Complexity of Sparse Polynomial Solving: Homotopy on Toric Varieties and the Condition Metric

Abstract

This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of n-variate polynomial equations is specified through n monomial bases. The natural locus for the roots of those systems is known to be a certain toric variety. This variety is a compactification of $$(\mathbb {C}\setminus \{0\})^n$$ , dependent on the monomial bases. A toric Newton operator is defined on that toric variety. Smale’s alpha theory is generalized to provide criteria of quadratic convergence. Two condition numbers are defined, and a higher derivative estimate is obtained in this setting. The Newton operator and related condition numbers turn out to be invariant through a group action related to the momentum map. A homotopy algorithm is given and is proved to terminate after a number of Newton steps which is linear on the condition length of the lifted homotopy path. This generalizes a result from Shub (Found Comput Math 9(2):171–178, 2009. https://doi.org/10.1007/s10208-007-9017-6 ).