An arithmetic circuit for the indicator function?
Any degree $d$ polynomial (over a field) has at most $d$ roots (unless it is identically zero). As the indicator function has $q-1$ roots, one gets that it must have degre $\geq q-1$.
There are generalizations of this to multivariate polynomials (the best known is Schwartz-Zippel). You could potentially try to generalize to non-fields, but one quickly runs into odd behavior --- say over $\mathbb{Z}/6\mathbb{Z}$ the linear function $f(x) = 3x$ has $\geq 1$ root. This is because it is &quot;identically zero in a CRT component&quot; though, but I do not know of the proper generalization to avoid issues such as this.