Abstract

The term “discriminant” is well known in relation with low degree univariate equations or ordinary differential equations and it has a well defined geometrical meaning naturally connected to the way our vision grasps geometrical shapes. The discriminant of polynomials with specified shape (when it exists) is an (irreducible) polynomial, whose variables are in coefficients of the assigned shape, vanishing when the corresponding polynomial has multiple roots. Finding the discriminant or at least estimating its degree is a classical, well explored problem in mathematics which still faces many challenges. Besides its importance in computational mathematics, the discriminant locus carries deep and interesting geometrical and combinatorial insights and is one example of fruitful interplay between classical projective algebraic geometry and convex real geometry. This interplay will be the main theme of the talk. Besides the classical theory of discriminants, natural generalisations associated to systems of polynomial equations will be presented.