We study the problem of representing symmetric Boolean functions as symmetric polynomials over \mathbb{Z}_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 \left\lceil {\log _p d} \right\rceil digits of the weight in base p and the other is given the first \left\lceil {\log _p d} \right\rceil 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 \Omega (n) on symmetric polynomials weakly representing classes of Mod_r and Threshold functions. Previously the best known lower bound for symmetric polynomials weakly representing any function over \mathbb{Z}_m of degree 0(n) strongly representing¹. 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 0(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.