5 - Straight-line Programs

Summary

Abstract

Solving symbolically polynomial equation systems when intermediate and final polynomials are represented in the usual dense encoding turns out to be very inefficient: the sizes of the systems one can deal with do not respond to realistic needs. Evaluation representations appeared in this frame a decade ago as a new possibility to treat new families of problems.

We present a survey of the most recent complexity results for different polynomial problems when polynomials are encoded by evaluation (straight-line) programs. We also show surprising mathematical by-products, such as new mathematical invariants and results, that appeared as a consequence of the search of good algorithms.

Introduction

There are several geometric questions that naturally arise when we are faced to a system of polynomial multivariate equations: do the given equations have at least a common root in an algebraic closure of the base field? If this is so, is there a finite or infinite number of them? What is the dimension of the solution variety? How to describe it in a more tractable manner?

Two major lines have been proposed to answer this kind of questions: numerical analysis which responds with approximate solutions, and computational algebra with its symbolic procedures giving exact solutions. In this paper we deal with this second aspect, although the evaluation methods we describe tend a natural bridge to the numerical point of view.