Related papers: Computing the tight closure in dimension two

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…

In the literature, there is no known general method (formula) to compute the Zariski closure of an ``infinite'' subset of the prime spectrum. This problem indeed deals with the prime ideals of an infinite direct product of nonzero…

In this paper, we present an interior point algorithm with a full-Newton step for solving a linearly constrained convex optimization problem, in which we propose a generalization of the work of Kheirfam and Nasrollahi…

In this paper, we consider the problem of minimizing a general homogeneous quadratic function, subject to three real or four complex homogeneous quadratic inequality or equality constraints. For this problem, we present a sufficient and…

Girth-regular graphs with equal girth, regular degree and chromatic index are studied for the determination of 1-factorizations with each 1-factor intersecting every girth cycle. Applications to hamiltonian decomposability and to…

We characterize the rings in which the equality $(\tau I:\tau)= I^*$ holds for every ideal $I \subset R$. Under certain assumptions, these rings must be either weakly F-regular or one-dimensional.

In this article we study F-pure thresholds (and, more generally, F-thresholds) of homogeneous polynomials in two variables over a field of characteristic p>0. Passing to a field extension, we factor such a polynomial into a product of…

Let $M$ be a representation of an acyclic quiver $Q$ over an infinite field $k$. We establish a deterministic algorithm for computing the Harder-Narasimhan filtration of $M$. The algorithm is polynomial in the dimensions of $M$, the weights…

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s) \oplus k(-2s+1)$, where $s \geq3$ is some…

We study the degree of non-homogeneous lattice ideals over arbitrary fields, and give formulae to compute the degree in terms of the torsion of certain factor groups of Z^s and in terms of relative volumes of lattice polytopes. We also…

The purpose of this note is to revisit the results of arXiv:1407.4324 from a slightly different perspective, outlining how, if the integral closures of a finite set of prime ideals abide the expected convexity patterns, then the existence…

We present an efficient and accurate algorithm for solving the Poisson equation in spherical polar coordinates with a logarithmic radial grid and open boundary conditions. The method employs a divide-and-conquer strategy, decomposing the…

In this paper we present and test a full discretization of all elements of the Calder\'on Calculus (layer potentials and integral operators) for the Helmholtz equation in smooth closed curves in the plane. The resulting integral equations…

Let K be a field and let S = K[x_1, ..., x_n] be a polynomial ring. Consider a homogenous ideal I in S. Let t_i denote reg(Tor_i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t_i for i > n/2 purely in…

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from…

We introduce a natural notion of depth that applies to individual cutting planes as well as entire families. This depth has nice properties that make it easy to work with theoretically, and we argue that it is a good proxy for the practical…

Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time…

We solve some computational problems for triangulated closed three-dimensional manifolds using groups of simplicial homology and cohomology modulo 2. Two efficient algorithms for computing the intersection numbers of 1- and 2-dimensional…

We analyse the complexity of solving the discrete logarithm problem and of testing the principality of ideals in a certain class of number fields. We achieve the subexponential complexity in $O(L(1/3,O(1)))$ when both the discriminant and…

In this paper, we consider a finiteness problem of saturated subsheaves of a hermitian locally free sheaf on an arithmetic variety. As an application, we could prove the unique existence of an arithmetic Harder-Narasimham filtration.