On the Degree of Boolean Functions as Polynomials over $\mathbb{Z}_m$

TL;DR

This paper studies the minimal degree of polynomial representations of Boolean functions over rings _m, establishing tight bounds for prime power moduli and nearly optimal bounds for symmetric functions with composite moduli.

Contribution

It provides new lower bounds on the modulo-_m degree of Boolean functions, including tight bounds for prime power moduli and nearly optimal bounds for symmetric functions with composite moduli.

Findings

For prime power moduli, the degree is at least k(p-1).

For composite moduli with two primes, the degree is at least a fraction of n.

Bounds are tight or nearly tight for large n.

Abstract

Polynomial representations of Boolean functions over various rings such as $\mathbb{Z}$ and $\mathbb{Z}_m$ 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 $m\ge2$, each Boolean function has a unique multilinear polynomial representation over ring $\mathbb Z_m$. The degree of such polynomial is called modulo-$m$ degree, denoted as $\mathrm{deg}_m(\cdot)$. In this paper, we investigate the lower bound of modulo-$m$ degree of Boolean functions. When $m=p^k$ ($k\ge 1$) for some prime $p$, we give a tight lower bound that $\mathrm{deg}_m(f)\geq k(p-1)$ for any non-degenerated function $f:\{0,1\}^n\to\{0,1\}$, 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 $f:\{0,1\}^n\to\{0,1\}$ that $\mathrm{deg}_m(f) \geq \frac{n}{2+\frac{1}{p-1}+\frac{1}{q-1}}$.