Related papers: F-thresholds, integral closure and convexity

We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive…

In this paper we investigate the question of normality for special monomial ideals in a polynomial ring over a field. We first include some expository sections that give the basics on the integral closure of a ideal, the Rees algebra on an…

We show that for ideals primary to a maximal ideal in a normal domain of finite type over the complex numbers, its tight closure is contained inside the continuous closure.

In this paper we show that the sets of $F$-jumping coefficients of ideals form discrete sets in certain graded $F$-finite rings. We do so by giving a criterion based on linear bounds for the growth of the Castelnuovo-Mumford regularity of…

The purpose of this article is to present, in a simple way, an analytic approach to special numbers and polynomials. The approach is based on the derivative polynomials. The paper is, to some extent, a review article, although it contains…

In this article, we investigate F-pure thresholds of polynomials that are homogeneous under some N-grading, and have an isolated singularity at the origin. We characterize these invariants in terms of the base p expansion of the…

Let I\subset K[x,y] be a <x,y>-primary monomial ideal where K is a field. This paper produces an algorithm for computing the Ratliff-Rush closure I for the ideal I=<m_0,...,m_{n}> whenever m_{i} is contained in the integral closure of the…

In this work we define a numerical invariant called $F$-volume. This number extends the definition of $F$-threshold of a pair of ideals $I$ and $J$, $c^J(I)$ to a sequence of ideals $J$, $I_1, \ldots, I_t$. We obtain several properties that…

This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The main result in…

In this paper, we prove that the set of all $F$-pure thresholds of ideals with fixed embedding dimension satisfies the ascending chain condition. As a corollary, given an integer $d$, we verify the ascending chain condition for the set of…

In this article we give an explicit formula for the jumping numbers of an ideal of finite colenght in a two-dimensional regular local ring with an algebraically closed residue field. For this purpose, we associate a certain numerical…

We provide a formula for $F$-thresholds of a Thom-Sebastiani type polynomial over a perfect field of prime characteristic. This result extends the formula for the $F$-pure threshold of a diagonal hypersurface. We also compute the first test…

In this paper we give upper and lower bounds as well as a heuristic estimate on the number of vertices of the convex closure of the set $$ G_n=\left\{(a,b) : a,b\in \Z, ab \equiv 1 \pmod{n}, 1\leq a,b\leq n-1\right\}. $$ The heuristic is…

The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains…

Given an ideal J on a smooth variety in characteristic zero, we estimate the F-jumping numbers of the reductions of J to positive characteristic in terms of the jumping numbers of J and the characteristic. We apply one of our estimates to…

The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…

In this note, we derive a formula for the F-pure threshold of diagonal hypersurfaces over a perfect field of prime characteristic. We also calculate the associated test ideal at the F-pure threshold, and give formulas for higher jumping…

We examine functions representing the cumulative probability of a binomial random variable exceeding a threshold, expressed in terms of the success probability per trial. These functions are known to exhibit a unique inflection point. We…

In this article, we define three new operations on ideals which generalize integral closure and Frobenius closure of ideals, whose definitions incorporate an auxiliary ideal and a real parameter. These additional ingredients are common in…

We prove that the $F$-jumping numbers of the test ideal $\tau(X; \Delta, \ba^t)$ are discrete and rational under the assumptions that $X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+\Delta$ is…