Functional Norms and Condition Numbers for Polynomial Systems

Numerical algorithms are commonly analysed in terms of a condition number. In numerical linear algebra the norms involved in the definition of these condition numbers are carefully picked in each case to minimise complexity. In the analysis of algorithms dealing with polynomial systems, in contrast, the so called Bombieri-Weyl norm dominates the literature. The goal of this project is to introduce other norms in the numerical analysis of polynomial systems.Following the lead of numerical linear algebra, we want to consider a version of the spectral norm and analyse some of the best-known numerical algorithms for polynomial systems, both over the real and the complex numbers. We expect to be able to show that this new choice of norm leads to a marked improvement in the cost of the algorithm at hand.