  A graded K-algebra R has property N_p if it is generated in degree 1, has
relations in degree 2 and the syzygies of order less or equal to p on the
relations are linear. The Green-Lazarsfeld index of R is the largest p such
that it satisfies the property N_p. Our main results assert that (under a mild
assumption on the base field K) the c-th Veronese subring of a polynomial ring
has Green-Lazarsfeld index greater or equal to c+1. The same conclusion also
holds for an arbitrary standard graded algebra, provided c is sufficiently
large.
