Related papers: Elliptic determinant evaluations and the Macdonald…

The existence of certain Fq-spaces of differential forms of the projective line over a field K containing Fq leads us to prove an identity linking the determinant of the Moore matrix of n indeterminates with the determinant of the Moore…

Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight bounds on the maximal number of non-degenerate roots F can have over L, with all…

A characterization of the minimal $\mathcal{W}$-algebras associated with the Deligne exceptional series at level $-h^\vee/6$ is obtained by using one-parameter family of modular linear differential equations of order $4$. In particular, the…

We develop an analog of the exponential families of Wilf in which the label sets are finite dimensional vector spaces over a finite field rather than finite sets of positive integers. The essential features of exponential families are…

In this paper, certain mixed special polynomial families associated with Appell sequences are introduced and their properties are established. Further, operational rules providing connections between these families and the known special…

In this paper, we define a set of good basic invariants for the elliptic Weyl group for the elliptic root system. For an elliptic root system of codimension $1$, we show that a set of good basic invariants gives a set of flat invariants…

We consider partial symmetric Toeplitz matrices where a positive definite completion exists. We characterize those patterns where the maximum determinant completion is itself Toeplitz. We then extend these results with positive definite…

We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verify Macdonald's normalization conjectures for these…

Any maximal root subsystem of a finite crystallographic reduced root system is either a closed root subsystem or its dual is a closed root subsystem in the dual root system. In this article, we classify the maximal root subsystems of an…

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…

Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…

In this paper, another proof of Pell identities is presented by using the determinant of tridiagonal matrices. It is calculated via the Laplace expansion.

We present a variation and generalization of a determinant evaluation of Wilf (math.CO/9809120). His result concerns a matrix whose entries are the coefficients of powers of a given power series; we replace the powers by repeated…

We extend several relative perturbation bounds to Hermitian matrices that are possibly singular, and also develop a general class of relative bounds for Hermitian matrices. As a result, corresponding relative bounds for singular values of…

Using multiple q-integrals and a determinant evaluation, we establish a nonterminating 8-phi-7 summation for the root system C_r. We also give some important specializations explicitly.

Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The obtained combinatorial structure fits perfectly into an existing…

Building on work of J. Robinson and A. Shlapentokh, we develop a general framework to obtain definability and decidability results of large classes of infinite algebraic extensions of $\mathbb{F}_p(t)$. As an application, we show that for…

Let G be a finite group. For each integral representation $\rho$ of G we consider $\rho-$decomposable principally polarized abelian varieties; that is, principally polarized abelian varieties (X,H) with $\rho(G)-$action, of dimension equal…

This paper introduces and systematically studies Weyl-type, Witt-type, and non-associative algebras defined over expolynomial rings -- commutative rings generated by exponential functions $e^{\alpha x}$, exponentials of exponentials $e^{\pm…

Weyl groups are ubiquitous, and efficient algorithms for them -- especially for the exceptional algebras -- are clearly desirable. In this paper we provide several of these, addressing practical concerns arising naturally for instance in…