O KERNEL FOR NON-PSEUDOCONVEX TUBE DOMAINS IN C 2

We consider the Szego kernel associated with domains in C 2 given by = {(z,w) : Imw > b(Rez)} for b a non-convex polynomial of even degree with positive leading coefficient. Such domains are not pseudoconvex. We give a precise description of a subset of × on which the kernel and all of its derivatives are finite. We show, in particular, that for such domains, the Szego kernel has singularities off the diagonal of @ × @ as well as points on the diagonal at which it is finite.