Symmetric polynomials over Zm and simultaneous communication protocols

We study the problem of representing symmetric Boolean functions as symmetric polynomials over Zm. 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 dlogp de digits of the weight in base p and the other is given the first dlogq de 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 (n) on symmetric polynomials weakly representing classes of Modr and Threshold functions. Previously the best known lower bound for symmetric polynomials weakly representing any function overZm was n 1t [1] where t is the number of distinct prime factors of m. We show there exist symmetric polynomials over Zm of degree o(n) strongly representing1 Threshold for 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. Research supported in part by NSF CCR-0105639 yResearch supported in part by NSF CCR-0002299 zAlso with Telcordia. Supported in part by NSF CCR-0002299 1By strong representation, we mean f(X) = 0 ) P (x) = 0 and f(X) 6= 0 ) P (X) 6= 0