相关论文: Is every matrix similar to a polynomial in a compa…

Given an arbitrary monic polynomial $f$ over a field $F$ of characteristic 0, we use companion matrices to construct a polynomial $M_f\in F[X]$ of minimum degree such that for each root $\alpha$ of $f$ in the algebraic closure of $F$,…

We study the Smith forms of matrices of the form $f(C_g)$ where $f(t),g(t)\in R[t]$, $C_g$ is the companion matrix of the (monic) polynomial $g(t)$, and $R$ is an elementary divisor domain. Prominent examples of such matrices are circulant…

Let $f,g\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z< C,D>$ to be a sublattice of finite index in the full…

We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial f(x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of…

Suppose $F$ is an infinite field and let $f \in F\{X_1, \dots,X_m\}$ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers $d$ and $m'$ such that, if $F$ is closed under taking $d$th…

Let $R$ be a commutative ring and $g(t) \in R[t]$ a monic polynomial. The commutative ring of polynomials $f(C_g)$ in the companion matrix $C_g$ of $g(t)$, where $f(t)\in R[t]$, is called the Companion Ring of $g(t)$. Special instances…

Let $p$ be a prime and $n$ a positive integer. As the first main result, we present a deterministic algorithm for deciding whether the matrix algebra $\mathbb{F}_p[A_1,\dots,A_t]$ with $A_1,\dots,A_t \in \mathrm{GL}(n,\mathbb{F}_p)$ is a…

As is well known, any complex cyclic matrix $A$ is similar to the unique companion matrix associated with the minimal polynomial of $A$. On the other hand, a cyclic matrix over a division ring $\mathbb F$ is similar to a companion matrix of…

Let $X, Y$ be smooth algebraic varieties of the same dimension. Let $f, g : X \to Y$ be finite polynomial mappings. We say that $f, g$ are equivalent if there exists a regular automorphism $\Phi \in Aut(X)$ such that $f = g\circ \Phi$. Of…

A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…

We determine all F,G in C[X] of degree at least 2 for which the semigroup generated by F and G under composition is not the free semigroup on the letters F and G. We also solve the same problem for F,G in X^2 C[[X]], and prove partial…

Let $K$ be a field, $R=K[x, y]$ the polynomial ring and $\mathcal{M}(K)$ the set of all pairs of square matrices of the same size over $K.$ Pairs $P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $\mathcal{M}(K)$ are called similar if…

Let $c(x_1,...,x_d)$ be a multihomogeneous central polynomial for the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of positive characteristic $p$. We show that there exists a multihomogeneous polynomial $c_0(x_1,...,x_d)$…

This paper concerns the enumeration of isomorphism classes of modules of a polynomial algebra in several variables over a finite field. This is the same as the classification of commuting tuples of matrices over a finite field up to…

Let $\mathcal{R} = \mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$ of characteristic 0. Consider $n$ algebraically independent elements $g_1, \dots, g_n$ in $\mathcal{R}$. Let $\mathcal{S}$ denote…

For a field $R$ of characteristic $p\ge 0$ and a matrix $c$ in the full $n\times n$ matrix algebra $M_n(R)$ over $R$, let $S_n(c,R)$ be the centralizer algebra of $c$ in $M_n(R)$. We show that $S_n(c,R)$ is a Frobenius-finite,…

We characterize polynomials having the same set of nonzero cyclic resultants. Generically, for a polynomial $f$ of degree $d$, there are exactly $2^{d-1}$ distinct degree $d$ polynomials with the same set of cyclic resultants as $f$.…

Polynomial $n\times n$ matrices $A(\lambda)$ and $B(\lambda)$ over a field $\mathbb F $ are called semi-scalar equivalent if there exist a nonsingular $n\times n$ matrix $P$ over the field $\mathbb F $ and an invertible $n\times n$ matrix…

Let $\mathbb{F}$ be a field. We show that given any $n$th degree monic polynomial $q(x)\in \mathbb{F}[x]$ and any matrix $A\in\mathbb{M}_n(\mathbb{F})$ whose trace coincides with the trace of $q(x)$ and consisting in its main diagonal of…

For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in…