26 research outputs found
A Note on the Hyperbolicity Cone of the Specialized V\'amos Polynomial
The specialized V\'amos polynomial is a hyperbolic polynomial of degree four
in four variables with the property that none of its powers admits a definite
determinantal representation. We will use a heuristical method to prove that
its hyperbolicity cone is a spectrahedron.Comment: Notable easier arguments and minor correction
Determinantal representations of semi-hyperbolic polynomials
We prove a generalization of the Hermitian version of the Helton-Vinnikov
determinantal representation of hyperbolic polynomials to the class of
semi-hyperbolic polynomials, a strictly larger class, as shown by an example.
We also prove that certain hyperbolic polynomials affine in two out of four
variables divide a determinantal polynomial. The proofs are based on work
related to polynomials with no zeros on the bidisk and tridisk.Comment: 14 pages, revisio
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Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability
Continuing the tradition initiated in the MFO workshops held in 2014 and 2017, this workshop was dedicated to the newest developments in real algebraic geometry and polynomial optimization, with a particular emphasis on free non-commutative real algebraic geometry and hyperbolic programming. A particular effort was invested in exploring the interrelations with free probability. This established an interesting dialogue between researchers working in real algebraic geometry and those working in free probability, from which emerged new exciting and promising synergies