Algorithms and their complexities

Summary

This chapter is intended as a brief survey of the different notions and results that arise when we try to compute the algebraic complexity of algorithms solving polynomial equation systems. Although it is essentially self-contained, many of the definitions, problems and results we deal with also appear in many other chapters of this book. We start by considering algorithms which use the dense representation of multivariate polynomials. Some results about the algebraic complexities of the effective Nullstellensatz, of quantifier elimination processes over algebraically closed fields and of the decomposition of algebraic varieties when considering this model are stated. Then, it is shown that these complexities are essentially optimal in the dense representation model. This is the reason why a change in the encoding of polynomials is needed to get better upper bounds for the complexities of new algorithms solving the already mentioned tasks. The straight-line program representation for multivariate polynomials is defined and briefly discussed. Some complexity results for algorithms in the straight-line program representation model are mentioned (an effective Nullstellensatz and quantifier elimination procedures, for instance). A description of the Newton-Hensel method to approximate roots of a system of parametric polynomial equations is made. Finally, we mention some new trends to avoid large complexities when trying to solve polynomial equation systems.