Related papers: A Vanishing Conjecture on Differential Operators w…

Let $z=(z_1, ..., z_n)$ and $\Delta=\sum_{i=1}^n \fr {\p^2}{\p z^2_i}$ 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…

Let $z=(z_1, ..., z_n)$ and $\Delta=\sum_{i=1}^n \frac {\partial^2}{\partial z^2_i}$ 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…

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…

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…

In this paper, we show that the GVC (generalized vanishing conjecture) holds for the differential operator $\Lambda=(\partial_x-\Phi(\partial_y))\partial_y$ and all polynomials $P(x,y)$, where $\Phi(t)$ is any polynomial over the base…

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)…

We show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lam^m f^m = 0] \Longrightarrow \forall_{m \gg 0} [\Lam^m (g f^m) = 0]$$ for a fixed differential operator $\Lam \in k[\partial]$ follows from a special case of it,…

Many questions in number theory concern the nonvanishing of determinants of square matrices of logarithms (complex or p-adic) of algebraic numbers. We present a new conjecture that states that if such a matrix has vanishing determinant,…

We prove the equivalence of the Jacobian Conjecture (JC(n)) and the Conjecture on the cardinality of the set of fixed points of a polynomial nilpotent mapping (JN(n)) and prove a series of assertions confirming JN(n).

We prove the rank-4 case of the conjecture of Ha-Hai-Nghia for the invariant subspace of the truncated polynomial ring $\mathcal{Q}_m(n)=\mathbb{F}_q[x_1,\dots,x_n]/(x_1^{q^m},\dots,x_n^{q^m}),$ under a new, explicit technical hypothesis.…

The Jacobian conjecture in dimension $n$ asserts that any polynomial endomorphism of $n$-dimensional affine space over a field of zero characteristic, with the Jacobian equal 1, is invertible. The Dixmier conjecture in rank $n$ asserts that…

Let $\mathcal H$ be the class of algebras verifying Han's conjecture. In this paper we analyse two types of algebras with the aim of providing an inductive step towards the proof of this conjecture. Firstly we show that if an algebra…

Orthogonal polynomials $P_{n}(\lambda)$ are oscillating functions of $n$ as $n\to\infty$ for $\lambda$ in the absolutely continuous spectrum of the corresponding Jacobi operator $J$. We show that, irrespective of any specific assumptions on…

Our goal is to settle the following faded problem: The Jacobian Conjecture (JC_n): If f_1,..,f_n are elements in a polynomial ring k[X_1,..,X_n] over a field k of characteristic 0 such that det(\partial f_i/ \partial X_j) is a nonzero…

In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\geq 1$ and give some…

There is a growing body of work on proving hardness results for average-case estimation problems by bounding the low-degree advantage (LDA) - a quantitative estimate of the closeness of low-degree moments - between a null distribution and a…

In this paper we study a long-standing conjecture of Huneke and Wiegand which is concerned with the torsion submodule of certain tensor products of modules over one-dimensional local domains. We utilize Hochster's theta invariant and show…

A classical result by Casten-Holland and Matano asserts that constants are the only positive and stable solutions to semilinear elliptic PDEs subject to homogeneous Neumann boundary condition in bounded convex domains. In other terms, this…

We have studied a faded problem, the Jacobian Conjecture ~: \noindent {\sf The Jacobian Conjecture $(JC_n)$}~: If $f_1, \cdots, f_n$ are elements in a polynomial ring $k[X_1, \cdots, X_n]$ over a field $k$ of characteristic $0$ such that…

Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables . Assume that this polynomial satisfies the property : \\ $|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) :…