Related papers: Discriminants et sommes de carr\'es
We generalize several important results from the perturbation theory of linear operators to the setting of semisimple orthogonal symmetric Lie algebras. These Lie algebras provide a unifying framework for various notions of matrix…
We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the…
We prove that a $q$-deformation $\Disc k\X q$ of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of $\Disc k\X q$…
Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute…
The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and…
We provide a complete structure theorem for involutory matrices. This yields a new approach to principal angles between subspaces and provide a series of nice formulae for these angles.
In [Trace identities and $\bf {Z}/2\bf {Z}$-graded invariants, {\it Trans. Amer. Math. Soc. \bf309} (1988), 581--589] we generalized the first and second fundamental theorems of invariant theory from the general linear group to the general…
The authors introduce generalized Campanato space with regularized condition over non-homogeneous space, and study its basic properties including the John-Nirenberg inequality and equivalent characterizations. As applications, the…
Two-dimensional linear spaces of symmetric matrices are classified by Segre symbols. After reviewing known facts from linear algebra and projective geometry, we address new questions motivated by algebraic statistics and optimization. We…
We present a generalization of the Newton-Girard identities, along with some applications. As an addendum, we collect many evaluations of symmetric polynomials to which these identities apply.
Given a smooth proper morphism $f\colon X\rightarrow S$, we introduce a certain derived category where morphisms are permitted to be $\mathcal{O}_S$-linear differential operators. We then prove a generalisation of Serre duality that applies…
In one variable, there exists a satisfactory classification of commutative rings of differential operators. In several variables, even the simplest generalizations seem to be unknown and in this report we give examples and pose questions…
In math.CO/0109093 the author obtained a formula for the value of an irreducible symmetric group character indexed by a partition of rectangular shape. In the present paper this formula is (conjecturally) generalized to arbitrary shapes.
We generalize the notion of S-equivalence, previously defined for semistable vector bundles, to points in arbitrary algebraic stacks and use it to describe the identification of points when passing to the moduli space. As applications, we…
We consider a generalized Gauss sum supported on matrices over a number field. We evaluate this Gauss sum and relate it to the number of totally isotropic subspaces of related quadratic spaces. Then we consider a further generalization of…
This paper presents a comprehensive description of the coordinate rings and Poisson brackets associated with the fourth Calogero-Moser space and invariant commuting pairs of matrices of size four. As an application, we compute their…
The continue fractions of quadratic surds are periodic, according to a theorem by Lagrange. Their periods may have differing types of symmetries. This work relates these types of symmetries to the symmetries of the classes of the…
Here we develop a technique of computing the invariants of $n-$ary forms and systems of forms using the discriminants of corresponding multilinear forms built of their partial derivatives, which should be cosidered as analogues of classical…
The paper gives a wide range, uniform, local approximation of symmetric binomial distribution. The result clearly shows how one has to modify the the classical de Moivre--Laplace normal approximation in order to give an estimate at the tail…
This paper investigates the equivalence reduction for several classes of multivariate polynomial matrices and their Smith forms, establishing some criteria for such reduction. In particular, we employ algebra isomorphisms as a key tool to…