Given generic real symmetric matrices A, B, C of size n x n, it is of interest to
study the set S of positivesemidefinite matrices of the form Id + x A + y B + z C,
where x, y, z are some real numbers.
The set S is a closed convex set in R^3, called a spectrahedron. The Zariski closure
of the Eucllidean boundary of S is an algebraic surface
{(x,y,z): det(Id+ x A+y B+ z C)=0}, which turns out to be always singular. A natural
question in real algebraic geometry is to understand (for a fixed n) possible
restrictions on the numbers P, Q of real singular points, respectively, of those
singularities that lie on S. In my talk I will discuss this problem for quintic
spectrahedra (n=5) and present a complete classification of pairs (P,Q), obtained in a
joint work with Taylor Brysiewicz and Mario Kummer.
