On the Degree of Boolean Functions as Polynomials over Z_m
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Polynomial representations of Boolean functions over various rings such as Z and Z_m have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including communication complexity, circuit complexity, learning theory, coding theory and so on. For any integer m>2, each Boolean function has a unique multilinear polynomial representation over ring Z_m. The degree of such polynomial is called modulo-m degree, denoted as deg_m(·). In this paper, we discuss the lower bound of modulo-m degree of Boolean functions. When m=p^k (k> 1) for some prime p, we give a tight lower bound that deg_m(f)≥ k(p-1) for any non-degenerated function f:{0,1}^n→{0,1}, provided that n is sufficient large. When m contains two different prime factors p and q, we give a nearly optimal lower bound for any symmetric function f:{0,1}^n→{0,1} that deg_m(f) ≥n/2+1/p-1+1/q-1. The idea of the proof is as follows. First we investigate properties of polynomial representation of MOD function, then use it to span symmetric Boolean functions to prove the lower bound for symmetric functions, when m is a prime power. Afterwards, Ramsey Theory is applied in order to extend the bound from symmetric functions to non-degenerated ones. Finally, by showing that deg_p(f) and deg_q(f) cannot be small simultaneously, the lower bound for symmetric functions can be obtained when m is a composite but not prime power.
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