Before revision:  Third, we prove a sparse-matrix version of Putinar's Positivstellensatz: if P (x) has chordal sparsity and is positive definite on a compact semialgebraic set \mathcal{K}=\{x:
After revision:  Third, we prove that if P is positive definite on a compact semialgebraic set \mathcal{K}=\{x: