Branch points of homotopies: Distribution and probability of failure

Abstract

Homotopy continuation is a standard method used in numerical algebraic geometry for solving multivariate systems of polynomial equations. Techniques such as the so-called gamma trick yield trackable homotopies with probability one. However, since numerical algebraic geometry employs numerical path tracking methods, being close to a branch point may cause concern with finite precision computations. This paper provides a systematic study of branch points of homotopies to elucidate how branch points are distributed and use this information to study the probability of failure when using finite precision. Several examples, including a system arising in kinematics, with various start systems are included to demonstrate the theoretical analysis.