Abstract
Define the MOD m -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 MOD m -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 MOD n function is 0 if the number of input ones is a multiple ofn and is one otherwise. We show that the MOD m -degree of both the MOD n and\(\neg MOD_n\) functions isN Ω(1) exactly when there is a prime dividingn but notm. The MOD m -degree of the MOD m function is 1; we show that the MOD m -degree of\(\neg MOD_m\) isN Ω(1) ifm is not a power of a prime,O(1) otherwise. A corollary is that there exists an oracle relative to which the MOD m P classes (such as ⊕P) have this structure: MOD m P is closed under complementation and union iffm is a prime power, and MOD n P is a subset of MOD m P iff all primes dividingn also dividem.
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Barrington, D.A.M., Beigel, R. & Rudich, S. Representing Boolean functions as polynomials modulo composite numbers. Comput Complexity 4, 367–382 (1994). https://doi.org/10.1007/BF01263424
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DOI: https://doi.org/10.1007/BF01263424