On the Betti polynomials of certain graded ideals

Let S = K[ x1,..., xn] be apolynomialringovera eld K and I be anonzero gradedidealof S. Then, for t >> 0, theBettinumber ss q( S/ It) is apolynomial in t, whichisdenotedby BI q( t). Itisprovedthat BI q( t) is vanishedorof degree l ( I) - 1 provided I is amonomialidealgeneratedinasingledegree or grade( mR( I)) = codim( mR( I)) where m = ( x1,..., xn) and R( I) is theRees ringof I. Onelowerboundfortheleadingcoe cientofBI q( t) is given. When I is aBorelprincipalmonomialideal, BI q( t) is calculatedexplicitly.