Related papers: Containment problem for points on a reducible coni…

We consider the ideal of inner $2$-minors $I_{\mathcal{P}}$ of a finite set of cells $\mathcal{P}$, which we call the cell ideal of $\mathcal{P}$. A nice interpretation for the height of an unmixed ideal $I_{\mathcal{P}}$, in terms of the…

Let $R$ be a smooth affine algebra over an infinite perfect field $k$. Let $I\subset R$ be an ideal, $\omega_I:(R/I)^n\to I/I^2$ a surjective homomorphism and $Q_{2n}\subset \mathbb{A}^{2n+1}$ be the smooth quadric defined by the equation…

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for…

This paper concerns the exponentiation of monomial ideals. While it is customary for the exponentiation operation on ideals to consider natural powers, we extend this notion to powers where the exponent is a positive real number. Real…

Symbolic powers are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph…

We give an elementary proof prove of the preservation of the Noetherian condition for commutative rings with unity $R$ having at least one finitely generated ideal $I$ such that the quotient ring is again finitely generated, and $R$ is…

We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic…

This paper proves a corner occupying theorem for the two-dimensional integral rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given integral rectangles into an integral rectangular container…

A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr\"obner bases for symmetric ideals in the infinite…

For an ideal $I$ in a polynomial ring over a field, a monomial support of $I$ is the set of monomials that appear as terms in a set of minimal generators of $I$. Craig Huneke asked whether the size of a monomial support is a bound for the…

Let $(R,\mathfrak{m})$ be a regular local ring or a polynomial ring over a field, and let $I$ be an ideal of $R$ which we assume to be graded if $R$ is a polynomial ring. Let astab$(I)$ resp. $\overline{\rm astab}(I)$ be the smallest…

We consider Hamiltonian with $N$ point interactions in $\R^d, d=2,3,$ all with the same coupling constant, placed at vertices of an equilateral polygon $\PP_N$. It is shown that the ground state energy is locally maximized by a regular…

In this paper we generalize the theorem of Ein-Lazarsfeld-Smith (concerning the behavior of symbolic powers of prime ideals in regular rings finitely generated over a field of characteristic 0) to arbitrary regular rings containing a field.…

Given a finite set of points $\Gamma$ in $\mathbb P^{k-1}$ not all contained in a hyperplane, the "fitting problem" asks what is the maximum number $hyp(\Gamma)$ of these points that can fit in some hyperplane and what is (are) the…

There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among…

The following two decision problems capture the complexity of comparing integers or rationals that are succinctly represented in product-of-exponentials notation, or equivalently, via arithmetic circuits using only multiplication and…

Let (R,m) be a regular local ring with prime ideals p and q such that p+q is m-primary and dim(R/p)+dim(R/q)=dim(R). It has been conjectured by Kurano and Roberts that p^{(n)} \cap q \subseteq m^{n+1} for all positive integers n. We discuss…

This paper is a continuation of a previous work by the author and G. Puninski where iterated intersections of powers of ideals were studied in rings of iterated differential polynomials. We present a method which can be used to show that…

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. Let $I+J$ denote the ideal generated by $I$ and $J$ in $A\otimes_k B$. We prove the following expansion for the symbolic powers: $$(I+J)^{(n)} =…

We discuss the problem of determining reduction number of a polynomial ideal I in n variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of I and requires computation…