Finding all solutions to a system of polynomial equations

Finding All Solutions to a System of Polynomial Equations

Given a polynomial equation of degree d over the complex domain, the Fundamental Theorem of Algebra tells us that there are d solutions, assuming that the solutions are counted by multiplicity. These solutions can be approximated by deforming a standard «th degree equation into the given equation, and following the solutions through the deformation. This is called the homotopy method. The Funda...

Finding All Hypergeometric Solutions of Linear Diierential Equations Finding All Hypergeometric Solutions of Linear Diierential Equations Finding All Hypergeometric Solutions of Linear Diierential Equations

Hypergeometric sequences are such that the quotient of two successive terms is a xed rational function of the index. We give a generalization of M. Petkov sek's algorithm to nd all hypergeometric sequence solutions of linear recurrences, and we describe a program to nd all hypergeometric functions that solve a linear diierential equation. Solutions hyperg eom etriques des equations dii erentiel...

Robustly finding all solutions to nonlinear systems of equations

This paper focuses on finding all solutions to nonlinear algebraic systems of equations in a given domain. First, scattered points are computed that can be considered as a representative sample of the solution manifold of a given underdetermined nonlinear system of equations. If this underdetermined system is then augmented by further equations such that the system becomes well-defined, the sam...

Computing All Solutions to Polynomial Equations in Economics

On Finding Small Solutions of Modular Multivariate Polynomial Equations

Abst rac t . Let P(x) -0 (rood N) be a modular multivariate polynomial equation, in m variables, and total degree k with a small root x0. We show that there is an algorithm which determines c(~ 1) integer polynomial equations (in m variables) of total degree polynomial in cmklog N, in time polynomial in craklog N, such that each of the equations has xo as a root. This algorithm is an extension ...