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 $A_p(n)=ct(P^n)\mathrm{m}\mathrm{o}\mathrm{d}p$, 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 $(A_p(n){)}_{n\ge 0}$ 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.