Spectrahedral representations of hyperbolic plane curves

par Naldi

jeudi 5 juil. 2018, 11:00 → 11:30 Europe/Paris

Salle XR203 (Bâtiment XLIM)

Every real homogeneous polynomial F in three variables admits a symmetric determinantal representation F = det(x*A+y*B+z*C) with A, B, C symmetric matrices. In case F is hyperbolic with respect to [1 0 0] then the Helton-Vinnikov Theorem (solving a 1956 conjecture by Peter Lax) implies that A can be chosen to be positive definite, and A, B, C to be of size d = deg F, leading to an explicit small certificate of hyperbolicity for the curve F = 0. The analogous certificate for more than three variables is false, but for a hyperbolic polynomial F there could be another hyperbolic polynomial H such that H*F has a definite determinantal representation (Generalized Lax Conjecture). In this talk I will show an explicit construction giving such a representation in the case of three variables, with G a product of linear forms and the size of the determinant at most quadratic on deg F (but that can become equal to deg F in some cases). Joint work with Daniel Plaumann (TU Dortmund) and Mario Kummer (TU Berlin).