Random sampling in computational algebra: Helly numbers and violator spaces. (November 2016)

• Record Type: Journal Article
• Authors: De Loera, Jesús A.; Petrović, Sonja; Stasi, Despina
• Is Part Of: Journal of symbolic computation. Volume 77(2016)
• Page Start: 1
• Page End: 15
• Publication Date: 2016-11
• Subjects: 68W20 -- 68R05 -- 12Y05 -- 13P10 -- 14Q10 -- 08A40; Violator spaces -- Ideal generators -- Solving polynomial systems -- Randomized algorithm in algebra -- Large sparse systems of equations; 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.2016.01.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems and finding small generating sets of graded ideals. The cornerstone of our work is showing that the theory of violator spaces of Gärtner et al. applies to polynomial ideal problems. To show this, one utilizes a Helly-type result for algebraic varieties. The resulting algorithms have expected runtime linear in the number of input polynomials, making the ideas interesting for handling systems with very large numbers of polynomials, but whose rank in the vector space of polynomials is small (e.g., when the number of variables and degree is constant).