The polynomial method and restricted sums of congruence classes

We present a simple and general algebraic technique for obtaining results in Additive Number Theory, and apply it to derive various new extensions of the Cauchy-Davenport Theorem. In particular we obtain, for subsets A(0), A(1), ..., A(k) of the finite field Z(p), a tight lower bound on the minimum possible cardinality of {a(0) + a(1), + ... + a(k): a(i) is an element of A(i), a(i) not equal a(j) for 0 less than or equal to i < j less than or equal to k} as a function of the cardinalities of the sets A(i). (C) 1996 Academic Press, Inc.