Freiman ideals and the number of generators of powers of monomial ideals

Let mu(l) denote the number of generators of a monomial ideal I. It is well known that mu(l(k)) < mu(l(k+1)) for k >> 0: In this paper we construct monomial ideals I in F[x, y] such that mu(l(k+1)) < mu(l(k)) for all k <= l, given any positive integer l. Also, we extend some results of Eliahou et al. by constructing monomial ideals in R = F[x(1), ..., x(n)] with mu(l(2)) < mu(l) and investigate mu(l(k)) for monomial ideals in R. Furthermore, we generalize the definition of Freiman ideals given in Herzog and Zhu and extend some results with simpler proofs. In particular, we give a complete characterization of Freiman ideals of maximum height in R.