A Geometric Framework for Condition Numbers

Abstract

Solving equations—linear, algebraic, differential, difference, analytic, Diophantine …—is arguably the most central problem in mathematics. A case of this problem that can be efficiently tackled is that of linear systems of equations. What could be considered the level of difficulty immediately above that for linear systems, the case of quadratic, or more generally, polynomial equations, is substantially more complicated. Even for polynomials in one variable, classical results of Abel and Galois deprive us of any hope to actually compute their zeros. The best we can do is to approximate them (and a number of algorithms compute these approximations quite efficiently).

For systems of multivariate polynomials we need to add complexity obstructions. The first that meets the eye is the possibly large number of solutions. A system of n quadratic equations in n variables has (generically) 2ⁿ solutions in complex space $\mathbb {C}^{n}$. But each polynomial in the system has $\frac{1}{2}(n^{2}+3n+2)$ coefficients, and therefore the whole system is specified with Θ(n ³) coefficients. If we were to compute approximations for all its zeros, the size of the output would be exponential in the input size!

A focal theme in this third part of the book is that of systems of polynomial equations and algorithms that approximate solutions of these systems. These algorithms have a “numeric” character, and it goes without saying that their analyses strongly rely on appropriate condition numbers. But the nature of these systems and their solutions suggests a view of their condition numbers within a more general framework than we encountered previously. The present chapter introduces this framework and provides some motivating (but also interesting per se) examples.