjecture. A connection of the image conjecture, and hence also the Jacobian conjecture, with multidimensional Laplace transformations of polynomials is also discussed.

math.CV↗

arXiv2010

A Deformation of Commutative Polynomial Algebras in Even Numbers of Variables

We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [Z3] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture [Ke] is reduced to an open problem on this deformation of polynomial algebras.

math.AC↗

arXiv2009

Generalizations of the Image Conjecture and the Mathieu Conjecture

We first propose a generalization of the image conjecture [Z3] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture [Ma] from integrals of $G$-finite functions over reductive Lie groups $G$ to integrals of polynomials over open subsets of $\mathbb R^n$ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of Duistermaat and van der Kallen's theorem [DK] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly-introduced notion.

math.CV↗

arXiv2008

Some Properties of and Open Problems on Hessian Nilpotent Polynomials

In the recent progress [BE1], [M], [Z1] and [Z2], the well-known Jacobian conjecture ([BCW], [E]) has been reduced to a problem on HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent) and their (deformed) inversion pairs. In this paper, we prove several results on HN polynomials, their (deformed) inversion pairs as well as the associated symmetric polynomial or formal maps. We also propose some open problems for further study of these objects.

math.CV↗

arXiv2007

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 $Λ$ and $Λ$-nilpotent polynomials $P$ (the polynomials $P(z)$ satisfying $Λ^m P^m=0$ 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 $Λ$-nilpotent polynomials and the associated VC for 2nd order homogeneous differential operators $Λ$. 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 $Λ$-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 $Λ$-nilpotent polynomials for the 2nd order homogeneous full rank differential operators $Λ$ with constant coefficients.

math.CV↗

arXiv2007

Differential Operator Specializations of Noncommutative Symmetric Functions

Let $K$ be any unital commutative $\mathbb Q$-algebra and $z=(z_1, ..., z_n)$ commutative or noncommutative free variables. Let $t$ be a formal parameter which commutes with $z$ and elements of $K$. We denote uniformly by $\kzz$ and $\kttzz$ the formal power series algebras of $z$ over $K$ and $K[[t]]$, respectively. For any $α\geq 1$, let $\cDazz$ 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, for any fixed $α\geq 1$ and $F_t\in \ataz$, we introduce five sequences of differential operators of $\kzz$ and show that their generating functions form a $\mathcal N$CS (noncommutative symmetric) system [Z4] over the differential algebra $\cDazz$. Consequently, by the universal property of the $\mathcal N$CS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [GKLLRT], we obtain a family of Hopf algebra homomorphisms $\cS_{F_t}: {\mathcal N}Sym \to \cDazz$ $(F_t\in \ataz)$, which are also grading-preserving when $F_t$ satisfies certain conditions. Note that, the homomorphisms $\cS_{F_t}$ above can also be viewed as specializations of NCSFs by the differential operators of $\kzz$. Secondly, we show that, in both commutative and noncommutative cases, this family $\cS_{F_t}$ (with all $n\geq 1$ and $F_t\in \ataz$) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions ([Ge], [MR], [S]) are also discussed.

math.CO↗