On Restricting No-Junta Boolean Function and Degree Lower Bounds by Polynomial Method

Let F∗ n be the set of Boolean functions depending on all n variables. We prove that for any f ∈ F∗ n, f |xi=0 or f |xi=1 depends on the remaining n − 1 variables, for some variable xi. This existent result suggests a possible way to deal with general Boolean functions via its subfunctions of some restrictions. As an application, we consider the degree lower bound of representing polynomials over finite rings. Let f ∈ F∗ n and denote the exact representing degree over the ring Zm (with the integer m > 2) as dm(f). Let m = Π r i=1p ei i , where pi’s are distinct primes, and r and ei’s are positive integers. If f is symmetric, then m · dp1 1 (f) · · · dper r (f) > n. If f is non-symmetric, by the second moment method we prove almost always m · dp1 1 (f) · · · dper r (f) > lg n− 1. In particular, asm = pq where p and q are arbitrary distinct primes, we have dp(f)dq(f) = Ω(n) for symmetric f and dp(f)dq(f) = Ω(lg n − 1) almost always for non-symmetric f . Hence any n-variate symmetric Boolean function can have exact representing degree o( √ n) in at most one finite field, and for non-symmetric functions, with o( √ lgn)-degree in at most one finite field.