Computing the topology of a real algebraic plane curve whose defining equations are available only “by values”

This paper is devoted to introducing a new approach for computing the topology of a real algebraic plane curve presented either parametrically or defined by its implicit equation when the corresponding polynomials which describe the curve are known only “by values”. This approach is based on the replacement of the usual algebraic manipulation of the polynomials (and their roots) appearing in the topology determination of the given curve with the computation of numerical matrices (and their eigenvalues). Such numerical matrices arise from a typical construction in Elimination Theory known as the Bézout matrix which in our case is specified by the values of the defining polynomial equations on several sample points

A Proof of the Main Theorem on Bezoutians

We give a self-contained proof that the nullity of the Bezoutian matrix associated with a pair of polynomials f and g equals the number of their common zeros counting multiplicities

Statistical and structured optimisation : methods for the approximate GCD problem.

The computation of polynomial greatest common divisors (GCDs) is a fundamental problem in algebraic computing and has important widespread applications in areas such as computing theory, control, image processing, signal processing and computer-aided design (CAD)

Vector spaces of linearizations for matrix polynomials: A bivariate polynomial approach

We revisit the important paper [D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971-1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials. We also show that every pencil in the double ansatz space is intrinsically connected to a Bézout matrix, which we use to prove the eigenvalue exclusion theorem. In addition our exposition allows for any degree-graded basis, the monomials being a special case. MATLAB code is given to construct the pencils in the double ansatz space for matrix polynomials expressed in any orthogonal basis