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

The exponential of an NxN matrix can always be expressed as a matrix polynomial of order N-1. In particular, a general group element for the fundamental representation of SU(N) can be expressed as a matrix polynomial of order N-1 in a…

This is the first of three papers dealing with the XX finite quantum chain with arbitrary, not necessarily hermitian, boundary terms. This extends previous work where the periodic or diagonal boundary terms were considered. In order to find…

For any real diagonalizable matrix with complex eigenvalues we provide a real, coordinate free decomposition with a clear geometric interpretation.

Let $X$ be a germ of holomorphic vector field at the origin of ${\bf C}^n$ and vanishing there. We assume that $X$ is a "nondegenerate" good perturbation of a singular completely integrable system. The latter is associated to a family of…

If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional…

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 and $\mathrm{M}(2,K)$ be the $2\times 2$ matrix algebra over $K$ and $\mathrm{GL}(2,K)$ be the invertible elements in $\mathrm{M}(2,K)$. We explore the image of polynomials with constants, namely…

This paper studies the existence of non trivial $\mu_p$ actions on a canonically polarized surface X defined over an algebraically closed field of characteristic p>0. In particular, an explicit function $f(K_X^2)$ is obtained such that if…

We develop techniques for computing the AK invariant of domains with arbitrary characteristic. As an example, we show that for any field $K$ the ring $K[X,Y,Z,T] / (X + X^2 Y + Z^2 + T^3)$ is not isomorphic to a polynomial ring over $K$.

Let $\K$ be a field of characteristic zero and $\Kbar$ be an algebraic closure of $\K$. Consider a sequence of polynomials$G=(g\_1,\dots,g\_s)$ in $\K[X\_1,\dots,X\_n]$, a polynomial matrix $\F=[f\_{i,j}] \in \K[X\_1,\dots,X\_n]^{p \times…

We describe solutions of the matrix equation $\exp(z(A-I_n))=A$, where $z \in {\mathbb C}$. Applications in quantum computing are given. Both normal and nonnormal matrices are studied. For normal matrices, the Lambert W-function plays a…

We provide an internal characterization of those finite algebras (i.e., algebraic structures) $\mathbf A$ such that the number of homomorphisms from any finite algebra $\mathbf X$ to $\mathbf A$ is bounded from above by a polynomial in the…

The so-called polynomial equations play an important role both in algebra and in the theory of functional equations. If the unknown functions in the equation are additive, relatively many results are known. However, even in this case, there…

It is proved that the Jacobian of a k-endomorphism of k[x_1,...,x_n] over a field k of characteristic zero taking every tame coordinate to a coordinate, must be a nonzero constant in k. It is also proved that the Jacobian of an…

In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the X-rank decomposition --- a powerful…

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…

A C*-algebra is said to be K-stable if its nonstable K-groups are naturally isomorphic to the usual K-theory groups. We study continuous $C(X)$-algebras, each of whose fibers are K-stable. We show that such an algebra is itself K-stable…

Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by…

Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly…

Let k be a field, n a positive integer, X a generic nxn matrix over k (i.e., a matrix (x_{ij}) of n^2 independent indeterminates over the polynomial ring k[x_{ij}]), and adj(X) its classical adjoint. It is shown that if char k=0 and n is…