Functional norms, condition numbers and numerical algorithms in algebraic geometry

Abstract

In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand in order to optimize accuracy or computational complexity. In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of $L_p$ norms for numerical algebraic geometry, with an emphasis on $L_{\infty}$. This classical idea yields strong improvements in the analysis of the number of steps performed by numerous iterative algorithms. In particular, we exhibit three algorithms where, despite the complexity of computing $L_{\infty}$-norm, the use of $L_p$-norms substantially reduces computational complexity: a subdivision-based algorithm in real algebraic geometry for computing the homology of semialgebraic sets, a well-known meshing algorithm in computational geometry, and the computation of zeros of systems of complex quadratic polynomials (a particular case of Smale's 17th problem).