arXiv:2601.21209v2

Positive characteristic analogues of finite algebraic numbers

Daichi Matsuzuki, Honami Sakamoto, Jun Ueki

math.NT11R5811B3711R4511A0711B3911J93

Paper metadata

Original submission: January 29, 2026
Latest revision: July 5, 2026 · version 2
Claimed MSC: 11R58, 11B37, 11R45, 11A07, 11B39, 11J93
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 21 pages. Section 9 added in v2

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, and provide further scopes.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The Frobenius-evaluation characterization, algebra and proper-inclusion results, Frobenian-set and polynomial-root consequences, density supremum, and Artin-motive period statement are supported. The characteristic-two construction used for the density theorem needs an omitted prime choice, but that choice gives a complete verified repair without changing the theorem.

Theorems 1.13, 1.16, Corollary 1.19, and Theorem 1.20Correct

Foundational characterization and algebraic consequences

Pages 4–14 · Theorems 1.13, 1.16, Corollary 1.19, Theorem 1.20, and proofs · arXiv:2601.21209v2

The equivalence between separable recurrence data, conjugation-equivariant Frobenius evaluation, and matrix coefficients is established by diagonalizing the recurrence over a finite Galois extension and applying additive and multiplicative Galois descent. Closure under sums and products follows after passing to a common Galois extension. The explicit idempotent and inseparable constructions separate all four inclusions. Chebotarev applied to the identity conjugacy class gives the $K$ -rational root in Theorem 1.20, and characteristic functions of conjugacy-stable subsets give the Frobenian-set converse.

Full paper, version 2 ↗

Theorem 1.21Correct

The density supremum is correct after repairing the characteristic-two wreath extension

Pages 14–17 · Lemmas 7.3–7.6 and proof of Theorem 1.21 · arXiv:2601.21209v2

Lemmas 7.3–7.5 convert the two prime densities into proportions of subsets of a finite Galois group and show that wreath extensions make the smaller proportion approach the larger one. In characteristic two, choose a prime of $K$ that splits completely in a Galois closure containing the Hilbert class field of $L$ , and let $\vartheta$ generate one resulting principal prime of $L$ . The conjugate prime ideals are then distinct. In any nonzero $\mathbb{F}_2$ -linear combination of the elements $\sigma(\vartheta^{-(2i-1)})$ , a conjugate valuation sees a unique largest odd pole order, whereas every negative valuation of $x^2+x$ is even. This proves the required Artin–Schreier independence and supplies the stated wreath extension. With that verified repair, the limiting group-count argument proves both assertions of Theorem 1.21.

Full paper, version 2 ↗

Proposition 8.2Correct

Artin $t$ -motive periods generate the separable closure

Pages 17–18 · Proposition 8.2 · arXiv:2601.21209v2

For a finite Galois extension $L/K$ , the matrix of Galois conjugates of a $K$ -basis is invertible by nondegeneracy of the trace form and satisfies the required Frobenius matrix relation. Its transpose is therefore a rigid analytic trivialization. The period matrix is the Frobenius of its inverse; its entries generate the same field as the Frobenius matrix itself, and the relation $\Psi=\tau(\Psi)\Phi^{-1}$ recovers every chosen basis element. Taking a basis containing an arbitrary $a\in K^{\mathrm{sep}}$ proves the converse inclusion, while the forward inclusion is immediate from the defining matrices over finite separable extensions.

Full paper, version 2 ↗

02Proofs2 reported findingsContains incorrect or incomplete proofs

The principal arguments are complete except for the characteristic-two branch of Lemma 7.6, where distinctness is substituted for linear independence and conjugate prime ideals are not controlled. A split-prime choice gives a rigorous repair. One contradictory preimage identity in the final group argument is a removable notation error.

Lemma 7.6, characteristic twoIncomplete as written

The claimed Artin–Schreier independence does not follow from the chosen element

Pages 15–16 · characteristic-two branch of Lemma 7.6 · arXiv:2601.21209v2

The proof chooses any principal-prime generator $\vartheta$ , shows only that $\vartheta^{-(2j-1)}-\vartheta^{-(2i-1)}$ is not in the image of $x\mapsto x^2+x$ , and then declares the entire tuple $(\sigma(\vartheta^{-(2i-1)}))_{\sigma,i}$ linearly independent in $L/(x^2+x)L$ . Pairwise distinct cosets do not imply linear independence, and if a nontrivial $\sigma$ fixes the chosen prime then conjugate entries may repeat. Repair classification: Verified repair. Choose the base prime to split completely in a Galois closure containing the Hilbert class field, so all conjugate prime ideals are distinct and principal. At a prime supporting a nonempty linear combination, the largest occurring pole has odd order, while an element $x^2+x$ has even negative valuation. This proves independence of the full tuple and the rest of the Kummer-style splitting argument goes through.

Full paper, version 2 ↗

Proof of Theorem 1.21Typo

The displayed identity $S'_2=\pi^{-1}(S_2)$ should be deleted

Page 17 · proof of Theorem 1.21, paragraph applying Lemmas 7.5–7.6 · arXiv:2601.21209v2

For $\xi\in\Gamma'$ , the image $\pi(C_{\Gamma'}(\xi))$ can be strictly smaller than $C_\Gamma(\pi(\xi))$ , so the newly defined set $S'_2$ is not generally $\pi^{-1}(S_2)$ . Indeed, making this centralizer image smaller is exactly the mechanism used in the next sentence to enlarge $S'_2$ toward $S'_1$ . Deleting the equality with $\pi^{-1}(S_2)$ is the unique local correction; the definition of $S'_2$ , Lemma 7.5, and all subsequent inequalities then agree and the argument is unchanged.

Full paper, version 2 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2601.21209v2
Authors listed: Daichi Matsuzuki, Honami Sakamoto, Jun Ueki
Audit date: August 15, 2026

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Positive characteristic analogues of finite algebraic numbers

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Foundational characterization and algebraic consequences

The density supremum is correct after repairing the characteristic-two wreath extension

Artin ttt-motive periods generate the separable closure

The claimed Artin–Schreier independence does not follow from the chosen element

The displayed identity S2′=π−1(S2)S'_2=\pi^{-1}(S_2)S2′​=π−1(S2​) should be deleted

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Artin $t$ -motive periods generate the separable closure

The displayed identity $S'_2=\pi^{-1}(S_2)$ should be deleted