On the Betti Polynomials of certain graded ideals

Let S = K[ x1,..., xn] be a polynomial ring over a field K and I be anonzero graded ideal of S. Then, for t >> 0, the Betti number ss q( S/ I_t) is a polynomial in t, which is denotedby B_Iq( t). It is proved that B_I q( t) is vanishedorof degree l ( I) - 1 provided I is a monomial ideal generated in a single degree or grade( mR( I)) = codim( mR( I)) where m = ( x1,..., xn) and R( I) is theRees ringof I. One lowe rbound for the leading coefficient of B_Iq( t) is given. When I is a Borel principal monomial ideal, B_I q( t) is calculated explicitly.