Related papers: A Shape Lemma for Ideals of Differential Operators
We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the…
Through examples, we illustrate how to compute differential operators on a quotient of an affine semigroup ring by a radical monomial ideal, when working over an algebraically closed field of characteristic 0.
We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of $\mathbb{Z}$-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a $p$-derivation…
The aim of this paper is to unveil an unexpected relationship between the normal form of a polynomial with respect to a polynomial ideal and the more geometric concept of orthogonality. We present a new way to calculate the normal form of a…
We prove some algebraic results on the ring of matrix differential operators over a differential field in the generality of non-commutative principal ideal rings. These results are used in the theory of non-local Poisson structures.
Vanishing polynomials are polynomials over a ring which output $0$ for all elements in the ring. In this paper, we study the ideal of vanishing polynomials over specific types of rings, along with the closely related ring of polynomial…
Proving statements about linear operators expressed in terms of identities often leads to finding elements of certain form in noncommutative polynomial ideals. We illustrate this by examples coming from actual operator statements and…
This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an…
A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for…
In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic…
We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these…
In this thesis we are interested in describing some homological invariants of certain classes of monomial ideals. We will pay attention to the squarefree and non-squarefree lexsegment ideals.
The algebraic notion of a differential operator on a module over a commutative ring is not extended to a module over a noncommutative ring.
We present an approach for construction of functional bases of differential invariants for some infinite-dimensional algebras with coefficients of generating operators depending on arbitrary functions. An example for the…
Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this paper, we introduce the cozero-divisor graph $\acute{\Gamma}_I(R)$ of $R$ and obtain some related results.
We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. The associated special functions are eigenfunctions of some shape invariant operators. These operators can…
This paper grew out of the author's work on arXiv:2504.18460. Differential operators in the sense of Grothendieck acting between modules over a commutative ring can be interpreted as torsion elements in the bimodule of all operators with…
An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in…
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…