ON THE NEED OF CONVEXITY IN PATCHWORKING

Viro's construction of real smooth hypersurfaces uses regular (also ca lled convex or coherent) subdivisions of Newton polytopes. Nevertheles s, Viro's construction, sometimes called patchworking, can be applied as well to arbitrary subdivisions as a purely combinatorial procedure. Are the schemes coming from nonregular subdivisions, still topologica l types of some real smooth hypersurfaces? In the first part of this p aper we prove a combinatorial version of Hilbert's Lemma (a consequenc e of Bezout's Theorem) that bounds the depth of nests in a T-curve, an d we use this result and a previous work by I. Itenberg to answer the question affirmatively for T-curves of degree less than 6. According t o V. A. Rokhlin, a real algebraic scheme has complex orientation of ty pe I (alternatively of type IT) if any curve with this real scheme div ides (does not divide) its complexification. A real algebraic scheme h as indefinite type if there are type I and type II curves with that pa rticular scheme. In the second part of this paper, we describe a combi natorial algorithm, due to I. Itenberg and O. Viro, that allows one to determine the type of a T-curve. We then present a partial list of in definite schemes for T-curves of degrees 7 and 8. (C) 1998 academic Pr ess.