相关论文: Slope-determinant method, complex cellular structu…
Motivated by the analysis of the performance of the simplex method we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules which are fundamental for the following reasons: First, they…
Caro and Pasten gave an explicit upper bound on the number of rational points on a hyperbolic surface that is embedded in an abelian variety of rank at most one. We show how to use their method to produce a refined bound on the number of…
Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose…
We extend the theory of complex cells introduced by Binyamini and Novikov to the sharply o-minimal setting, obtaining cellular preparation and parameterization theorems which are polynomially effective in the degrees of the relevant sets.…
We study the topology of the real algebraic hypersurfaces in $\mathbb{P}^n$ that can be constructed via combinatorial patchworking using triangulations that are dilations by two of other triangulations. By examining the real critical points…
We give an asymptotic formula for the number of rational points of bounded height on algebraic varieties defined by systems of multihomogeneous diagonal equations. The proof uses the Hardy-Littlewood circle method and the hyperbola method…
In the first part of this paper, we establish some results around generalized Borel's Theorem. As an application, in the second part, we construct example of smooth surface of degree $d\geq 19$ in $\mathbb{CP}^3$ whose complements is…
In some cases, computational benefit can be gained by exploring the hyper parameter space using a deterministic set of grid points instead of a Markov chain. We view this as a numerical integration problem and make three unique…
We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity,…
The Novelli-Pak-Stoyanovskii algorithm is a sorting algorithm for Young tableaux of a fixed shape that was originally devised to give a bijective proof of the hook-length formula. We obtain new asymptotic results on the average case and…
We outline a method to compute rational models for the Hilbert modular surfaces Y_{-}(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in Q(sqrt{D}), via moduli…
We propose a framework of principal manifolds to model high-dimensional data. This framework is based on Sobolev spaces and designed to model data of any intrinsic dimension. It includes principal component analysis and principal curve…
We construct standard resolutions for analytic local modules on complex hypersurfaces using standard basis methods, with extensions to complete intersections. The algebraic version over arbitrary infinite fields is also suggested.…
We describe a class of sharply o-minimal structures, called analytically generated structures, whose definable sets and their complexity filtration are determined by the collection of definable complex cells. We prove a polynomially…
Let $X$ be a smooth projective hypersurface defined over $\mathbb{Q}$. We provide new bounds for rational points of bounded height on $X$. In particular, we show that if $X$ is a smooth projective hypersurface in $\mathbb{P}^n$ with $n\geq…
A Besicovitch-Rado-Kinney (BRK) set in $\mathbb{R}^n$ contains a hypersphere of every radius. In $\mathbb{F}_q^n$, BRK-type sets of degree $\ell$ analogously contain a family of $(n-1)$-dimensional surfaces, parametrized by a dilation…
We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic…
Higher-order numerical methods are used to find accurate numerical solutions to hyperbolic partial differential equations and equations of transport type. Limiting is required to either converge to the correct type of solution or to adhere…
We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of…
In this paper, we will give a uniform upper bound of the number of rational points of bounded height in non-singular curves by applying the global determinant method.