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We study a graded vector space of polynomials associated to a square matrix, defined by a finite difference condition along the rows. We show this space coincides with one defined by directional derivatives, and prove it is…

We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal $(x_1,x_2,\cdots,x_m)^n$ of a polynomial ring in $m$ variables. We also give a combinatorial description of the Alexander…

The polarizations of one relation of degree five and two relations of degree six minimally generate the ideal of relations among a minimal generating system of the algebra of multisymmetric polynomials in an arbitrary number of…

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…

Real Nullstellensatz is a classical result from Real Algebraic Geometry. It has recently been extended to quaternionic polynomials by Alon and Paran. The aim of this paper is to extend their Quaternionic Nullstellensatz to matrix…

This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.

We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…

We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of standard, tilting, and simple modules, as…

We derive the solvability conditions and a formula of a general solution to a Sylvester-type matrix equation over Hamilton quaternions. As an application, we investigate the necessary and sufficient conditions for the solvability of the…

Supervised dimensionality reduction has emerged as an important theme in the last decade. Despite the plethora of models and formulations, there is a lack of a simple model which aims to project the set of patterns into a space defined by…

Bipartite determinantal ideals are introduced by Illian and the author as a vast generalization of the classical determinantal ideals intensively studied in commutative algebra, algebraic geometry, representation theory and combinatorics.…

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible), using the theory of V-filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by…

We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in…

We prove that double Schubert polynomials have the Saturated Newton Polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of…

We study pairs of finitely generated modules over a principal ideal domain and their corresponding matrix representations. We introduce equivalence relations for such pairs and determine invariants and canonical forms.

In this paper we study primality and primary decomposition of certain ideals which are generated by homogeneous degree $2$ polynomials and occur naturally from determinantal conditions. Normality is derived from these results.

We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s of X to positive characteristic such that the action of the Frobenius morphism on the…

We consider generalizations of the Sylvester matrix equation, consisting of the sum of a Sylvester operator and a linear operator $\Pi$ with a particular structure. More precisely, the commutator of the matrix coefficients of the operator…

Let $F$ be an algebraically closed field of characteristic different from $2$. We show that the images of multilinear $*$-polynomials on $UT_2$ are homogeneous vector spaces. An analogous result holds for $UT_3$ endowed with non-trivial…