Communication complexity in distributed algebraic computation

Consideration is given to a situation in which two processors, P₁ and P₂, are to evaluate a collection of functions f₁,..., f]_s of two vector variables x, y under the assumption that processor P₁ (respectively, P₂) has access only to the value of the variable x (respectively, y) and the functional form of f₁, ..., f_s. Bounds on the communication complexity (the amount of information that has to be exchanged between the processors) are provided. An almost optimal bound is derived for the case of one-way communication when the functions are polynomials. Lower bounds for the case of two-way communication that improve on earlier bounds are also derived. As an application, the case in which x and y are n × n matrices and f(x,y) is a particular entry of the inverse of x + y is considered. Under a certain restriction on the class of allowed communication protocols, an Ω(n²) lower bound is obtained. The results are based on certain tools from classical algebraic geometry and field extension theory.