Related papers: Jacobian Conjecture and Nilpotency

We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a…

Let K be an algebraically closed field. We prove that a polynomial K-derivation $D$ in two variables is locally nilpotent if and only if the subgroup of polynomial K-automorphisms which commute with D admits elements whose degree is…

Let F be a linear unital map of a unital matrix algebra A over the complex numbers into the complex n by n matrices. Then F induces a linear unital map Fk of the k by k matrices over A into the complex nk by nk matrices by the action of F…

We give a detailed proof of the conjecture by Hohm and Zwiebach in double field theory. This result implies that their proposal for large gauge transformations in terms of the Jacobian matrix for coordinate transformations is, as required,…

We study Lie subalgebras $L$ of the vector fields $\mathrm{Vec}^{c}({\mathbb A}^{2})$ of affine 2-space ${\mathbb A}^{2}$ of constant divergence, and we classify those $L$ which are isomorphic to the Lie algebra $\mathfrak{aff}_{2}$ of the…

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…

We study the homotopy category $ K(\Inj A)$ of all injective modules over a finite dimensional algebra $A$ with discrete derived category. We give a classification of the indecomposable objects of $ K(\Inj A)$ for any radical square zero…

Let K be a field of characteristic zero, n>4 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group is doubly transitive simple non-abelian group. Let p be an odd prime, Z[\zeta_p] the ring of integers in the p-th…

Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a {\it retraction}, or {\it projection}) $\varphi:…

We consider non-degenerate graph immersions into affine space $\mathbb A^{n+1}$ whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and…

We prove that an endomorphism $f$ of affine space is injective on rational points if its B\'ezoutian is constant. Similarly, $f$ is injective at a given rational point if its reduced B\'ezoutian is constant. We also show that if the…

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 $p:\mathcal K_n\to \mathbb R^n$,…

We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras $A$ with a generalized Hopf action of an associative algebra $H$ with $1$ over an algebraically closed field of characteristic $0$ assuming only the…

In this paper, we contribute toward a classification of two-variable polynomials by classifying (up to an automorphism of $C^2$) polynomials whose Newton polygon is either a triangle or a line segment. Our classification has several…

Suppose that h in F[x,y,z], char F=2, defines a nodal cubic. In earlier papers we made a precise conjecture as to the Hilbert-Kunz functions attached to the powers of h. Assuming this conjecture we showed that a class of characteristic 2…

Let F be a finite field of characteristic 2 and h be the element x^3+y^3+xyz of F[[x,y,z]]. In an earlier paper we made a precise conjecture as to the values of the colengths of the ideals (x^q,y^q,z^q,h^j) for q a power of 2. We also…

Let D be a bounded convex domain in C^N, N\geq 2. We prove that a continous map F from bD to C^N extends holomorphically through D if and only if for every polynomial map P from C^N to C^N such that F+P has no zero on bD, the degree of…

We establish necessary and sufficient conditions for invertibility of symmetric three-by-three block matrices having a double saddle-point structure \fb{that guarantee the unique solvability of double saddle-point systems}. We consider…

A conjecture of Odoni stated over Hilbertian fields $K$ of characteristic zero asserts that for every positive integer $d$, there exists a polynomial $f\in K[x]$ of degree $d$ such that for every positive integer $n$, each iterate $f^{\circ…

Finite-dimensional Jacobian algebras are studied from the perspective of representation types. We establish that (like other representation types) the notions of $E$-finiteness and $E$-tameness are invariant under mutations of quivers with…