|Responsables :||F. Delon, M. Dickmann, D. Gondard|
|Email des responsables :||firstname.lastname@example.org|
|Adresse :||Sophie Germain|
|Orateur(s)||Victor Vinnikov - Ben Gurion University of the Neguev - Israel,|
|Titre||Hyperbolic polynomials and their determinantal representations|
|Horaire||14:00 à 15:45|
A homogeneous polynomial of degree d with real coefficients is called hyperbolic with respect to a point if any real line through this point intersects the corresponding hypersurface in d real points (counting multiplicities). Hyperbolic polynomials are in a sense the opposite of strictly positive polynomials: they have as many real zeroes as possible. Hyperbolic polynomials were first introduced by Garding in the study of linear hyperbolic PDEs with constant coefficient in the 1950s; he showed that a hyperbolic polynomial determines a convex cone, called a hyperbolicity cone. In recent years hyperbolic polynomials and hyperbolicity cones came to play an important in convex programming as well as combinatorics and other areas.
Much like a representation as a sum of squares certifies the positivity of a polynomial, its hyperbolicity is certified by a representation as a determinant of a matrix of linear forms, with the coefficient matrices of the linear forms satisfying some positivity conditions. I will describe some of what is known about the existence of these determinantal representations, usually "with denominators". One fruitful approach uses a Hermitian Positivstellensatz that gives a representation of a polynomial satisfying matrix positivity conditions as a weighted sum of hermitian squares.