Jean-Baptiste Cadart: Introduction to Rational Univariate Representation or How to Compute Roots From a Polynomial System

Abstract

Computational algebraic geometry is the study of varieties we can compute and of its algorithms.

The first traces of solving polynomial systems with several variables appeared in the first century CE, but the question of efficiently computing the roots is still open. Fabrice Rouillier introduced in 1999 the concept of the RUR, which is a representation of the roots of a 0-dimensional polynomial system (points) in the variables x₁,...,x_n by a univariate polynomial in a new variable T and an expression of the coordinates x₁,...,x_n as fractions of polynomials in T.

I will start by giving some basic notions of computational algebraic geometry to then give the strategies to compute such a RUR. I will also explain how to get in the first univariate polynomial the multiplicities of the roots of the system, using a method that was known, but never proved and expanded into details (before me).