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

Abstract

We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function $F:\{0,1{\}}^n\to \{0,1\}$, define its $k$-lift to be $F_k:\{0,1{\}}^n\to \{0,2^{k-1}\}$ by $F_k(x)=2^{k-F(x)}(\mathrm{mod}{2}^k)$. We consider the fractional agreement (which we refer to as ${\gamma }_{d,k}(F)$) of $F_k$ with degree $d$ polynomials from $\mathbb{Z}/2^k\mathbb{Z}[x_1,\dots ,x_n]$. 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 ${\gamma }_{d,k}(F)$ actually increases. The first is an infinite family of functions $F$ such that ${\gamma }_{2d,2}(F)-{\gamma }_{3d-1,1}(F)\ge \Omega (1)$. The second is an infinite family of functions $F$ such that ${\gamma }_{d,1}(F)\le \frac{1}{2}+o(1)$ -- as small as possible -- but ${\gamma }_{d,3}(F)\ge \frac{1}{2}+\Omega (1)$. - Increasing $k$ doesn't always help: Adapting a proof of Green [Comput. Complexity, 9(1):16-38, 2000], we show that irrespective of the value of $k$, the Majority function ${\mathrm{Maj}}_n$ satisfies ${\gamma }_{d,k}({\mathrm{Maj}}_n)\le \frac{1}{2}+\frac{O(d)}{\sqrt{n}}$. In other words, polynomials over $\mathbb{Z}/2^k\mathbb{Z}$ for large $k$ do not approximate the majority function any better than polynomials over $\mathbb{Z}/2\mathbb{Z}$. 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.

Cited by in corpus (5)

• Polynomial bound for the partition rank vs the analytic rank of tensors
• Non-classical polynomials and the inverse theorem
• On the Bias of Reed-Muller Codes over Odd Prime Fields
• An inverse theorem for Gowers norms of trace functions over prime fields
• Quadratic polynomials of small modulus cannot represent OR