  Given a generic semidefinite program, specified by matrices with rational
entries, each coordinate of its optimal solution is an algebraic number. We
study the degree of the minimal polynomials of these algebraic numbers.
Geometrically, this degree counts the critical points attained by a linear
functional on a fixed rank locus in a linear space of symmetric matrices. We
determine this degree using methods from complex algebraic geometry, such as
projective duality, determinantal varieties, and their Chern classes.
