Related papers: Polynomial Matrices, Splitting Subspaces and Krylo…

Let $f(x) \in \mathbb{C}[x]$ of degree $n$. We attach to $f$ a $\mathbb{C}$-vector space $W(f)$ which consists of complex polynomials $p(x)$ of degree at most $n - 2$ such that $f(x)$ divides $f"(x)p(x) - f'(x) p'(x)$. The space $W(f)$…

Let \Delta be a finite sequence of n vectors from a vector space over any field. We consider the subspace of \operatorname{Sym}(V) spanned by \prod_{v \in S} v, where S is a subsequence of \Delta. A result of Orlik and Terao provides a…

We investigate point arrangements $v_i\in\mathbb R^d,i\in \{1,...,n \}$ with certain prescribed symmetries. The arrangement space of $v$ is the column span of the matrix in which the $v_i$ are the rows. We characterize properties of $v$ in…

We analyse the Krylov solvability of inverse linear problems on Hilbert space $\mathcal{H}$ where the underlying operator is compact and normal. Krylov solvability is an important feature of inverse linear problems that has profound…

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a $(2+1)$-dimensional Riemann (curved) spacetime.…

A linear code over $\mathbb{F}_q$ with the Hamming metric is called $\Delta$-divisible if the weights of all codewords are divisible by $\Delta$. They have been introduced by Harold Ward a few decades ago. Applications include subspace…

Let n be a positive integer, and let R be a finitely presented (but not necessarily finite dimensional) associative algebra over a computable field. We examine algorithmic tests for deciding (1) if every n-dimensional representation of R is…

Let $r$ be a nonconstant noncommutative rational function in $m$ variables over an algebraically closed field $K$ of characteristic 0. We show that for $n$ large enough, there exists an $X\in M_n(K)^m$ such that $r(X)$ has $n$ distinct and…

We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the…

We consider the category $\mathcal S(n)$ of all pairs $X = (U,V)$, where $V$ is a finite-dimensional vector space with a nilpotent operator $T$ with $T^n = 0$, and $U$ is a subspace of $V$ such that $T(U) \subseteq U$. Our main interest in…

We are interested in contractible n-manifolds M which "split" as M = A union B where A,B, and A intersect B are all homeomorphic to Euclidean n-space (such M are called open n-splitters) or A,B, and A intersect B are all homeomorphic to the…

The Orbit Problem asks whether the orbit of a point under a matrix reaches a given target set. When the target is a single point, the problem was shown to be decidable in polynomial time by Kannan and Lipton. This decidability result was…

It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: if two points in the direct sum of the $G$--modules $W$ and $m$ copies of $V$ can be separated by polynomial…

We denote by $\mathcal{P}_q$ the vector space of functions from a finite field $\mathbb{F}_q$ to itself, which can be represented as the space $\mathcal{P}_q := \mathbb{F}_q[x]/(x^q-x)$ of polynomial functions. We denote by $\mathcal{O}_n…

This is a survey on a notion of invariant operators, or Fourier multipliers on Hilbert spaces. This concept is defined with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. In particular this…

Evaluating a polynomial on a set of points is a fundamental task in computer algebra. In this work, we revisit a particular variant called trimmed multipoint evaluation: given an $n$-variate polynomial with bounded individual degree $d$ and…

Let $T$ be a bounded quaternionic normal operator on a right quaternionic Hilbert space $\mathcal{H}$. We show that $T$ can be factorized in a strongly irreducible sense, that is, for any $\delta >0$ there exist a compact operator $K$ with…

We discuss the concept of invariant subspaces for unbounded linear operators, point out some shortcomings of known definitions, and propose our own.

The multi-point Taylor polynomial, which is the general, unique and of minimum degree ($mk+m-1$) polynomial $P_{k,m}(x)$ which interpolates a function's derivatives in multiple points is presented in its explicit form. A proof that this…

A complete classification of linear differential operators possessing finite-dimensional invariant subspace with a basis of monomials is presented.