Complexity of Bezout's Theorem VI: Geodesics in the Condition (Number) Metric

TL;DR

This paper introduces a new complexity measure based on the length of paths in a condition metric, providing bounds on Newton steps needed for problem-solving and motivating the study of geodesics in this metric.

Contribution

It defines a novel condition metric and establishes a complexity measure for problem-solving paths, linking path length to computational effort.

Findings

New condition metric introduced

Upper bounds on Newton steps established

Study of geodesics in the condition metric motivated

Abstract

We introduce a new complexity measure of a path of (problems, solutions) pairs in terms of the length of the path in the condition metric which we define in the article. The measure gives an upper bound for the number of Newton steps sufficient to approximate the path discretely starting from one end and thus produce an approximate zero for the endpoint. This motivates the study of short paths or geodesics in the condition metric.