Complexity of Bezout’s Theorem IV: Probability of Success; Extensions

Abstract

We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in $n + 1$ complex variables of fixed degrees will find all the roots of the system. We also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context. Further complexity theory is developed.