Author:

Keywords:

SISTA, Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, multivariate polynomial division, oblique projection, multivariate polynomial elimination, QR decomposition, CS decomposition, sparse matrices, principal angles, GROBNER BASES, CS DECOMPOSITION, PERTURBATION, ALGORITHMS, SUBSPACES, MATRICES, ANGLES, 0102 Applied Mathematics, 0103 Numerical and Computational Mathematics, Numerical & Computational Mathematics, 4901 Applied mathematics

Abstract:

Multivariate polynomials are usually discussed in the framework of algebraic geometry. Solving problems in algebraic geometry usually involves the use of a Gröbner basis. This article shows that linear algebra without any Gröbner 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. © 2013 Society for Industrial and Applied Mathematics.