Communication complexity of algebraic computation

The authors consider a situation in which two processors P/sub 1/ and P/sub 2/ are to evaluate one or more functions f/sub 1/, . . ., f/sub s/ of two vector variables x and y, under the assumption that processor P/sub 1/ (respectively, P/sub 2/) has access only to the value of x (respectively, y) and the functional form of f/sub 1/, . . ., f/sub s/. They consider a continuous model of communication whereby real-valued messages are transmitted, and they study the minimum number of messages required for the desired computation. Tight lower bounds are established for the following three problems: (1) each f/sub i/ is a rational function and only one-way communication is allowed. (2) The variables x and y are matrices and the processors wish to solve the linear system (x+y)z=b for the unknown z. (3) The processors wish to evaluate a particular root of the polynomial equation Sigma (x/sub i/+y/sub i/)z/sup i/=0, where the sum is from i=0 to n-1.