The geometry of multivariate polynomial division and elimination

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.status: publishe

The canonical decomposition of Cnd and numerical Gröbner and border bases

© 2014 Society for Industrial and Applied Mathematics. This article introduces the canonical decomposition of the vector space of multivariate polynomials for a given monomial ordering. Its importance lies in solving multivariate polynomial systems, computing Gröbner bases, and solving the ideal membership problem. An SVD-based algorithm is presented that numerically computes the canonical decomposition. It is then shown how, by introducing the notion of divisibility into this algorithm, a numerical Gröbner basis can also be computed. In addition, we demonstrate how the canonical decomposition can be used to decide whether the affine solution set of a multivariate polynomial system is zero-dimensional and to solve the ideal membership problem numerically. The SVD-based canonical decomposition algorithm is also extended to numerically compute border bases. A tolerance for each of the algorithms is derived using perturbation theory of principal angles. This derivation shows that the condition number of computing the canonical decomposition and numerical Gröbner basis is essentially the condition number of the Macaulay matrix. Numerical experiments with both exact and noisy coefficients are presented and discussed.status: publishe

A geometrical approach to finding multivariate approximate LCMs and GCDs

In this article we present a new approach to compute an approximate least common multiple (LCM) and an approximate greatest common divisor (GCD) of two multivariate polynomials. This approach uses the geometrical notion of principal angles whereas the main computational tools are the Implicitly Restarted Arnoldi method and sparse QR decomposition. Upper and lower bounds are derived for the largest and smallest singular values of the highly structured Macaulay matrix. This leads to an upper bound on its condition number and an upper bound on the 2-norm of the product of two multivariate polynomials. Numerical examples are provided.publisher: Elsevier articletitle: A geometrical approach to finding multivariate approximate LCMs and GCDs journaltitle: Linear Algebra and its Applications articlelink: http://dx.doi.org/10.1016/j.laa.2012.12.043 content_type: article copyright: Copyright © 2013 Elsevier Inc. All rights reserved.status: publishe

Urachal neuroblastoma: first case report

Tumours of the urachus are exceptional in children. They represent 0.01% of all rumours and consist of mucosecretory adenocarcinoma and, more rarely, transitional cell carcinoma. We report a 6-month-old child with a urachal mass which, following biopsy, was shown to be a neuroblastoma