# ES Lecture by Prof. Dr. Bart De Moor

## Tidspunkt

27.06.2017 kl. 14.00 - 15.00

## Beskrivelse

Department of Electronic Systems is pleased to announce the following ES lecture:

“Back to the Roots: Solving Polynomial Systems with Numerical Linear Algebra Tools”

by Prof. Dr. Bart De Moor, ESAT-STADIUS, Department of Electrical Engineering, KU Leuven, Belgium

Abstract
Finding the roots of a set of multivariate polynomials has numerous applications in geometry and optimization, system and control theory, modeling and identification, statistics and bioinformatics, and many other scientific disciplines. It is an old yet fascinating problem, that has intrigued scientists throughout the ages, starting with the Greeks, over Fermat and Descartes, Newton, Leibniz, Bezout and many many others. It all started with trying to find the roots of a polynomial equation with real coefficients in one unknown.

In the beginning of the 19-th century, formulas were known for the roots of polynomials up to degree 4, but it was Galois who showed that no general formulas exist for degree 5 and higher. This implies that roots of polynomials of degree higher than 4 in general can only be found using (iterative) numerical algorithms. Soon thereafter, Sylvester and other contemporains started research on how to find the roots of sets of multivariate polynomials. Sylvester derived an elimination algorithm, in which he eliminates variables one by one, ending up with a ‘characteristic equation’ in one variable only. If one then has obtained the roots of this last equation, one can then ‘back-substitute’ root-by-root into the other equations, and hence in principle find all roots. Sylvester’s algorithm is the equivalent of Gaussian elimination for linear equations, but more importantly, his results imply that finding the roots of a polynomial system is an eigenvalue problem!

Later on, in the 20th century, there was a booming mathematical development, which gave birth to a discipline that today is called algebraic geometry, with a fabulous rich history, to which famous mathematicians, such as Hilbert, but also many others, contributed. It also led to the machinery of Gröbner bases and the like, which today are ubiquitous in books and symbolic methods in algebraic geometry, with numerous applications in fields like geometric design, combinatorics and integer programming, coding theory, robotics, etc…

We will however not talk about these developments, but return to the very roots and early days of the problem. In this talk, we will elaborate on a research program, the objective of which is to translate the many – symbolic - algorithms from algebraic geometry, into numerical linear algebra algebra algorithms. Our talk develops ideas on three complementary levels: - Geometric linear algebra, which deals with column and row vector spaces, dimensions, orthogonality, kernels.

CV
Bart De Moor was born Tuesday July 12, 1960 in Halle, Belgium. In 1983, he obtained his Master Degree in Electrical Engineering at the KU Leuven, Belgium, and a PhD in Engineering at the same university in 1988. He spent 2 years as a Visiting Research Associate at Stanford University (1988-1990) at the departments of EE (ISL, Prof. Kailath) and CS (Prof.Golub). Currently, he is a full professor at the Department of Electrical Engineering (http://www.esat.kuleuven.be/english) of the KU Leuven in the research group ESAT - STADIUS, Stadius Centre for Dynamical Systems, Signal Processing and Data Analytics

His research interests are in numerical linear algebra and optimization, system theory and system identification, quantum information theory, control theory, data-mining, information retrieval and bio-informatics (for books and research publications, see the publication search engine at http://homes.esat.kuleuven.be/~sistawww/cgi-bin/pub.pl).

Full details on his CV can be found at http://www.kuleuven.be/wieiswie/nl/person/00008904

Refreshments will be served after the lecture.

All are welcome!

Free of charge

### Arrangør

Department of Electronic Systems