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

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 ).