Related papers: The matrix equation $XA-AX=f(X)$ when $A$ is diago…

The L'vov-Kaplansky conjecture states that the image of a multilinear noncommutative polynomial $f$ in the matrix algebra $M_n(K)$ is a vector space for every $n \in {\mathbb N}$. We prove this conjecture for the case where $f$ has degree…

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and…

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…

We describe algebraic curves $ X : F(x, y) = 0 $ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S \subset k$ such that, for every $y \in S$, the roots of the…

Let X be a complex nonsingular affine algebraic variety, K a holomorphically convex subset of X, and Y a homogeneous variety for some complex linear algebraic group. We prove that a holomorphic map f:K-->Y can be uniformly approximated on K…

Let $K$ be a field and $\sigma$ an automorphism of $K$ of order $n$.Employing a nonassociative algebra, we study the eigenspace of a bounded skew polynomial $f\in K[t;\sigma]$. We mainly treat the case that $K/F$ is a cyclic field extension…

The purpose of this paper is to study finite-dimensional Lie algebras over a field k of characteristic zero which admit a commutative polarization (CP). Among the many results and examples, it is shown that, if k is algebraically closed,…

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to…

Newton's method for solving the matrix equation $F(X)\equiv AX-XX^TAX=0$ runs up against the fact that its zeros are not isolated. This is due to a symmetry of $F$ by the action of the orthogonal group. We show how differential-geometric…

Assume that B(X) is the algebra of all bounded linear operators on a complex Banach space X, and let W in B(X) is such that cl(W(X)) is not equal to X or W=zI, where z is a complex number and I is the identity operator. We show that if f:…

The polynomial-time computability of the permanent over fields of characteristic 3 for k-semi-unitary matrices (i.e. square matrices such that the differences of their Gram matrices and the corresponding identity matrices are of rank k) in…

Let $K$ be a fixed field. We attach to each column-finite quiver $E$ a von Neumann regular $K$-algebra $Q(E)$ in a functorial way. The algebra $Q(E)$ is a universal localization of the usual path algebra $P(E)$ associated with $E$. The…

Let $\mathbb{K}$ be an algebraically closed field, and $A \subset \mathbb{K}[x_{1}, \ldots, x_n]$ be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of $\mathbb{K}$-algebras \[…

In this paper, we introduce and develop the method of diagonalization of functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to show that the equations of the form $\Gamma_r(n)+k=m^2$ has a finite number of solutions…

This paper shows that algebraic (in)dependence is encoded in Milnor K-theory of fields. As an application, we show that the isomorphism type of a field is determined by its Milnor K-theory, up to purely inseparable extensions, in most…

Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient…

In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with…

We investigate the space $X$ of unitary hermitian matrices over $\frp$-adic fields through spherical functions. First we consider Cartan decomposition of $X$, and give precise representatives for fields with odd residual characteristic,…

The Nichols algebras of diagonal type with finite root system are either of standard, super or (yet) unidentified type. A concrete description of the defining relations of all those Nichols algebras was given in \cite{A-exp presentation}.…

We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a black-box, i.e., by a function to compute its matrix-vector product. The methods apply to…