Abstract

. Define the MODm -degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 0-1 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm -degree of the OR of N variables is O( r p N ), where r is the number of distinct prime factors of m. This is optimal in the case of representation by symmetric polynomials. The MOD n function is 0 if the number of input ones is a multiple of n and is one otherwise. We show that the MODm -degree of both the MOD n and :MOD n functions is N\Omega\Gamma1/ exactly when there is a prime dividing n but not m. The MODm -degree of the MODm function is 1; we show that the MODm -degree of :MODm is N\Omega\Gamma30 if m 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 \PhiP) have this structure: MODmP is closed under complementation and union iff m is a prime power, and...

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