One-way communication complexity of computing a collection of rational functions

We consider the problem of evaluating a collection of rational functions f(hook)₁(x, y), f(hook)₂(x, y), ⋯, f(hook)_s(x, y) (x ∈ R^m, y ∈ Rⁿ) using two processors P₁ and P₂, assuming that processor P₁ (respectively P₂) has access to input x (respectively y) and the functional form of f(hook). We establish, by way of algebraic field extension theory, an almost optimal lower bound on the one-way communication complexity (i.e. the minimum number of real-valued messages that have to be exchanged). Our result strengthens the early result of Abelson in several directions.