\documentclass[12pt]{article}
\title{Numerical techniques in Groebner basis
and polynomial system solving}
\author{Carlo Traverso}
\date{Conference at COCOA-VI, Torino, 3 June 1999}
\begin{document}
\maketitle
We discuss some experiments concerning the possibility to use inexact
arithmetics in Buchberger algorithm for the computation of Groebner
bases, in view of finding the numerical solutions of a polynomial
system. These experiments have been made using the C++ library
PoSSoLib, developed in the POSSO and FRISCO ESPRIT projects.
The key issues are the following:
\begin{itemize}
\item the hybrid arithmetic, combining an integer mod $p$ and a float
(usually a bigfloat); other floating arithmetics usable in
Buchberger algorithm;
\item infinitesimals, numerical conditioning of
the problem of finding a Groebner basis or a border basis;
\item comparisons of different algorithms from the numerical
viewpoint: Buchberger, FGLM, F4;
\item how to introduce numerical techniques (pivoting) in Groebner
basis computation;
\item from a Groebner basis to the solutions: where is the best
cutting point for purely numerical techniques?
\end{itemize}
\end{document}