arXiv subjects
Publications and source records attributed to Arno van den Essen.
We show how a counterexample to the Jacobian Conjecture can be found by elementary mathematics. The example we find is, apart from linear coordinate changes the example found by Levent Alpöge.
The theorem of Duistermaat and Van der Kallen from 1998 proved the first case of the Mathieu conjecture. Using the theory of Mathieu-Zhao spaces, we can reformulate this theorem as $\operatorname{Ker} L$ is a Mathieu-Zhao space where $L$ is the linear map \begin{align*} L\colon {\bf C}[X_1,\ldots,X_n,X_1^{-1},\ldots,X_n^{-1}] \to C,\ f \mapsto f_0\end{align*}. In this paper, we generalize this result (for $n = 1$) to all non-trivial linear maps $L\colon C[X,X^{-1}] \to C$ such that $\{X^n \mid |n|\geq N\} \subset \operatorname{Ker} L$ for some $N \geq 1$.
Some cases of the LFED Conjecture, proposed by the second author [Z3], for certain integral domains are proved. In particular, the LFED Conjecture is completely established for the field of fractions $k(x)$ of the polynomial algebra $k[x]$, the formal power series algebra $k[[x]]$ and the Laurent formal power series algebra $k[[x]][x^{-1}]$, where $x=(x_1, x_2, \dots, x_n)$ denotes $n$ commutative free variables and $k$ a field of characteristic zero. Furthermore, the relation between the LFED Conjecture and the Duistermaat-van der Kallen Theorem [DK] is also discussed and emphasized.
We describe all Mathieu-Zhao spaces of $k[x_1,\cdots,x_n]$ ($k$ is an algebraically closed field of characteristic zero) which contains an ideal of finite codimension. Furthermore we give an algorithm to decide if a subspace of the form $I+kv_1+\cdots+kv_r$ is a Mathieu-Zhao space, in case the ideal $I$ has finite codimension.
This is the note for the four lectures given by the author in the ``International Short-School/Conference on Affine Algebraic Geometry and the Jacobian Conjecture" at Chern Institute of Mathematics, Nankai University, Tianjin, China. July 14-25, 2014. The aim of this lectures is to give an introduction to the theory of Mathieu subspaces. We will not treat all topics in their most general setting, but will restrict to certain classes of "nice" rings. As we shall see even for these rings there is a lot we don't know yet!
The classification of the nilpotent Jacobians with some structure has been an object of study because of its relationship with the Jacobian Conjecture. In this paper we classify the polynomial maps in dimension $n$ of the form $H = (u(x,y), u_2(x,y,x_3), \ldots, u_{n-1}(x,y,x_n), h(x,y))$ with $JH$ nilpotent. In addition we prove that the maps $X + H$ are invertible, which shows that for this kind of maps the Jacobian Conjecture is verified.
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 .
In this paper we present an unexpected link between the Factorial Conjecture and Furter's Rigidity Conjecture. The Factorial Conjecture in dimension $m$ asserts that if a polynomial $f$ in $m$ variables $X_i$ over $\C$ is such that ${\cal L}(f^k)=0$ for all $k\geq 1$, then $f=0$, where ${\cal L}$ is the $\C$-linear map from $\C[X_1,...,X_m]$ to $\C$ defined by ${\cal L}(X_1^{l_1}... X_m^{l_m})=l_1!... l_m!$. The Rigidity Conjecture asserts that a univariate polynomial map $a(X)$ with complex coefficients of degree at most $m+1$ such that $a(X)=X$ mod $X^2$, is equal to $X$ if $m$ consecutive coefficients of the formal inverse of $a(X)$ are zero.
In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. In the case R is a Dedekind Q-algebra, some stronger results are obtained. A key element in the proof is a theorem which yields the following corollary: Over an Artinian ring A all two-dimensional polynomial automorphisms having Jacobian determinant one are stably tame, and are tame if A is a Q-algebra. Another crucial ingredient, of interest in itself, is that stable tameness is a local property: If an automorphism is locally tame, then it is stably tame.
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.
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 $k[x, y]$.
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 $A$.
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.
In this paper we discuss a general framework in which we present a new conjecture, due to Wenhua Zhao, the Image Conjecture. This conjecture implies the Generalized Vanishing Conjecture and hence the Jacobian Conjecture. Crucial ingredient is the notion of a Mathieu space: let $k$ be a field and $R$ a commutative $k$-algebra. A $k$-linear subspace $M$ of $R$ is called a Mathieu subspace of $R$, if the following holds: let $f\in R$ be such that $f^m\in M$, for all $m\geq 1$, then for every $g\in R$ also $gf^m\in M$, for almost all $m$, i.e. only finitely many exceptions. Let $A$ be the polynomial ring in $ζ=ζ_1, ...,ζ_n$ and $z_1, ...,z_n$ over $\mathbb C$. The Image Conjecture (IC) asserts that $\sum_i(\partial_{z_i}-ζ_i)A$ is a Mathieu subspace of $A$. We prove this conjecture for $n=1$. Also we relate (IC) to the following Integral Conjecture: if $B$ is an open subset of $\mathbb R^n$ and $σ$ a positive measure, such that the integral over $B$ of each polynomial in $z$ over $\mathbb C$ is finite, then the set of polynomials, whose integral over $B$ is zero, is a Mathieu subspace of $\mathbb C[z]$. It turns out that Laguerre polynomials play a special role in the study of the Jacobian Conjecture.
Algebraic actions of unipotent groups $U$ actions on affine $k-$varieties $X$ ($k$ an algebraically closed field of characteristic 0) for which the algebraic quotient $X//U$ has small dimension are considered$.$ In case $X$ is factorial, $O(X)^{\ast}=k^{\ast},$ and $X//U$ is one-dimensional, it is shown that $O(X)^{U}$=$k[f]$, and if some point in $X$ has trivial isotropy, then $X$ is $U$ equivariantly isomorphic to $U\times A^{1}(k).$ The main results are given distinct geometric and algebraic proofs. Links to the Abhyankar-Sathaye conjecture and a new equivalent formulation of the Sathaye conjecture are made.
We prove that for a polynomial $f\in k[x,y,z]$ equivalent are: (1)$f$ is a $k[z]$-coordinate of $k[z][x,y]$, and (2) $k[x,y,z]/(f)\cong k^{[2]}$ and $f(x,y,a)$ is a coordinate in $k[x,y]$ for some $a\in k$. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate $f\in k[x,y,z]$ which is also a $k(z)$-coordinate, is a $k[z]$-coordinate. We discuss a method for constructing automorphisms of $k[x,y,z]$, and observe that the Nagata automorphism occurs naturally as the first non-trivial automorphism obtained by this method - essentially linking Nagata with a non-tame $R$-automorphism of $R[x]$, where $R=k[z]/(z^2)$.
Let $A$ be the polynomial ring over $k$ (a field of characteristic zero) in $n+1$ variables. The commuting derivations conjecture states that $n$ commuting locally nilpotent derivations on $A$, linearly independent over $A$, must satisfy $A^{D_1,...,D_m}=k[f]$ where $f$ is a coordinate. The conjecture can be formulated as stating that a $(G_m)^n$-action on $k^{n+1}$ must have invariant ring $k[f]$ where $f$ is a coordinate. In this paper we prove a statement (theorem \ref{CDH2}) where we assume less on $A$ ($A$ is a {\sc UFD} over $k$ of transcendence degree $n+1$ satisfying $A^*=k$) and prove less ($A/(f-α)$ is a polynomial ring for all but finitely many $α$). Under certain additional conditions (the $D_i$ are linearly independent modulo $(f-α)$ for each $α\in k$) we prove that $A$ is a polynomial ring itself and $f$ is a coordinate. This statement is proven even more generally by replacing ``free unipotent action of dimension $n$'' for ``$G_a^n$-action''. We make links with the (Abhyankar-)Sataye conjecture and give a new equivalent formulation of the Sataye conjecture.
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 $m>>0$.