Bit complexity for multi-homogeneous polynomial system solving Application to polynomial minimization

TL;DR

This paper develops bit complexity estimates for solving multi-homogeneous polynomial systems and applies these results to polynomial minimization, under genericity assumptions ensuring finite solutions and maximal Jacobian rank.

Contribution

It provides the first bit complexity bounds for solving multi-homogeneous systems and applies them to polynomial minimization problems.

Findings

Bit complexity is quadratic in the number of solutions.

Complexity is linear in the input system's height.

Probabilistic algorithm with analyzed success probability.

Abstract

Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input system under some genericity assumptions. The assumptions essentially imply that the Jacobian matrix of the system under study has maximal rank at the solution set and that this solution set if finite. The algorithm is probabilistic and a probability analysis is provided. Next, we apply these results to the problem of optimizing a linear map on the real trace of an algebraic set. Under some genericity assumptions, we provide bit complexity estimates for solving this polynomial minimization problem.