Representing Boolean functions as polynomials modulo composite numbers

AbstractDefine the MODm-degree of a boolean functionF to be the smallest degree of any polynomialP, over the ring of integers modulom, such that for all 0–1 assignments $$\vec x$$ , $$F(\vec x) = 0$$ iff $$P(\vec x) = 0$$ . We obtain the unexpected result that the MODm-degree of the OR ofN variables is $$O(\sqrt[\tau ]{N})$$ , wherer is the number of distinct prime factors ofm. This is optimal in the case of representation by symmetric polynomials. The MODn function is 0 if the number of input ones is a multiple ofn and is one otherwise. We show that the MODm-degree of both the MODn and $$\neg MOD_n$$ functions isNΩ(1) exactly when there is a prime dividingn but notm. The MODm-degree of the MODm function is 1; we show that the MODm-degree of $$\neg MOD_m$$ isNΩ(1) ifm is not a power of a prime,O(1) otherwise. A corollary is that there exists an oracle relative to which the MODmP classes (such as ⊕P) have this structure: MODmP is closed under complementation and union iffm is a prime power, and MODnP is a subset of MODmP iff all primes dividingn also dividem.