Length approximations for independently generated ideals

We study ideals primary to the maximal ideal of a commutative Noetherian lo cal ring. When such an ideal is generated by elements which are independent in the sense of C. Lech, we prove a lower bound on the length of the quoti ent ring in terms of the orders of the generators. As a corollary we obtain a substantial partial result on Lech's conjecture on the multiplicities of a flat couple of local rings. (C) 2001 Academic Press.