OAI.Paper256.main_result

The theorem states that, for the ambient real vector space of triples consisting of two real symmetric 4x4 matrices X and Z and a vector y in R^3, six things hold. First, this space has real dimension 23 (10+10+3). Second, a distinguished polynomial in these 23 coordinates, defined as det(det(X)·Z − φ_y(adj X)) where adj is the adjugate and φ_y(M) is a 4x4 matrix built from M and the 3x3 matrix Q(y) with diagonal entries y_i²+y_{i+1}² (indices mod 3) and off-diagonal entries −y_i y_j, is homogeneous of degree 20, and, third, its total degree is exactly 20. Fourth, it takes the value 1 at the base point where X and Z are both the identity matrix and y=0. Fifth, for every point x, the univariate polynomial obtained by substituting t·(base point) − x for the coordinates has only real roots, so any complex root has zero imaginary part; this is the hyperbolicity of the polynomial with respect to the base point. Sixth, for every positive integer N and every real-linear map L from the ambient space to the real symmetric N×N matrices, the cone of points x for which all complex roots t of that line polynomial are real and nonnegative is not equal to the set of x for which L(x) is positive semidefinite; that is, the hyperbolicity cone has no spectrahedral representation of any size.