Related papers: The Jacobian Conjecture is stably equivalent to th…

We conjecture that the injective dimension of the Jacobson radical equals the global dimension for Artin algebras. We provide a proof of this conjecture in case the Artin algebra has finite global dimension and in some other cases.

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…

A conjecture of Miyanishi says that an endomorphism of an algebraic variety, defined over an algebraically closed field of characteristic zero, is an automorphism if the endomorphism is injective outside a closed subset of codimension at…

We describe the complete set of pairwise non-isomorphic irreducible modules S(a) over the algebra R given by the defining relation xy-yx=yy, and the rule how they could be glued to indecomposables. Namely, we show that Ext_k^1(S(a),S(b))=0,…

Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surface $Y$ is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms on $Y$ quotiented by the torsion-free…

It is well known that the integral identity conjecture is of prime importance in Kontsevich-Soibelman's theory of motivic Donaldson-Thomas invariants for non-commutative Calabi-Yau threfolds. In this article we consider its numerical…

We present an alternative definition of the twisted 1-loop invariant in terms of the Jacobian of Ptolemy coordinates. As an application, we prove that the twisted 1-loop invariant is equal to the adjoint twisted Alexander polynomial for all…

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism $f: M \to\mathbb{R}^n$ is bijective if and only if $H_{n-1}(M)=0$ and the pre-image of…

In this note, we show that, if the Druzkowski mappings $F(X)=X+(AX)^{*3}$, i.e. $F(X)=(x_1+(a_{11}x_1+...+a_{1n}x_n)^3,...,x_n+(a_{n1}x_1+...+a_{nn}x_n)^3)$, satisfies $TrJ((AX)^{*3})=0$, then $rank(A)\leq 1/2(n+\delta)$ where $\delta$ is…

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…

We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When…

The Jacobian algebra arising from a consistent dimer model is a bimodule $3$-Calabi-Yau algebra, and its center is a $3$-dimensional Gorenstein toric singularity. A perfect matching of a dimer model gives the degree making the Jacobian…

Let us denote by $\mathcal K_n$ the hyperspace of all convex bodies of $\mathbb R^n$ equipped with the Hausdorff distance topology. An affine invariant point $p$ is a continuous and Aff(n)-equivariant map $p:\mathcal K_n\to \mathbb R^n$,…

It is shown that in dimension at least three a local diffeomorphism of Euclidean n-space into itself is injective provided that the pull-back of every plane is a Riemannian submanifold which is conformal to a plane. Using a similar…

We give two characterizations of Jacobians of curves with involution having fixed points in the framework of two particular cases of Welter's trisecant conjecture. The geometric form of each of these characterizations is the statement that…

We generalize Abel's classical theorem on linear equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact oriented Riemannian $n$--manifold, or more generally for any $d$--cycle $Z$ relative…

We investigate a known problem whether a Sobolev homeomorphism between domains in $\mathbb{R}^n$ can change sign of the Jacobian. The only case that remains open is when $f\in W^{1,[n/2]}$, $n\geq 4$. We prove that if $n\geq 4$, and a…

In the paper {\em The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic}, J. Pure Appl. Algebra 212 (2008), no. 10, 2320-2337, see also arXiv:math/0604477, Vladimir Bavula states…

Ahmadi-Shparlinski conjectured that every ordinary, geometrically simple Jacobian over a finite field has maximal angle rank. Using the L-Functions and Modular Forms Database, we provide two counterexamples to this conjecture in dimension…

Using the local bijectivity of Keller maps, we give a proof of two-dimensional Jacobian conjecture.