arXiv:2608.17686v1

On the $p$ -adic Wirsing problem

Anup B Dixit

math.NT11J8211J6111J13

Paper metadata

Original submission: August 18, 2026
Claimed MSC: 11J82, 11J61, 11J13
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: 18 pages. Comments welcome

Abstract

For a real transcendental number $\xi$ , let $\omega_n^*(\xi)$ denote the supremum of all $\omega$ for which there exist infinitely many real algebraic numbers $\alpha$ of degree $\leq n$ satisfying $|\xi-\alpha|\leq H(\alpha)^{-\omega -1}$ , where $H(\alpha)$ is the naive height of the minimal polynomial of $\alpha$ . A celebrated result of Wirsing gives the uniform lower bound $\omega_n^*(\xi)\geq\frac{n+1}{2}$ , which was improved significantly in a recent work of Poëls to $\frac{n}{2-\log 2}$ . In this paper, we establish a $p$ -adic counterpart of Poëls's result. Let $p$ be a prime and $\xi\in\mathbb{Q}_p$ be transcendental. Let $\omega_{n,p}^*(\xi)$ be the supremum of all real numbers $\omega$ for which there exist infinitely many algebraic numbers $\alpha \in \mathbb{Q}_p$ of degree $\leq n$ such that $|\xi-\alpha|_p\leq H(\alpha)^{-\omega -1}$ . We show that $\omega^*_{n,p}(\xi)\geq\frac{n}{2-\log 2}-1$ . This improves the known lower bounds in the $p$ -adic setting, namely the analogue of Wirsing's theorem, due to Morrison and Teulié.

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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.

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Detailed mathematical audit

Generated August 19, 2026

01Statements4 reported findingsCorrect

The stated lower bound for approximation of a transcendental $p$ -adic number by algebraic numbers of degree at most $n$ is correct. The polynomial-exponent reduction, congruence-lattice construction, generalized-resultant estimate, and Hensel step jointly cover every prime $p$ , every $n\geq2$ , and every transcendental $\xi\in\mathbb Q_p$ .

Theorem 1.1Correct

$p$ -adic Wirsing lower bound

Pages 2 and 15–17 · Theorem 1.1 and its proof · arXiv:2608.17686v1

After scaling the target into $\mathbb Z_p$ , the proof constructs infinitely many $\alpha_j\in\mathbb Q_p$ of degree at most $n$ satisfying $|\xi-\alpha_j|_p\ll H(\alpha_j)^{-F_n+\eta}$ . The local determinant estimate supplies the algebraic roots, Lemma 5.1 verifies the strong Hensel hypothesis, and Poëls's minimization theorem gives $F_n\geq n/(2-\log 2)$ . The definition of $\omega^*_{n,p}$ therefore yields $\omega^*_{n,p}(\xi)\geq F_n-1\geq\frac{n}{2-\log 2}-1.$ The reduction preserves degree and changes height only by constants depending on the fixed scaling factor, so the full stated domain is covered.

Poëls, Theorem 8.1 ↗

Lemma 2.4Correct

Polynomial approximation quality and irreducible extraction

Pages 4–5 · Lemma 2.4 · arXiv:2608.17686v1

The pigeonhole construction gives the Dirichlet threshold $\tau_{n,p}(\xi)\geq n$ . Multiplicativity of $\Delta_\xi(P)=M(P)|P(\xi)|_p$ , together with the uniform lower bound for Mahler-measure-one factors, shows that any positive limiting quality is attained along unbounded irreducible factors. Finally, the bounded-degree comparison $\log M(P)=\log H(P)+O_n(1)$ gives $\tau_{n,p}(\xi)=w_{n,p}(\xi)$ .

Propositions 3.3 and 3.4Correct

Independent families with uniformly small $p$ -adic values

Pages 8–10 · Propositions 3.3–3.4 · arXiv:2608.17686v1

The index identity $[\mathbb Z^{n+1}:\Lambda_h(\xi)]=p^h$ and Minkowski's second theorem give the required product of successive minima. The chosen shortest vector has height comparable with the irreducible seed and is coprime to it. Bounded additions of the seed preserve independence, height order, the product estimate, and the common $p$ -adic value bound. Extracting seeds along the defining limsup then gives $\max_i|P_i^{(j)}(\xi)|_p\ll H_{1,j}^{-1}H_{2,j}^{-x_j+o(1)}$ with $x_j\geq n$ and bounded along the selected sequence.

Proposition 4.4Correct

Local generalized resultant produces nearby algebraic roots

Pages 13–15 · Proposition 4.4 · arXiv:2608.17686v1

The generalized-resultant basis has a nonzero integer coefficient determinant $D$ with $|D|_\infty\ll_n B_{k,j}$ . The product formula and the unit-determinant divided-Taylor transformation imply that some basis polynomial $S$ satisfies $|S'(\xi)|_p\gg_n(\delta_jB_{k,j})^{-1}$ . Thus $\delta_j^3B_{k,j}^2\to0$ gives the strict Hensel inequality and a root with $|\xi-\alpha_j|_p\ll\delta_j^2B_{k,j}$ . The auxiliary integer roots stay a fixed positive $p$ -adic distance from the transcendental target, so the nearby root belongs to an original degree-at-most- $n$ polynomial and has the claimed height.

02Proofs5 reported findingsCorrect

The proof is correct and complete. Its four material links—successive minima, good-seed extraction, the generalized-resultant basis, and the determinant/Hensel optimization—have compatible constants, quantifiers, and height exponents.

Lemmas 3.1–3.2 and Proposition 3.3Correct and complete

Congruence-lattice minima and the coprime family

Pages 6–9 · Lemmas 3.1–3.2 and Proposition 3.3 · arXiv:2608.17686v1

Evaluation modulo $p^h$ is surjective because constants represent every residue, so the lattice determinant is exactly $p^h$ . Minkowski's second theorem in the coefficient cube gives $\prod_i\lambda_i(h)\asymp_n p^h$ , and a prescribed shortest vector can be extended at the attained minima. Maximality of $h$ gives both $Y/p<\lambda_1(h)\leq Y$ and the exact value $|Q(\xi)|_p=p^{-h}$ . The height-divisibility estimate proves $P$ and $Q$ are coprime, and replacing one successive-minimum vector by $P$ changes the product only by a uniform factor.

Proposition 3.4Correct and complete

Choice of seeds realizing the limiting quality

Pages 9–10 · Proposition 3.4 · arXiv:2608.17686v1

For an irreducible seed sequence with $\nu_\xi(P^{(j)})=\tau-o(1)$ , the product estimate bounds $\Delta_\xi(Q_j)$ above by the inverse Mahler-measure product. The definition of finite $\tau$ bounds the same quantity below by $M(Q_j)^{-\tau-o(1)}$ , forcing $y_j\leq\tau+o(1)$ . This bounds $x_j$ , makes the height/Mahler comparison error vanish, and yields the claimed common value estimate for every member of the family.

Proposition 4.2Correct and complete

Generalized-resultant basis and determinant height

Pages 11–13 · Proposition 4.2 · arXiv:2608.17686v1

Padding the two coprime polynomials to degree $n$ at distinct bounded integer roots preserves coprimality and controls height. Poëls's intermediate-dimension lemma then supplies one new monomial multiple at each stage, producing exactly $N+1=2(n-k+2)+(k-2)$ independent integer polynomials. Their $p$ -adic values remain at most $\delta$ because $\xi\in\mathbb Z_p$ , and the Leibniz determinant bound gives precisely $H_1^{n-k+2}H_2^{n-k+2}H_3\cdots H_k$ .

Poëls, generalized resultants and dimension estimate ↗

Proposition 4.4Correct and complete

Product formula, divided derivatives, and Hensel's lemma

Pages 13–15 · Proposition 4.4 · arXiv:2608.17686v1

The divided-Taylor coordinate matrix is upper triangular with unit diagonal, so it preserves the integer determinant. Every determinant term contains one row-zero entry bounded by $\delta$ and one derivative entry, while the higher divided derivatives are $p$ -adic integral. Comparing the resulting upper bound for $|D|_p$ with the product-formula lower bound isolates a polynomial satisfying the strong Hensel inequality. Root provenance and Mahler-height divisibility then give both the degree and height conclusions without losing a parameter-dependent factor.

Lemma 5.1 and proof of Theorem 1.1Correct and complete

Uniform Hensel margin and final exponent optimization

Pages 15–17 · Lemma 5.1 and proof of Theorem 1.1 · arXiv:2608.17686v1

Choosing $k_j$ at the maximum defining $F(x_j,\mathbf a_j)$ gives a positive uniform gap $D_{k_j,j}-\frac{a_{1,j}+x_j}{2}\geq\rho_n+o(1),$ because $F$ is bounded below by both $n/(2-\log2)$ and its $k=2$ term. This makes $\delta_j^3B_{k_j,j}^2$ decay by a fixed power of $H_j$ . The identity $D_{k,j}=A_k(x_j,\mathbf a_j)+1+o(1)$ then transfers the minimizing ratio to the height of the root. The finitely many possible indices $k_j$ make the proposition's fixed- $k$ constants uniform, and the approximants are distinct because their distance to the transcendental target tends to zero.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.17686v1
Authors listed: Anup B Dixit
Audit date: August 19, 2026

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On the ppp-adic Wirsing problem

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

ppp-adic Wirsing lower bound

Polynomial approximation quality and irreducible extraction

Independent families with uniformly small ppp-adic values

Local generalized resultant produces nearby algebraic roots

Congruence-lattice minima and the coprime family

Choice of seeds realizing the limiting quality

Generalized-resultant basis and determinant height

Product formula, divided derivatives, and Hensel's lemma

Uniform Hensel margin and final exponent optimization

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On the $p$ -adic Wirsing problem

$p$ -adic Wirsing lower bound

Independent families with uniformly small $p$ -adic values