相关论文: Puiseux power series solutions for systems of equa…

This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order…

The problem of extrapolating the series in powers of small variables to the region of large variables is addressed. Such a problem is typical of quantum theory and statistical physics. A method of extrapolation is developed based on…

In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the…

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…

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are…

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…

We investigate the Voronoi diagrams with respect to an asymmetric tropical distance function, also for infinite point sets. These turn out to be much better behaved than the tropical Voronoi diagrams arising from the standard tropical…

The method of Taylor series expansion is used to develop a numerical solution to the reactor point kinetics equations. It is shown that taking a first order expansion of the neutron density and precursor concentrations at each time step…

Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial…

In this work, we illustrate and explore the use of Taylor series as solutions of differential equations. For a large a number of classes of differential equations in the literature, there are plenty of sources where the well known Taylor…

This article considers the problem of solving a system of $n$ real polynomial equations in $n+1$ variables. We propose an algorithm based on Newton's method and subdivision for this problem. Our algorithm is intended only for nondegenerate…

We present a new algorithm for computing hyperexponential solutions of ordinary linear differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic…

In this paper we develop a symbolic algorithm to calculate multivariate power series expansions of univariate polynomials over general base rings. We use this to give a complete power series algorithm to calculate the dual intersection…

One way of constructing explicit expressions of solutions of integrable systems of Partial Differential Equations (PDEs) goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to…

We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof…

In this paper, we present a generic parametrization of generically zero-dimensional parametric polynomial systems. More specifically, we study the specialization properties of the Rational Univariate Representation and derive bounds on the…

A polynomial homotopy is a family of polynomial systems in one parameter, which defines solution paths starting from known solutions and ending at solutions of a system that has to be solved. We consider paths leading to isolated singular…

We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the…

The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given…

We construct algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice of an Abhyankar valuation and are…