This will be an algebraic talk, where some issues of classical
19th century algebra (of Bezout, Sylvester,...) will be mixed with
certain aspects of the theory of orthogonal polynomials. In fact,
the roots of the talk are even more ancient: we shall discuss
some new properties of ... Euclid's algorithm.

Abstract:
We prove a quadratic expression for the Bezoutian of two univariate
polynomials in terms of the remainders for the Euclidean algorithm.
In case of two polynomials of the same degree, or of consecutive
degrees, this allows us to interpret their Bezoutian as the Christoffel-
Darboux kernel for a finite family of orthogonal polynomials, arising
from the Euclidean algorithm. As a consequence, we give orthogonality
properties of remainders, and reproducing properties of Bezoutians.