EXTENDED ABSTRACT. This talk is based on joint work with V. Vin-nikov. Historical Background. In this extended abstract we shall denote by P d the complex projective d-dimensional space and by P d (R) the real projective d-dimensional space. By G(, d) we shall denote the Grassmannian of-dimensional linear subspaces in P d. In fact G(, d) = Gr(+ 1, d + 1), the Grassmannian of l + 1 dimensional spaces in C d+1. The study of hyperbolic polynomials originated with partial differential equations.

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... ear hyperbolic partial differential equations are equations for which the Cauchy problem can be locally solved for non-characteristic hyperplanes. Lars Gärding published a pioneering work defining hyperbolic polynomials (and thus hyperbolic hyper-planes in P d). A homogeneous polynomial, p ∈ R[x 0 ,. .. , x d ], is hyperbolic if there exists a real point a such that p(a) = 0 and for every other real point x the roots of p(ta + x) are real. We will say that a point a is a witness to the hyperbolicity of p. The cone over the set of all witnesses is called the hyperbolicity cone of p. A closed set C ⊂ R d+1 is called an algebraic interior of degree d if there exists a polynomial of degree m such that C is the closure of a connected component of {x ∈ R m | p(x) > 0} and m is the minimal degree among all polynomials that satisfy the above condition for C. In fact hyperbolicity cones of homogeneous polynomials provide the first examples of algebraic interiors. Another example is provided by LMI representable sets, i.e., sets C such that there exist symmetric real matrices A 0 ,. .. , A d , with A 0 positive definite and C = {x ∈ R d+1 | x 0 A 0 +. .. + x d A d ≥ 0}. A natural question arises whether all algebraic interiors admit an LMI representation with size equal the degree of the algebraic interior. A positive answer was provided by J. W. Helton and V. Vinnikov based on earlier works of V. Vinnikov on determinantal representations. A negative answer for higher dimensions was provided, using different methods, by several authors. In particular using matroid theory P. Branden proved that even if one allows to take powers of your polynomial still there won't be any LMI representations available. Main Results. In this work we chose to focus on a slightly different question, namely hyperbolicity and determinantal representability of subvarieties of codimen-sion greater that 1. The idea originated from the fact that P d = Gr(1, d + 1). If one considers X ⊂ P d defined over the reals and of codimension , then a natural notion of hyperbolicity arises. We say that X is hyperbolic if there exists a real − 1-dimensional plane V ⊂ P d , such that for every real-dimensional plane V ⊂ U ⊂ P d , we have that U intersects X only at real points. The incidence correspondence is the set Σ ⊂ G(− 1, d) × P d , defined as follows: Σ = {(V, x) | x ∈ V } .