An upper bound on the number of relevant variables in a bounded degree Boolean function on the Hamming graph

In this work, we prove that any Boolean function of degree $d$ on ${\mathbb{Z}}_q^n$, $q\ge 3$, has at most $m_q{q}^d$ relevant variables, where $m_q=\frac{2q(q^2+4q+1)}{(q-1{)}^4}$. For $q\in \{3,4,5,6,7\}$, we improve this bound to $2.854\cdot 3^d$, $1.749\cdot 4^d$, $1.263\cdot 5^d$, $0.994\cdot 6^d$, and $0.814\cdot 7^d$, respectively.