Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces

Autor:	De Loera, Jesús A., Petrović, Sonja, Stasi, Despina
Rok vydání:	2015
Předmět:	Computer Science - Discrete Mathematics Mathematics - Commutative Algebra Mathematics - Algebraic Geometry Mathematics - Combinatorics
Druh dokumentu:	Working Paper
Popis:	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 G\"artner 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). Comment: Minor edits, added two references; results unchanged
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/1503.08804 Zobrazit plný text záznamu View this record from Arxiv