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Positive characteristic analogues of finite algebraic numbers

Daichi Matsuzuki, Honami Sakamoto, Jun Ueki

TL;DR

This work develops a positive-characteristic analogue ${\mathcal P}^0_{{\mathcal A}_K}$ of Rosen’s ring of finite algebraic numbers by working in the function-field setting ${K=\mathbb{F}_q(\theta)}$ with ${\mathcal A}_K=(\prod_P R/(P))/(\bigoplus_P R/(P))$. It establishes a robust triad of equivalent characterizations for elements of ${\mathcal A}_K$—(1) linear recurrences with separable eigenpolynomials, (2) Frobenius-data coming from finite Galois extensions, and (3) matrix coefficients of the ${\mathcal A}_K$-valued Frobenius automorphism—mirroring Rosen’s original results. The paper proves that ${\mathcal P}^0_{{\mathcal A}_K}$ is a $K$-subalgebra and sits strictly inside the separable and algebraic closures, with explicit transcendental and separable examples distinguishing the various layers. It also derives density results for primes via Chebotarev-type arguments and outlines how Artin $t$-motives illuminate the relation between this finite analogue and the separable closure $K^{\rm sep}$. These findings extend the finite-analogue paradigm to positive characteristic and open avenues toward a finite-period theory parallel to classical motives. The work thus deepens the analogy between number-theoretic and function-field settings in the study of finite arithmetic invariants and their densities.

Abstract

J.~Rosen introduced the ring $\mathcal{P}^0_{\mathcal{A}}$ of so-called finite algebraic numbers, which may be seen as an analogue of certain periods in the ring $\mathcal{A}=\prod_p \mathbb{Z}/p\mathbb{Z} /\bigoplus_p \mathbb{Z}/p\mathbb{Z}$, $p$ running through all prime numbers. In this article, we introduce its positive characteristic analogue $\mathcal{P}^0_{\mathcal{A}_K}$ over the rational function field $K=\mathbb{F}_q(θ)$, $q$ being a prime power, and study foundational properties.

Positive characteristic analogues of finite algebraic numbers

TL;DR

This work develops a positive-characteristic analogue of Rosen’s ring of finite algebraic numbers by working in the function-field setting with . It establishes a robust triad of equivalent characterizations for elements of —(1) linear recurrences with separable eigenpolynomials, (2) Frobenius-data coming from finite Galois extensions, and (3) matrix coefficients of the -valued Frobenius automorphism—mirroring Rosen’s original results. The paper proves that is a -subalgebra and sits strictly inside the separable and algebraic closures, with explicit transcendental and separable examples distinguishing the various layers. It also derives density results for primes via Chebotarev-type arguments and outlines how Artin -motives illuminate the relation between this finite analogue and the separable closure . These findings extend the finite-analogue paradigm to positive characteristic and open avenues toward a finite-period theory parallel to classical motives. The work thus deepens the analogy between number-theoretic and function-field settings in the study of finite arithmetic invariants and their densities.

Abstract

J.~Rosen introduced the ring of so-called finite algebraic numbers, which may be seen as an analogue of certain periods in the ring , running through all prime numbers. In this article, we introduce its positive characteristic analogue over the rational function field , being a prime power, and study foundational properties.
Paper Structure (23 sections, 32 theorems, 39 equations)

This paper contains 23 sections, 32 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Rosen's ring ${\mathcal{P}}^0_{{\mathcal{A}}}$ of finite algebraic numbers
  4. Positive characteristic analogue ${\mathcal{P}}^0_{{\mathcal{A}}_K}$ (our results)
  5. Frobenius evaluation map
  6. Proof of \ref{['thm.def.AK']} (1) $\Leftrightarrow$ (2)
  7. Preliminaries
  8. Main proof
  9. Proof of \ref{['thm.def.AK']} (2) $\Leftrightarrow$ (3)
  10. Proof of \ref{['thm.subset.AK']}
  11. ${\mathcal{P}}^0_{{\mathcal{A}}_K}$ is a $K$-subalgebra of ${\mathcal{A}}_K$
  12. Proper inclusions $K\subsetneq {\mathcal{P}}^0_{{\mathcal{A}}_K}\subsetneq {\mathcal{C}}^{\rm sep}_{{\mathcal{A}}_K}\subsetneq {\mathcal{C}}^{\rm alg}_{{\mathcal{A}}_K}$
  13. "Transcendental" elements
  14. In ${\mathcal{A}}$
  15. In ${\mathcal{A}}_K$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\alpha\in {\mathcal{A}}$. The following conditions are equivalent. (1) There is a linear recurrent sequence $(a_n)_n$ over ${\mathbb{Q}}$, namely, $(a_n)_n\in {\mathbb{Q}}^{\mathbb{N}}$ satisfying a linear recurrence relation over ${\mathbb{Q}}$, such that $\alpha=[(a_p\ {\rm mod}\ p)_p]$. (2)

Theorems & Definitions (75)

  • Theorem 1.1: (Rosen Rosen2020JNT)
  • Remark 1.2
  • Definition 1.3: (Rosen2020JNT+)
  • Theorem 1.4: (Rosen Rosen2020JNT, Rosen--Takeyama--Tasaka--Yamamoto RosenTakeyamaTasakaYamamoto2024JNT, Anzawa--Funakura AnzawaFunakura2024)
  • Remark 1.5
  • Example 1.6: (Rosen Rosen2020JNT)
  • Theorem 1.7: (Skolem Skolem1934EGDG, Mahler Mahler1935ETKF, Lech Lech1953AM)
  • Definition 1.8
  • Corollary 1.9: (cf. Rosen Rosen2020JNT)
  • Theorem 1.10: (Rosen Rosen2020JNT)
  • ...and 65 more