Lecture 12 : Hyperbolic Polynomials , Interlacers , and Sums of Squares

It is clear Ce(f) is invariant under scaling by R>0 and open. Further, this cone is convex. Its closure is basic semialgebraic with all faces exposed, but we shall not elaborate on these points. The first examples of hyperbolic polynomials are familiar from optimization. Example 12.2. Let f = ∏n i=1 xi ∈ R[x1, . . . , xn]n and e = (1, . . . , 1). Then f is hyperbolic with respect to e and the hyperbolicity cone Ce(f) is the positive orthant R>0. Example 12.3. Let X = (xij) be an m ×m symmetric matrix of n = ( m+1 2 ) indeterminantes. Let f = det(X) ∈ R[xij ]m and e = I be the identity matrix. Then f is hyperbolic with respect to e and the hyperbolicity cone Ce(f) is the set of positive definite m×m matrices. To prove this, note that here f(te− a) = det(tI −A) is the characteristic polynomial of the real symmetric matrix A, so has all real roots, namely the eigenvalues of A.