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Automatic Bounds on Constant Term Sequences Modulo Primes

Justin Offutt

TL;DR

This work shows that the conjectured bound $n_0 < p^{\deg(P)}$ for the first zero of the constant-term sequence $A_p(n)=\operatorname{ct}(P^n)\bmod p$ does not hold in general for univariate Laurent polynomials. It introduces an automaton-based bound $n_0< p^{\kappa(P,p)}$, where $\kappa(P,p)$ is the number of states in the minimal DFA computing $A_p(n)$, and proves a digit-wise bound leveraging base-$p$ representations. The authors support the theory with randomized experiments, yielding explicit counterexamples (including $P(t)=4t^2+6t+1+6t^{-1}$ over $\mathbb{F}_7$ with $n_0=225$) and ten additional instances, showing the practical relevance of the $\kappa(P,p)$ bound. The paper also provides a practical framework for computing $\kappa(P,p)$ and advocates it as a more reliable alternative to the general Rowland–Zeilberger bound, with implications for understanding zeros in modular constant-term sequences.

Abstract

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.

Automatic Bounds on Constant Term Sequences Modulo Primes

TL;DR

This work shows that the conjectured bound for the first zero of the constant-term sequence does not hold in general for univariate Laurent polynomials. It introduces an automaton-based bound , where is the number of states in the minimal DFA computing , and proves a digit-wise bound leveraging base- representations. The authors support the theory with randomized experiments, yielding explicit counterexamples (including over with ) and ten additional instances, showing the practical relevance of the bound. The paper also provides a practical framework for computing and advocates it as a more reliable alternative to the general Rowland–Zeilberger bound, with implications for understanding zeros in modular constant-term sequences.

Abstract

This paper provides counterexamples to a previously conjectured upper bound on the first index at which a zero appears in constant term sequences of the form , where . The conjecture posited that the first zero must occur at some index . We prove an automaton state-based bound for univariate polynomials , where is the automaticity of over . We support our theoretical results with randomized experiments on low degree Laurent polynomials and propose the based bound as a practical alternative to the general worst case bound arising from the Rowland Zeilberger construction.
Paper Structure (11 sections, 2 theorems, 12 equations, 4 figures)

This paper contains 11 sections, 2 theorems, 12 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background and Definitions
  3. A Motivating Example
  4. Automata and Base-$p$ Representation of Constant Term Sequences
  5. Theorem and Conjecture on the First Zero of $A_p(n)$
  6. Methodology
  7. Results
  8. An Explicit Counterexample to Conjecture \ref{['conj']}
  9. Ten Additional Counterexamples to Conjecture \ref{['conj']}
  10. Automatic Bound on First Zero
  11. Conclusion

Key Result

Proposition 2.2

Let $P$ be any Laurent polynomial and $p$ be prime. There exists a $B_{P,p} \in \mathbb{N}$, depending on $p$ and $\deg(P)$, such that if there exists some $n \in \mathbb{N}$ with $p \mid \mathop{\mathrm{ct}}\nolimits\left[P^n\right]$, then there exists an $n_0 \in \mathbb{N}$ with $n_0 < B_{P,p}$ s

Figures (4)

  • Figure 1: Finite automaton computing a 2-automatic sequence $a(n) \in \mathbb{F}_2$. Input is the base-2 expansion of $n$, read from least to most significant digit.
  • Figure 2: Shortest zero values across primes for various randomly generated Laurent polynomials. Each subplot represents a different polynomial, with prime numbers on the x-axis and the first zero index on the y-axis.
  • Figure 3: Shortest zero values across primes for $P(t) = 32t^2 + 13t + 1 + 27t^{-1} + 35t^{-2}$ with violations marked.
  • Figure 4: Range where the shortest zero must occur: inputs with base-$p$ length exactly $\kappa$ digits lie between $p^{\kappa-1}$ and $p^\kappa$.

Theorems & Definitions (6)

  • Definition 2.1
  • Proposition 2.2: Kohen, Proposition 6, Section 4
  • proof
  • Conjecture 2.3
  • Proposition 5.1
  • proof