Symmetric polynomials over /spl Zopf//sub m/ and simultaneous communication protocols

We study the problem of representing symmetric Boolean functions as symmetric polynomials over /spl Zopf//sub m/. We show an equivalence between such representations and simultaneous communication protocols. Computing a function f on 0 - 1 inputs with a polynomial of degree d modulo pq is equivalent to a two player simultaneous protocol for computing f where one player is given the first [log/sub p/d] digits of the weight in base q. This reduces the problem of proving bounds on the degree of symmetric polynomials to proving bounds on simultaneous communication protocols. We use this equivalence to show lower bounds of /spl Omega/(n) on symmetric polynomials weakly representing classes of Mod/sub r/ and Threshold functions. We show there exist symmetric polynomials over /spl Zopf//sub m/ of degree o(n) strongly representing Threshold c for c constant, using the fact that the number of solutions of certain exponential Diophantine equations are finite. Conversely, the fact that the degree is o(n) implies that some classes of Diophantine equations can have only finitely many solutions. Our results give simplifications of many previously known results and show that polynomial representations are intimately related to certain questions in number theory.