LALO 60: Matrices and Polynomials in Computer Algebra: Algorithms and Software

The University of Western Ontario
London, Canada, July 22-24, 2024

We are grateful to the following colleagues who have accepted to deliver keynote presentations at LALO 60:

Juan Marie-Françoise Roy, University of Rennes, France
Title: Algebraic winding numbers

Abstract: We study in detail the properties of the algebraic winding number proposed in a paper by M. Eisermann with respect to complex root counting in rectangles, based on Cauchy indices computations for real polynomials. We also propose a new algebraic winding number which computes the number of complex roots of a polynomial in a rectangle under no assumptions, including roots on edges or vertices with appropriate counting. We extend both winding numbers to rational functions, obtaining then an algebraic version of the argument principle for rectangles. Joint work with Daniel Perrucci

Gema Gema María Díaz-Toca, Universidad de Murcia, Spain
Title: Bezoutians and Subresultants

Abstract: As it could not be otherwise, in this talk, I will discuss the two main pillars on which my work with Lalo is based: the Bezout matrix and Subresultants. I will mention their main characteristics, the relationship between these two tools, and the multiple applications we have developed over the years. For example, computing parametric GCD of polynomials, approximate GCD of polynomials, topology of plane curves, topology by values, and so on. It is worth noting that Subresultants are one of the main tools in computer algebra for dealing with polynomials. They provide fraction-free algorithms for computing the greatest common divisor of two polynomials, with good behavior under specialization, and they have multiple properties over integral domains. On the other hand, the Bezout matrix plays an important role in many fields of numerical and computer algebra as well, including elimination theory, stability theory, and control theory.

Tomas Tomás Recio Muñiz, Universidad Antonio de Nebrija, Spain

Title: A Real Mechanical Geometer

Abstract

Jane Jane Breen, Ontario Tech University, Canada
Title: Maximum spread of graphs and bipartite graphs

Abstract: Given a graph G, let $\lambda_1$ and $\lambda_n$ be the maximum and minimum eigenvalues of its adjacency matrix and define the spread of G to be $\lambda_1-\lambda_n$. In this talk we discuss solutions to a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. Our proofs use techniques from the theory of graph limits (graphons) and numerical analysis, including a computer-assisted proof of a finite-dimensional eigenvalue problem using both interval arithmetic and symbolic computations.

Juan Juan Rafael Sendra Pons, CUNEF, Spain
Title: Working in geometry with radicals of polynomials

Abstract

>Jürgen Jürgen Gerhard, Maplesoft, Canada
Title: Recent progress for polynomial computations in Maple

Abstract: I will discuss two recent additions to Maple in the area of polynomial computations: a new multivariate complex root solver, and a new implementation for computing the minimal polynomial of the real (or complex) part of an algebraic number.