On Diophantine equations over the integer rings of quadratic fields

Zhi-Wei Sun

REVIEW 2 major objections 3 minor 13 references

16 unknowns suffice for undecidability over every quadratic field's integer ring

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2026-08-05 04:22 UTC pith:LLGQ4SBO

load-bearing objection The 16-variable imaginary quadratic theorem looks like a real advance; the 15-variable real quadratic theorem is not proved — Section 3 omits the key details. the 2 major comments →

arxiv 2608.03992 v1 pith:LLGQ4SBO submitted 2026-08-04 math.NT math.LO

On Diophantine equations over the integer rings of quadratic fields

Zhi-Wei Sun This is my paper

classification math.NT math.LO MSC 11U0503D3503D2511D0911R11

keywords Hilbert's Tenth Problemquadratic fieldsundecidabilityDiophantine equationsalgebraic integersinteger ringsvariable bounds

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that Hilbert's Tenth Problem remains undecidable when the integer ring of any quadratic field is the domain and the number of unknowns is fixed at 16; for real quadratic fields, the paper claims 15 unknowns suffice. The proof reduces a known undecidable 10-variable problem over the integers, with the last variable required to be nonzero, to the solvability of a single constructed polynomial equation over the ring of integers of K. The construction uses polynomial identities that force certain ring elements to be ordinary integers and that package a divisibility condition together with two square conditions into one equation. A sympathetic reader should care because it pins undecidability to a fixed, small number of variables, uniform across all quadratic fields.

Core claim

The central claim is a uniform bound on Hilbert's Tenth Problem over quadratic integer rings. For any positive squarefree d, with K = Q(sqrt(-d)), there is no algorithm that decides, for arbitrary P in Z[x1,...,x16], whether P=0 has a solution in O_K (Theorem 1.1). For d>1 squarefree and K=Q(sqrt(d)), the same is claimed for arbitrary P in Z[x1,...,x15] (Theorem 1.2). The proof of Theorem 1.1 gives an explicit reduction: given a 10-variable polynomial P, the paper builds a 16-variable polynomial over O_K that has a solution over O_K iff P has an integer solution with the tenth variable nonzero. Theorem 1.2 is stated to follow by the same method with one variable saved, though that proof is n

What carries the argument

The engine is the quartic polynomial F(A1,A2,S,T,m) = (T - mS)^4 - 2(A1+A2)S^2(T-mS)^2 + (A1-A2)^2 S^4. Lemma 2.2 states that for A1≠A2 and T≠0, the equation F=0 in m over O_K holds exactly when A1 and A2 are squares in O_K and S divides T in O_K. Because a single polynomial equation can therefore express a conjunction of two square conditions and a divisibility condition, the paper can translate the full 10-variable integer problem into one equation over O_K. Auxiliary lemmas force elements like Z1 and Z2 to be integers via Pell-type equations and congruences from a Lucas-type sequence, and Lemma 2.7 converts 'z1,...,z10 are all integers' into an equivalent rationality condition.

Load-bearing premise

The 15-variable claim for real quadratic fields rests on the unproved assertion that the proof of Theorem 1.2 can be completed in a way similar to Theorem 1.1 using exactly 15 unknowns; if the analogous encoding requires one more auxiliary variable, that bound is not established.

What would settle it

Write out the omitted proof of Theorem 1.2 with a full variable count. If the analogue of Lemma 2.6 for real quadratic fields requires an additional auxiliary variable, the constructed polynomial has 16 unknowns, refuting the stated 15-variable bound. More concretely, exhibit any 15-variable polynomial in the claimed form whose solvability fails to match the 10-variable integer problem.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

Any algorithm that could decide solvability of the constructed 16-variable equations over O_K would solve the 10-variable integer problem, which is known to be algorithmically impossible; so no such algorithm exists.
The bound is uniform in d: the same 16-variable construction works for every imaginary quadratic field, and the claimed real-quadratic bound is 15.
The reduction makes the undecidability of Hilbert's Tenth Problem over O_K effective with an explicit small number of variables, strengthening the earlier negative answer for quadratic rings.
Because the constructed polynomials have integer coefficients, the undecidability holds even when coefficients are restricted to the base ring Z rather than O_K.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same encoding scheme could give explicit variable bounds for higher-degree number fields if one can build an analogous 'Z1,Z2 integrality' lemma using embeddings and units; the paper only treats quadratic fields.
The 15-variable real-quadratic result depends on the omitted Section 3; if the analogous encoding requires one additional auxiliary variable, the real-quadratic bound would be 16 rather than 15.
The choice of constants 2 and 3 in Lemmas 2.1 and 2.5 is tied to the fact that -1/2 and -1/3 are not algebraic integers in quadratic fields; varying these constants may yield other small-variable encodings.
For imaginary quadratic fields other than Q(i) and Q(sqrt(-3)), the unit group is just {±1}, which simplifies the integrality forcing; this may let the 16-variable bound drop further for specific d.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The 16-variable imaginary quadratic theorem looks like a real advance; the 15-variable real quadratic theorem is not proved — Section 3 omits the key details.

read the letter

The paper has two advertised results: 16 unknowns for imaginary quadratic fields and 15 for real quadratic fields. The first is a genuine advance and the proof is laid out in detail. The second is the headline improvement, and it is not proved. Section 3 ends with 'We omit the details.' That is a load-bearing omission, not a cosmetic one.

The imaginary quadratic proof reduces Sun's 10-variable theorem over Z (with last variable nonzero) to a 16-variable polynomial over O_K. The chain of lemmas — Lemma 2.5 for membership in Z, Lemma 2.6 for pairs via Pell sequences, Lemma 2.7 for rationality of a sum — is internally consistent on my reading. The extra six auxiliary variables are counted explicitly, and the reduction checks out. If Theorem 1.3 and Lemma 2.7 are sound (they are cited published results, not restated here), then Theorem 1.1 holds.

The real quadratic section is where the paper falls short. Theorem 3.1 gives a clean rationality criterion: with Y = 2∏(x_j^2+1) and W = Y^{n+1} + Σ x_j Y^{n-j}, W ∈ Q forces all x_j ∈ Z. The proof, using the two real embeddings and the gap Lemma 3.1, is elegant. But to get Theorem 1.2 you need to encode W ∈ Q as a polynomial equation over O_K while counting variables. That step is not shown. The stress-test note is right: if the encoding needs one extra auxiliary variable, the 15-unknown claim fails. The author says it can be done 'in a way similar to the proof of Theorem 1.1,' but the analogue is not automatic. In Theorem 1.1, Lemma 2.6 is used to force denominators into Z; in the real case you need a different mechanism to handle the x_j^2+1 factors. None is provided.

So my verdict: Theorem 1.1 is a solid incremental contribution, worth refereeing. Theorem 1.2 is a conjecture or at best an unverified claim until the details appear. The paper's citation pattern is fine — self-citations are to prior published work, not a restatement of the present claim. The reliance on the author's own 10-variable theorem is legitimate but does increase the verification burden.

For whom: specialists in Hilbert's tenth problem and Diophantine definability. I would not cite the 15-variable result yet. I would send it to a serious referee, with the explicit instruction that the omitted proof of Theorem 1.2 must be supplied. If the author can fill the gap, this becomes a clean paper; until then, only the 16-variable theorem carries weight.

Referee Report

2 major / 3 minor

Summary. The paper addresses Hilbert's Tenth Problem (HTP) over the ring of integers O_K of a quadratic field K. The main claimed results are: (Theorem 1.1) for every imaginary quadratic field K, no algorithm decides whether an arbitrary polynomial P(z_1,...,z_16) with integer coefficients has a solution in O_K; and (Theorem 1.2) for every real quadratic field K, 15 unknowns suffice for undecidability. The proof strategy reduces Sun's 10-variable undecidable problem over Z with the last variable nonzero (Theorem 1.3) to solvability of a constructed polynomial over O_K. Section 2 gives a detailed proof of Theorem 1.1, using auxiliary lemmas including Lemma 2.7 (Matiyasevich–Sun) to encode integrality of the 10 Z-variables. Section 3 develops tools for real quadratic fields, proves Theorem 3.1 (a rational-integrality criterion), gives three lemmas of Denef type, and then states that Theorem 1.2 follows 'in a way similar to the proof of Theorem 1.1' with details omitted.

Significance. If the results are correct, the paper would substantially improve the known bounds for undecidability of HTP over quadratic integer rings: it would give a uniform 16-unknown bound for all quadratic fields and a 15-unknown bound for real quadratic fields, improving on the prior 20-unknown result over Z[i] of Matiyasevich–Sun and the 18-unknown AI-assisted result of Ding–Li. The proof of Theorem 1.1 is detailed and machine-checkable in principle, and Theorem 3.1 is an elegant rationality criterion for real quadratic fields. However, the advertised 15-unknown theorem for real quadratic fields is not actually proved in the manuscript: Section 3 ends with an omitted proof. Since the variable count is the central contribution, this gap is load-bearing and prevents acceptance of the paper in its current form.

major comments (2)

[Section 3, final paragraph] Theorem 1.2 is not proved. The last paragraph of Section 3 says 'Based on the above lemmas and Theorem 3.1, we can prove Theorem 1.2 completely in a way similar to the proof of Theorem 1.1. We omit the details.' This is a load-bearing omission, not a cosmetic one. Theorem 3.1 only characterizes when W∈Q for W defined in (3.2); it does not by itself provide a polynomial equation over O_K in 15 unknowns. To obtain Theorem 1.2 one must: (i) encode the rationality of W as a polynomial condition over O_K, (ii) incorporate the condition z_10≠0, and (iii) count the auxiliary variables. No analogue of Lemma 2.6 or of the F-polynomial encoding (Lemma 2.2) is supplied for the real quadratic case, and Lemmas 3.3–3.5 are never used. The claimed 15-unknown bound cannot be verified from the manuscript; if the analogous encoding requires even one more auxiliary variable, the stated improvement fails.
[Section 2, Lemma 2.6] The proof of Lemma 2.6 has mislabeled directions and a typo. It says 'We first prove the “only if” direction' but then proves the 'if' direction, and later 'we have z_2∈□' should presumably be 'z_2∈Z'. These are presentation issues, but they made inspection of the lemma more difficult and should be corrected. The lemma itself is plausible and appears to be used correctly in Theorem 1.1.

minor comments (3)

[Section 2, Lemma 2.2] The proof says 'This is [9, Lemma 3.1] if we replace the condition T≠0 by S≠0.' The statement retains T≠0, and the comparison to the cited lemma is unclear. Please restate the relation to the cited result.
[Section 3, Lemma 3.2] The phrase 'positive rational integer' is nonstandard; use 'positive integer' for clarity.
[References] Reference [4] is an arXiv preprint without a DOI or journal page; if it has been published or updated, please provide the full citation. Also, reference [8] is cited for Lemma 2.7, but the lemma is stated for a general quadratic field while the title of [8] mentions Z[i]; please confirm the exact scope of the cited theorem.

Circularity Check

0 steps flagged

No circular reduction: the O_K transfer is anchored in Sun's independent 10-variable theorem over Z; the real-quadratic 15-variable claim rests on an omitted proof, not on a circular step.

full rationale

I walked the reduction chain. The input is Theorem 1.3 (Sun [11]): no algorithm to decide P(z_1,...,z_10)=0 over Z with z_10≠0. That is a statement about Z, not about O_K, so it is independent of the present conclusion. The paper then constructs an explicit polynomial \tilde P and proves the equivalence (P solvable in Z with z_10≠0) ⇔ (\tilde P=0 solvable in O_K), using Lemmas 2.1–2.7. Some of these lemmas are cited from the author's prior work (e.g. Lemma 2.7 = [8, Theorem 1.2]), but they are proved in the paper or are prior published results about integrality in quadratic fields; none of them restates Theorem 1.1 or 1.2, and no fitted parameter is renamed as a prediction. No uniqueness theorem or ansatz is imported. Section 3 contains a genuine gap: 'Based on the above lemmas and Theorem 3.1, we can prove Theorem 1.2 completely in a way similar to the proof of Theorem 1.1. We omit the details.' This omission is load-bearing for the advertised 15-unknown real-quadratic bound and prevents the proof from being checked; however it is an omitted proof, not a circular derivation. Given the heavy self-citation and this unproved transfer, I assign a slightly elevated non-zero score, but no circular step was exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The reduction relies on several prior theorems, including the same author's 10-variable undecidability over Z (Theorem 1.3) and the integer-detection lemmas from [8] and [9]. These are cited, not derived here. No empirical free parameters or invented entities appear; the proof is a chain of explicit polynomial encodings.

axioms (5)

domain assumption Theorem 1.3 (Sun [11]): no algorithm decides whether P(z1,...,z10)=0 for some z1,...,z10 in Z with z10 != 0.
Used as the undecidable base problem in the reduction; stated in Section 1 but not proved here.
domain assumption Lemma 2.7, i.e. [8, Theorem 1.2]: for y=2*prod(3xk+1), y + sum xk/y^k in Q iff all xk in Z in imaginary quadratic fields.
Quoted from prior work by Matiyasevich and Sun; it is the bridge that lets the reduction detect integer variables in the imaginary quadratic case (Section 2, equation (2.8)).
domain assumption Lemma 3.3 (Denef [3]): solutions of x^2 - E y^2 = 1 over O_K have y^2 in N, and the congruence Y_{nk}^2 = k^2 Y_n^2 mod Y_n^4 holds.
This is the corresponding integer-detection tool for the real quadratic case (Section 3), cited from Denef.
domain assumption Lemma 2.2, i.e. [9, Lemma 3.1]: the polynomial F encodes two squares plus a divisibility condition.
The proof says it follows from a prior lemma by replacing a condition; used to compress multiple conditions into one polynomial in Theorem 1.1.
standard math Lemma 2.1: m is a nonzero integer iff m divides (2w+1)(3w+1) for some w in Z, said to follow from the Chinese remainder theorem.
Used to force z10 != 0 through a divisibility condition; the proof is only a reference to the CRT and a remark by Tung.

pith-pipeline@v1.3.0-daily-deepseek · 8706 in / 27044 out tokens · 257831 ms · 2026-08-05T04:22:34.638782+00:00 · methodology

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Cite this review

Pith. "Pith review of On Diophantine equations over the integer rings of quadratic fields." pith.science (2026). https://pith.science/paper/LLGQ4SBO

@misc{pith2026260803992,
  author       = {Pith},
  title        = {Pith review of: On Diophantine equations over the integer rings of quadratic fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LLGQ4SBO}},
  note         = {Machine review of arXiv:2608.03992}
}

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read the original abstract

Let $K$ be any quadratic number field, and let $O_K$ be the ring of algebraic integers in $K$. In 1975 J. Denef proved that Hilbert's Tenth Problem over $O_K$ has a negative solution. In this paper we establish the following undecidability result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{16})=0$ (with integer coefficients and 16 unknowns) has solutions over $O_K$. Moreover, when $K$ is a real quadratic field, we show that $15$ unknowns suffice for undecidability.

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages · 2 internal anchors

[1]

Alp¨ oge, M

L. Alp¨ oge, M. Bhargava, W. Ho and A. Shnidman,Rank stability in qua- dratic extensions and Hilbert’s tenth problem for the ring of integers of a number field, Invent. Math.243(2026), 1129–1139

work page 2026
[2]

Davis, H

M. Davis, H. Putnam and J. Robinson,The decision problem for exponential diophantine equations, Ann. of Math.74(1961), 425–436

work page 1961
[3]

Denef,Hilbert’s Tenth Problem for quadratic rings, Proc

J. Denef,Hilbert’s Tenth Problem for quadratic rings, Proc. Amer. Math. Soc.48(1975), 214–220

work page 1975
[4]

An AI Proof of 18-Variable Undecidability for Diophantine Equations over $\mathbb Z[i]$

Y. Ding and J. Li,An AI proof of 18-variable undecidability for Diophantine equations overZ[i], arXiv:2606.12776, 2026,

work page internal anchor Pith review Pith/arXiv arXiv 2026
[5]

Koymans and C

P. Koymans and C. Pagano, Hilbert’s tenth problem via additive combina- torics, arXiv:2412.01768, 2024

arXiv 2024
[6]

Matiyasevich,Enumerable sets are diophantine, Dokl

Y. Matiyasevich,Enumerable sets are diophantine, Dokl. Akad. Nauk SSSR 191(1970), 279–282; English translation with addendum, Soviet Math. Doklady11(1970), 354–357

work page 1970
[7]

Robinson,Reduction of an arbitrary diophantine equation to one in 13 unknowns, Acta Arith.27(1975), 521–553

Y Matiyasevich and J. Robinson,Reduction of an arbitrary diophantine equation to one in 13 unknowns, Acta Arith.27(1975), 521–553

work page 1975
[8]

On Diophantine equations over $\mathbb Z[i]$ with $52$ unknowns

Y. Matiyasevich and Z.-W. Sun,On Diophantine equations overZ[i]with52 unknowns, in: Mathematical Logic, Computability, Complexity and Ran- domness (edited by J. Brendle et al.), pp. 153–158, World Sci., 2026. See also arXiv:2002.12136

work page internal anchor Pith review Pith/arXiv arXiv 2026
[9]

Matiyasevich and Z.-W

Y. Matiyasevich and Z.-W. Sun,Undecidability on Diophantine equations overZ[i]with 20 unknowns, J. Number Theory290(2027), 210–219

work page 2027
[10]

Sun,Reduction of unknowns in Diophantine representations, Sci

Z.-W. Sun,Reduction of unknowns in Diophantine representations, Sci. China Ser. A35(1992), no. 3, 257–269. Available from the website http://maths.nju.edu.cn/∼zwsun/12d.pdf

work page 1992
[11]

Sun,Further results on Hilbert’s tenth problem, Sci

Z.-W. Sun,Further results on Hilbert’s tenth problem, Sci. China Math.64 (2021), 281–306

work page 2021
[12]

Sun, Fibonacci Numbers and Hilbert’s Tenth Problem (in Chinese), Harbin Institute of Technology Press, Harbin, 2024

Z.-W. Sun, Fibonacci Numbers and Hilbert’s Tenth Problem (in Chinese), Harbin Institute of Technology Press, Harbin, 2024

work page 2024
[13]

S. P. Tung,On weak number theories, Japan. J. Math. (N.S.)11(1985), 203–232. School of Mathematics, Nanjing University, Nanjing 210093, Peo- ple’s Republic of China Email address:zwsun@nju.edu.cn

work page 1985

[1] [1]

Alp¨ oge, M

L. Alp¨ oge, M. Bhargava, W. Ho and A. Shnidman,Rank stability in qua- dratic extensions and Hilbert’s tenth problem for the ring of integers of a number field, Invent. Math.243(2026), 1129–1139

work page 2026

[2] [2]

Davis, H

M. Davis, H. Putnam and J. Robinson,The decision problem for exponential diophantine equations, Ann. of Math.74(1961), 425–436

work page 1961

[3] [3]

Denef,Hilbert’s Tenth Problem for quadratic rings, Proc

J. Denef,Hilbert’s Tenth Problem for quadratic rings, Proc. Amer. Math. Soc.48(1975), 214–220

work page 1975

[4] [4]

An AI Proof of 18-Variable Undecidability for Diophantine Equations over $\mathbb Z[i]$

Y. Ding and J. Li,An AI proof of 18-variable undecidability for Diophantine equations overZ[i], arXiv:2606.12776, 2026,

work page internal anchor Pith review Pith/arXiv arXiv 2026

[5] [5]

Koymans and C

P. Koymans and C. Pagano, Hilbert’s tenth problem via additive combina- torics, arXiv:2412.01768, 2024

arXiv 2024

[6] [6]

Matiyasevich,Enumerable sets are diophantine, Dokl

Y. Matiyasevich,Enumerable sets are diophantine, Dokl. Akad. Nauk SSSR 191(1970), 279–282; English translation with addendum, Soviet Math. Doklady11(1970), 354–357

work page 1970

[7] [7]

Robinson,Reduction of an arbitrary diophantine equation to one in 13 unknowns, Acta Arith.27(1975), 521–553

Y Matiyasevich and J. Robinson,Reduction of an arbitrary diophantine equation to one in 13 unknowns, Acta Arith.27(1975), 521–553

work page 1975

[8] [8]

On Diophantine equations over $\mathbb Z[i]$ with $52$ unknowns

Y. Matiyasevich and Z.-W. Sun,On Diophantine equations overZ[i]with52 unknowns, in: Mathematical Logic, Computability, Complexity and Ran- domness (edited by J. Brendle et al.), pp. 153–158, World Sci., 2026. See also arXiv:2002.12136

work page internal anchor Pith review Pith/arXiv arXiv 2026

[9] [9]

Matiyasevich and Z.-W

Y. Matiyasevich and Z.-W. Sun,Undecidability on Diophantine equations overZ[i]with 20 unknowns, J. Number Theory290(2027), 210–219

work page 2027

[10] [10]

Sun,Reduction of unknowns in Diophantine representations, Sci

Z.-W. Sun,Reduction of unknowns in Diophantine representations, Sci. China Ser. A35(1992), no. 3, 257–269. Available from the website http://maths.nju.edu.cn/∼zwsun/12d.pdf

work page 1992

[11] [11]

Sun,Further results on Hilbert’s tenth problem, Sci

Z.-W. Sun,Further results on Hilbert’s tenth problem, Sci. China Math.64 (2021), 281–306

work page 2021

[12] [12]

Sun, Fibonacci Numbers and Hilbert’s Tenth Problem (in Chinese), Harbin Institute of Technology Press, Harbin, 2024

Z.-W. Sun, Fibonacci Numbers and Hilbert’s Tenth Problem (in Chinese), Harbin Institute of Technology Press, Harbin, 2024

work page 2024

[13] [13]

S. P. Tung,On weak number theories, Japan. J. Math. (N.S.)11(1985), 203–232. School of Mathematics, Nanjing University, Nanjing 210093, Peo- ple’s Republic of China Email address:zwsun@nju.edu.cn

work page 1985

On Diophantine equations over the integer rings of quadratic fields

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it