Related papers: Asymptotic Behavior of Differential Powers
This paper studies the unitary diagonalization of matrices over formal power series rings. Our main result shows that a normal matrix is unitarily diagonalizable if and only if its minimal polynomial completely splits over the ring and the…
The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie…
In this paper we consider graded ideals in a polynomial ring over a field and ask when such an ideal has the property that all of its powers have a linear resolution. In particular it is shown that all powers of a monomial ideal with…
We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a…
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…
We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible…
We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…
Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to…
We consider the problem of determining whether a monomial ideal is dominant. This property is critical for determining for which monomial ideals the Taylor resolution is minimal. We first analyze dominant ideals with a fixed least common…
Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let $I\subset \mathbb K[x_1,\ldots,x_n]$ be a monomial ideal and let $G(I)$ denote the (unique) minimal monomial generating set of $I$. How small can…
This article investigates the soft-interior and the soft-cover of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the…
We exhibit an example of a product of two proper monomial ideals such that the residue class ring is not Golod. We also discuss the strongly Golod property for rational powers of monomial ideals, and introduce some sufficient conditions for…
We translate the operations of polarization and depolarization from monomial ideals in a polynomial ring to abstract simplicial complexes. As a result, we explicitly describe the relation between the Koszul simplicial complex of a monomial…
Symmetric strongly shifted ideals are a class of monomial ideals which come equipped with an action of the symmetric group and are analogous to the well-studied class of strongly stable monomial ideals. In this paper we focus on algebraic…
We study a class of determinantal ideals that are related to conditional independence (CI) statements with hidden variables. Such CI statements correspond to determinantal conditions on a matrix whose entries are probabilities of events…
In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties…
A class of unitary operations generated by idealized, semiclassical fields is studied. The operations implemented by sharp potential kicks are revisited and the possibility of performing them by softly varying external fields is examined.…
We consider the polynomial ring in finitely many variables over an algebraically closed field of positive characteristic, and initiate the systematic study of ideals preserved by the action of the general linear group by changes of…
Let $k$ be a field, let $A$ and $B$ be polynomial rings over $k$, and let $S= A \otimes_k B$. Let $I \subseteq A$ and $J \subseteq B$ be monomial ideals. We establish a binomial expansion for rational powers of $I+J \subseteq S$ in terms of…
Let $I$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$. The asymptotic behaviour of the $\text{v}$-number of the powers of $I$ is investigated. Natural lower and upper bounds which are linear…