Related papers: Huneke's degree-computing problem
The main focus of these notes is recent work on linear systems in which line arrangements play a role, including problems such as semi-effectivity, containment problems of symbolic powers of homogeneous ideals in their powers, bounded…
Let I = (F_1,...,F_r) be a homogeneous ideal of R = k[x_0,...,x_n] generated by a regular sequence of type (d_1,...,d_r). We give an elementary proof for an explicit description of the graded Betti numbers of I^s for any s \geq 1. These…
The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two…
Some proofs of the problems of the basic statistics proposed for numeric symbolic data.
In this note, we show various versions of holomorphic Morse inequalities tensoring with a coherent sheaf.
In his book `Physics and Philosophy', Heisenberg suggested that the quantum world is one of ``potentialities or possibilities'' and that the classical realm is one of ``things or facts''. After ascertaining that his categories most…
We continue the study of counting complexity begun in [Buergisser, Cucker 04] and [Buergisser, Cucker, Lotz 05] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the…
Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously…
In this paper we explore inequalities between symmetric homogeneous polynomials of degree four of three real variables and three nonnegative real variables. The main theorems describe the cases in which the smallest possible coefficient is…
We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and…
We give improved uniform estimates for the rate of convergence to Plancherel measure of Hecke eigenvalues of holomorphic forms of weight 2 and level N. These are applied to determine the sharp cutoff for the non-backtracking random walk on…
Author presents a study of certain category of the integrals, which might look quite difficult to compute, but in fact are easily computable, because they do not depend on the parameter in the integrand. As simple and elementary the…
Herzog, Huneke, and Srinivasan have conjectured that for any homogeneous $k$-algebra, the multiplicity is bounded above by a function of the maximal degrees of the syzygies and below by a function of the minimal degrees of the syzygies. The…
Given a set of inequalities determined by homogeneous forms, the following intertwined results are established: (1) the volume of the real semi-algebraic domain determined by these inequalities is explicitly determined; it is shown to be…
Motivated by the work of Chudnovsky and the Eisenbud-Mazur Conjecture on evolutions, Harbourne and Huneke give a series of conjectures that relate symbolic and regular powers of ideals of fat points in $\mathbb P^n$. The conjectures involve…
The present work is concerned with characterizing some algebraic invariants of edge ideals of hypergraphs. To this aim, firstly, we introduce some kinds of combinatorial invariants similar to matching numbers for hypergraphs. Then we…
Let $\pi$ be a cohomological automorphic representation of $PGL(2)$ over a number field of arbitrary signature and assume that the local component of $\pi$ at a prime $\mathfrak{p}$ is the Steinberg representation. In this situation one can…
The extremal problem of hypergraph colorings related to Erd\H{o}s--Hajnal property $B$-problem is considered. Let $k$ be a natural number. The problem is to find the value of $m_k(n)$ equal to the minimal number of edges in an $n$-uniform…
We give a positive answer to the Huneke-Wiegand Conjecture for monomial ideals over free numerical semigroup rings, and for two generated monomial ideals over complete intersection numerical semigroup rings.
This article investigates under which conditions the symbolic powers of the extension of an ideal is the same as the extension of the symbolic powers. Our result generalizes the known scenarios. As an application, we prove formulas for the…