The Geometry of Multivariate Polynomial Division and Elimination | Zendy

Kim Batselier | Zendy; Philippe Dreesen | Zendy; Bart De Moor | Zendy

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The Geometry of Multivariate Polynomial Division and Elimination

Author(s) -

Kim Batselier,

Philippe Dreesen,

Bart De Moor

Publication year - 2013

Publication title -

siam journal on matrix analysis and applications

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.268

H-Index - 101

eISSN - 1095-7162

pISSN - 0895-4798

DOI - 10.1137/120863782

Subject(s) - mathematics , algebraic geometry , algebra over a field , gröbner basis , linear algebra , linear subspace , division (mathematics) , polynomial , multivariate statistics , basis (linear algebra) , system of polynomial equations , real algebraic geometry , geometry , algebraic number , pure mathematics , mathematical analysis , statistics , arithmetic

Multivariate polynomials are usually discussed in the framework of algebraic geometry. Solving problems in algebraic geometry usually involves the use of a Grobner basis. This article shows that linear algebra without any Grobner basis computation suffices to solve basic problems from algebraic geometry by describing three operations: multiplication, division, and elimination. This linear algebra framework will also allow us to give a geometric interpretation. Multivariate division will involve oblique projections, and a link between elimination and principal angles between subspaces (CS decomposition) is revealed. The main computational tool in this approach is the QR decomposition.

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