Papers › Automatic Bounds on Constant Term Sequences Modulo Primes
Automatic Bounds on Constant Term Sequences Modulo Primes
Justin Offutt
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This paper provides counterexamples to a previously conjectured upper bound on the first index n₀ at which a zero appears in constant term sequences of the form Aₚ(n) = ct(Pⁿ) p, where P(t) ∈ℤ[t, t⁻¹]. The conjecture posited that the first zero must occur at some index n₀ < pᵈᵉᵍ⁽ᴾ⁾. We prove an automaton state-based bound for univariate polynomials n₀ < p^(κ(P, p)), where κ(P, p) is the automaticity of (Aₚ(n))_(n ≥0) over 𝔽ₚ. We support our theoretical results with randomized experiments on low degree Laurent polynomials and propose the κ(P, p) based bound as a practical alternative to the general worst case bound arising from the Rowland Zeilberger construction.
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