On polynomial approximations over $\mathbb{Z}/2^k\mathbb{Z}$

We study approximation of Boolean functions by low-degree polynomials over the ring Z/2(k)Z. More precisely, given a Boolean function F : {0,1}(n) -> (0,1}, define its k-lift to be F-k : {0,1}(n) -> {0,2(k-1)} by F-k(x) = 2(k-F(x)) (mod 2(k)). We consider the fractional agreement (which we refer to as gamma(d,k)(F)) of F-k with degree d polynomials from Z/2(k)Z[x(1), ..., x(n)]. Our results are the following: Increasing k can help: We observe that as k increases, gamma(d,k)(F) cannot decrease. We give two kinds of examples where gamma(d,k)(P) actually increases. The first is an infinite family of functions F such that gamma(2d,2)(F) - gamma(3d-1,1)(F) >= Omega(1). The second is an infinite family of functions F such that gamma(d,1) (F) < 1/2 + o(1) - as small as possible - but gamma(d,3)(F) > 1/2 + Omega(1). Increasing k doesn't always help: Adapting a proof of Green [Compd. Complexity, 9(1):16-38, 2000], we show that irrespective of the value of k, the Majority function Maj(n) satisfies gamma(d,k)(Maj(n)) <= 1/2 + O(d)/root n. In other words, polynomials over Z/2(k)Z for large k do not approximate the majority function any better than polynomials over Z/2Z. We observe that the model we study subsumes the model of non-classical polynomials in the sense that proving bounds in our model implies bounds on the agreement of non-classical polynomials with Boolean functions. In particular, our results answer questions raised by Bhowmick and Lovett [In Proc. 30th Computational Complexity Conf., pages 72-87, 2015] that ask whether non-classical polynomials approximate Boolean functions better than classical polynomials of the same degree.