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

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 discuss 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 $\mathrm{deg}_m(f) \geq \frac{n}{2+\frac{1}{p-1}+\frac{1}{q-1}}$. The idea of the proof is as follows. First we investigate properties of polynomial representation of $\mathsf{MOD}$ function, then use it to span symmetric Boolean functions to prove the lower bound for symmetric functions, when $m$ is a prime power. Afterwards, Ramsey Theory is applied in order to extend the bound from symmetric functions to non-degenerated ones. Finally, by showing that $\mathrm{deg}_p(f)$ and $\mathrm{deg}_q(f)$ cannot be small simultaneously, the lower bound for symmetric functions can be obtained when $m$ is a composite but not prime power.