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

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 $m=p<sup>k$ ($k\ge 1$) for some prime $p$, we give a tight lower bound that $\mathrm{deg}_m(f)\geq k(p-1)$ for any non-degenerated function $f:{0,1}<sup>n\to{0,1}$, 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 $f:{0,1}<sup>n\to{0,1}$ that $$\mathrm{deg}_m(f) \geq \frac{n}{2+\frac{1}{p-1}+\frac{1}{q-1}}$$.