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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.19031 |
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| _version_ | 1866912349278961664 |
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| author | Offutt, Justin |
| author_facet | Offutt, Justin |
| contents | 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 $n_0 < p^{κ(P, p)}$, where $κ(P, p)$ 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 $κ(P, p)$ based bound as a practical alternative to the general worst case bound arising from the Rowland Zeilberger construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19031 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Automatic Bounds on Constant Term Sequences Modulo Primes Offutt, Justin Number Theory Combinatorics 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 $n_0 < p^{κ(P, p)}$, where $κ(P, p)$ 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 $κ(P, p)$ based bound as a practical alternative to the general worst case bound arising from the Rowland Zeilberger construction. |
| title | Automatic Bounds on Constant Term Sequences Modulo Primes |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2504.19031 |