Bit complexity for multi-homogeneous polynomial system solving—Application to polynomial minimization. (July 2018)

• Record Type: Journal Article
• Authors: Safey El Din, Mohab; Schost, Éric
• Is Part Of: Journal of symbolic computation. Volume 87(2018)
• Page Start: 176
• Page End: 206
• Publication Date: 2018-07
• Subjects: Polynomial system solving -- Multi-homogeneous systems -- Polynomial optimization; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2017.08.001
• Languages: English
• ISSNs: 0747-7171

Abstract

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 is 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.