Related papers: Counting generating spaces of matrices

A recent paper of Church, Ellenberg, and Farb uses topology and representation theory of the symmetric group to prove enumerative results about square-free polynomials and F-stable maximal tori of the general linear group over the algebraic…

We propose several techniques to construct complete permutation polynomials of finite fields by virtue of complete permutations of subfields. In some special cases, any complete permutation polynomials over a finite field can be used to…

We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + ax + by + c for q in K^* and a, b, c in K. We prove, that among such algebras there are precisely five isomorphism classes.…

In this note, we initiate a study of the finite-dimensional representation theory of a class of algebras that correspond to noncommutative deformations of compact surfaces of arbitrary genus. Low dimensional representations are investigated…

The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…

We obtain tight bounds for the minimal number of generators of an ideal with bounded-degree generators in a polynomial ring $K[X_1,\dots,X_n],$ as well as a sharp quantification of the maximum possible size of a minimal generating set of…

We consider dual frames generated by actions of countable discrete groups on a Hilbert space. Module frames in a class of modules over a group algebra are shown to coincide with a class of ordinary frames in a representation of the group.…

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full…

We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input…

It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on matrices that are invariants by the action of conjugation by general linear…

Let $M_n(\mathbb{Z})$ the ring of $n$-by-$n$ matrices with integral entries, and $n \geq 2$. This paper studies the set $G_n(\mathbb{Z})$ of pairs $(A,B) \in M_n(\mathbb{Z})^2$ generating $M_n(\mathbb{Z})$ as a ring. We use several…

Over a field of characteristic 0, the algebra of invariants of several $n\times n$ matrices under simultaneous conjugation by $GL_n$ is generated by traces of products of generic matrices. In this paper we have found, in terms of…

We compute the minimal cardinality of a covering (resp. an irredundant covering) of a vector space over an arbitrary field by proper linear subspaces. Analogues for affine linear subspaces are also given.

We define matrix models that converge to the generating functions of a wide variety of loop models with fugacity taken in sets with an accumulation point. The latter can also be seen as moments of a non-commutative law on a subfactor planar…

We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials over a finite…

In this paper, we construct explicitely polynomial automorphisms of affine n-space for certain n. More precisely, we construct algebraic subgroups of the general polynomial group GA_n(k) where k is an arbitrary base ring of characteristic…

In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the number of matrices whose rows…

C. F. Gauss discovered a beautiful formula for the number of irreducible polynomials of a given degree over a finite field. Assuming just a few elementary facts in field theory and the exclusion-inclusion formula, we show how one see the…

In this paper we study a relationship between systems of $n$ subspaces and representations of $*$-algebras generated by projections. We prove that irreducible nonequivalent $*$-representations of $*$-algebras $\mathcal P_{4,com}$ generate…