Length Approximations for Independently Generated Ideals

We study ideals primary to the maximal ideal of a commutative Noetherian local 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 quotient ring in terms of the orders of the generators. As a corollary we obtain a substantial partial result on Lechʹs conjecture on the multiplicities of a flat couple of local rings.