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 A₀, A₁, ..., A_k of the finite field Z_p, a tight lower bound on the minimum possible cardinality of {a₀ + a₁ + ⋯ + a_k: a_i∈ A_i, a_i ≠ a_j for 0 ≤ i < j ≤ k} as a function of the cardinalities of the sets A_i.