Automatic Bounds on Constant Term Sequences Modulo Primes

This paper provides counterexamples to a previously conjectured upper bound on the first index $n_0$ at which a zero appears in constant term sequences of the form $A_p(n) = ct(P^n) \mod p$, where $P(t) \in \mathbb{Z}[t, t^{-1}]$. The conjecture posited that the first zero must occur at some index $n_0 < p^{\text{deg}(P)}$. We prove an automaton state-based bound for univariate polynomials $n_0 < p^{\kappa(P, p)}$, where $\kappa(P, p)$ is the automaticity of $(A_p(n))_{n \geq 0}$ over $\mathbb{F}_p$. We support our theoretical results with randomized experiments on low degree Laurent polynomials and propose the $\kappa(P, p)$ based bound as a practical alternative to the general worst case bound arising from the Rowland Zeilberger construction.