Positive characteristic analogues of finite algebraic numbers

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.

💡 Research Summary

This paper develops a positive‑characteristic analogue of J. Rosen’s ring of finite algebraic numbers, denoted 𝒫⁰_𝒜, by constructing a corresponding ring 𝒫⁰_{A_K} inside the function‑field analogue A_K of the classical ring 𝒜. The classical setting uses the profinite product 𝒜 = (∏_p ℤ/pℤ)/(⊕_p ℤ/pℤ) over all rational primes p, and Rosen’s finite algebraic numbers are characterised by three equivalent conditions: (i) they arise from linear recurrent sequences over ℚ, (ii) they can be expressed via a Frobenius‑evaluation map attached to a finite Galois extension of ℚ, and (iii) they are ℚ‑linear combinations of matrix coefficients of a Frobenius automorphism on L⊗𝒜 for some finite Galois extension L/ℚ.

The authors replace the number‑field background by the rational function field K = 𝔽_q(θ) (q a power of a prime) and its polynomial ring R = 𝔽_q