On the Communication Complexity of Solving a Polynomial Equation

We consider the problem of evaluating a function f(x, y) (x E sm, y E Rn) using two processors P1 and P2, assuming that processor P1 (respectively, P2) has access to input z (respectively, y) and the functional form of f. We establish a new general lower bound on the communication complexity (i.e., the minimum number of real-valued messages that have to be exchanged). We then apply our result to the case where f(z, y) is defined as a root z of a polynomial equation Ad1(xi + yi)zi = 0 and obtain a lower bound of n. This is in contrast to the 2q(1) lower bound obtained by applying earlier results of Abelson. 'Research supported by the ONR under contract N00014-84-K-0519 (NR 649-003) and the ARO under contract DAAL03-86-K-0171. 2Operations Research Center and Laboratory for Information and Decision Systems, MIT, Cambridge, Mass. 02139.