Functional norms, condition numbers and numerical algorithms in algebraic geometry

TL;DR

This paper explores the use of $L_p$ norms, especially $L_{ infty}$, in numerical algebraic geometry to improve the efficiency of iterative algorithms for problems like homology computation, meshing, and polynomial zeros.

Contribution

It introduces the application of $L_p$ norms in algebraic geometry, demonstrating their advantages in reducing computational complexity of key algorithms.

Findings

Using $L_p$ norms improves algorithm efficiency

Significant reduction in computational complexity for key problems

Application to real and complex polynomial systems

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).