Related papers: A constructive elementary method for local resolut…
This paper formulates an elementary algorithm for resolution of singularities in a neighborhood of a singular point over a field of characteristic zero. The algorithm is composed of finite sequences of Newton polyhedra and monomial…
We use the resolution of singularities algorithm of [G4] to provide new estimates for exponential sums as well as new bounds on how often a function f(x) such as a polynomial with integer coefficients is divisible by various powers of a…
In this paper, a geometric resolution of singularities algorithm is developed. This method is elementary in its statement and proof, using explicit coordinate systems as much as possible. Each coordinate change used in the resolution…
We discuss to what extent the local techniques of resolution of singularities over fields of characteristic zero can be applied to improve singularities in general. For certain interesting classes of singularities, this leads to an embedded…
This article is an exposition of an elementary constructive proof of canonical resolution of singularities in characteristic zero, presented in detail in Invent. Math. 128 (1997), 207-302. We define a new local invariant and get an…
The elementary resolution of singularities algorithm of the author's earlier paper (math.CA/0609217) is developed further, replacing the quasibump functions in the blown up coordinates with the characteristic function of a rectangle times a…
We study the local behavior of weak solutions, with possible singularities, of nonlocal nonlinear equations. We first prove that sets of capacity zero are removable for weak solutions under certain integrability conditions. We then…
We propose the generalized quadrature methods for numerical solution of singular integral equation of Abel type. We overcome the singularity using the analytical calculation of the singular integral expression. The problem of solution of…
We prove the existence of resolution of singularities for arbitrary (not necessarily reduced or irreducible) excellent two-dimensional schemes, via permissible blow-ups. The resolution is canonical, and functorial with respect to…
Given a first-order autonomous algebraic ordinary differential equation, we present a method for computing formal power series solutions by means of places. We provide an algorithm for computing a full characterization of possible initial…
An algorithm for resolution of singularities in characteristic zero is described. It is expressed in terms of multi-ideals, that essentially are defined as a finite sequence of pairs, each one consiting of a sheaf of ideals and a positive…
In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a…
A convergence theorem is proved for a class of Nystrom methods for weakly singular integral equations on surfaces in three dimensions. Fredholm equations of the second kind as arise in connection with linear elliptic boundary value problems…
We present a new method for the numerical solution of singular integral equations on the real axis. The method's value stems from an explicit formula for the Cauchy integral of a complex exponential multiplied by a rational function.…
These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic…
We provide a procedure for resolving, in characteristic 0, singularities of a variety $X$ embedded in a smooth variety $Y$ by repeatedly blowing up the worst singularities, in the sense of stack-theoretic weighted blowings up. No history,…
We consider the uniqueness of solution (i.e., nonsingularity) of systems of $r$ generalized Sylvester and $\star$-Sylvester equations with $n\times n$ coefficients. After several reductions, we show that it is sufficient to analyze periodic…
The aim of this article is to introduce standard bases of ideals in polynomial rings with respect to a class of orderings which are not necessarily semigroup orderings. Our approach generalises the concept of standard bases with respect to…
We investigate the qualitative properties of a critical Hartree equation defined on punctured domains. Our study has two main objectives: analyzing the asymptotic behavior near isolated singularities and establishing radial symmetry of…
Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and…