Se p 20 08 THETA BODIES FOR POLYNOMIAL IDEALS

A polynomial ideal I ⊆ R[x] is THk-exact if every linear polynomial that is non-negative over VR(I), the real variety of I , is a sum of squares of polynomials of degree at most k modulo I . Lovász recognized that a graph is perfect if and only if the vanishing ideal of the characteristic vectors of its stable sets is TH1-exact, and asked for a characterization of ideals which are TH1-exact. We characterize finite point sets whose vanishing ideals are TH1-exact answering Lovász’s question for zero-dimensional varieties instead of ideals. Several properties and examples follow. Lovász’s question leads to a hierarchy of relaxations for the convex hull of VR(I) that generalizes Lovász’s theta body of a graph to a sequence of theta bodies for polynomial ideals. We prove that these theta bodies are a version of Lasserre’s relaxations, and are thus feasible regions of semidefinite programs. When VR(I) ⊆ {0, 1} , we show how these theta bodies relate to the Lovász-Schrijver relaxations of the convex hull of VR(I). As an application we derive a (new) canonical set of semidefinite relaxations for the cut polytope of an arbitrary graph. We also determine the structure of the first theta body of an arbitrary ideal which yields examples of non-zero-dimensional TH1-exact ideals.