Random sampling in computational algebra: Helly numbers and violator spaces

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• 作者：Jesú ; s A. De Loera^a ; ^{deloera@math.ucdavis.edu" class="auth_mail" title="E-mail the corresponding author} ; Sonja Petrović^b ; ^{sonja.petrovic@iit.edu" class="auth_mail" title="E-mail the corresponding author} ; Despina Stasi^b ; ^{despina.stasi@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：68W20 ; 68R05 ; 12Y05 ; 13P10 ; 14Q10 ; 08A40
• 刊名：Journal of Symbolic Computation
• 出版年：2016
• 出版时间：November-December 2016
• 年：2016
• 卷：77
• 期：Complete
• 页码：1-15
• 全文大小：465 K

文摘

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).