We study various notions of multivariate functions which map families of positive semidefinite matrices or of conditionally positive semidefinite matrices into matrices of the same type. LINEAR ALGEBRA AND ITS APPLICATIONS 221:83-102 (1995) @ Elsevier Science Inc., 1995 0024-3795/95/$9.50 655 Avenue of the Americas, New York, NY 10010 SSDI 0024-3795(93)00232-O 84 C. H. FITZGERALD ET AL.

In this project, we are interested in approximating permanents of positive semidefinite Hermitian matrices. Specifically, we find conditions on positive semidefinite Hermitian matrices such that we can generalize the algorithm described in Sections 3.6 3.7 of [1] to matrices satisfying these conditions.

Notation: The set of real symmetric n ×n matrices is denoted S . A matrix A ∈ S is called positive semidefinite if x Ax ≥ 0 for all x ∈ R, and is called positive definite if x Ax > 0 for all nonzero x ∈ R . The set of positive semidefinite matrices is denoted S and the set of positive definite matrices + n is denoted by S++. The cone S is a proper cone (i.e., closed, convex, pointed, and solid). +

Using elementary duality properties of positive semidefinite moment matrices and polynomial sum-of-squares decompositions, we prove that the convex hull of rationally parameterized algebraic varieties is semidefinite representable (that is, it can be represented as a projection of an affine section of the cone of positive semidefinite matrices) in the case of (a) curves; (b) hypersurfaces param...