The Mordell-Schinzel conjecture for cubic diophantine equations

J\'anos Koll\'ar; Jennifer Li

REVIEW 3 major objections 6 minor 13 references

The Mordell-Schinzel conjecture for cubic diophantine equations

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that every cubic Diophantine equation of the form $xyz=ax^3+by^3+c+a_2x^2+a_1x+b_2y^2+b_1y$ has infinitely many integer solutions.

desk verdict First fully proven case for degrees at least 3, but the load-bearing reduction rests on an unverified preprint and an uncertified computation; send to referees, expect revision. read the letter →

arxiv 2412.12080 v1 pith:TAL65VVX submitted 2024-12-16 math.NT math.AG

János Kollár , Jennifer Li This is my paper · ORCID

classification math.NTmath.AG MSC 11D2514G0514J50

keywords Mordell-SchinzelconjecturecubicDiophantineequationsintegralpointssurfacesinfiniteautomorphismgroupscompanionorbitgrowth

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves the cubic case of the Mordell-Schinzel conjecture: every equation of the form $xyz=ax^3+by^3+c+a_2x^2+a_1x+b_2y^2+b_1y$ with nonzero integer $a,b$ and arbitrary integer coefficients has infinitely many integer solutions. This settles a claim about equations $xyz=G(x,y)$ that was first made for all polynomials in 1952 but had known counterexamples in degree 2, leaving the cubic case as the natural next step. The proof shows that once one integer point is sufficiently large, symmetries of the underlying cubic surface generate infinitely many other integer points; the few remaining small cases are handled by explicitly listed large solutions. If the proof is right, a whole infinite family of cubic Diophantine equations is completely understood, with an explicit mechanism producing the solutions.

What carries the argument

The key machinery is the isomorphism groupoid $\Sigma_{A,B}$ attached to the surface and its three companion surfaces. It is generated by two involutions $\sigma_x$ and $\sigma_y$, given in rational form by $(x,y)\mapsto (B(y)/x,y)$ and $(x,y)\mapsto (x,A(x)/y)$, and the composition $\sigma_{A,B}=\sigma_y\circ\sigma_x\circ\sigma_y\circ\sigma_x$ is an infinite-order automorphism. The workhorse statement is Corollary 10: if a point satisfies $\max\{|x|,|y|\}>1+\max\{1,\sum|a_i|,\sum|b_j|\}$, then one of $\sigma_x,\sigma_y$ strictly increases the norm, so the $\Sigma_{A,B}$-orbit of the point is infinite. This reduces the whole theorem to finding one sufficiently large integral point: for most surfaces a trivial point $(\pm1,\pm1,z)$ works, and the few exceptions are settled by an explicit large solution.

What would settle it

The claim would be falsified by exhibiting any equation of the stated form with nonzero $a,b$ and integer coefficients that has only finitely many integer solutions; a more targeted check is to rerun the enumeration behind Proposition 12 and find a surface with $|a_1|+|a_2|\le6$ and $|b_1|+|b_2|\le6$ whose trivial solutions all have finite $\Sigma_{A,B}$-orbits but which is not isomorphic to one of the four listed surfaces.

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Extended reading notes

Core claim

The central claim of the paper is Theorem 3: for every integer choice of $a,b\ne 0$ and $a_1,a_2,b_1,b_2,c$, the cubic surface defined by $xyz=ax^3+by^3+c+a_2x^2+a_1x+b_2y^2+b_1y$ contains infinitely many integral points. The paper reduces the problem to the normalized case $a=b=c=1$ by sign changes. It then studies the surface together with its three companion surfaces, obtained by replacing $A(x)=x^3+a_2x^2+a_1x+1$ with $x^3A(1/x)$ and similarly for $B$. The isomorphisms between these surfaces generate the groupoid $\Sigma_{A,B}$, and an elementary estimate shows that any point with $\max\{|x|,|y|\}$ larger than a bound depending only on the coefficients has an infinite $\Sigma_{A,B}$-orbit. For the normalized cubics this covers all cases with $|a_1|+|a_2|\ge 7$ or $|b_1|+|b_2|\ge 7$; the finitely many remaining surfaces form three isomorphism classes, and for each the authors exhibit an explicit integral point with $\max\{|x|,|y|\}\ge 6$, whose orbit is then infinite and supplies infinitely many integral points.

Load-bearing premise

The proof depends on an unproved structural result about the automorphism groups of these cubic surfaces, namely that a single explicit automorphism generates a finite-index subgroup and that every infinite orbit is Zariski dense; if that result is wrong, the reduction to the finite list of exceptional surfaces collapses.

Editorial extensions

If this is right

Every cubic equation of the form $xyz=ax^3+by^3+c+a_2x^2+a_1x+b_2y^2+b_1y$ with $a,b\ne 0$ has infinitely many integer solutions, so the cubic case of the Mordell-Schinzel conjecture is settled.
Apart from three isomorphism classes listed in Proposition 12, the trivial points $(\pm1,\pm1,z)$ already generate infinitely many solutions through the automorphism group.
For the auxiliary equations $mxyz=x^3+y^3+1+a_2x^2+a_1x+b_2y^2+b_1y$ with fixed $m\ne\pm1$, the same orbit method proves infinitude whenever a single solution with $\max\{|x|,|y|\}>2+\max\{|a_1|+|a_2|,|b_1|+|b_2|\}$ exists.
The method does not directly extend to quartic equations, since the paper exhibits infinitely many quartic examples where all trivial-solution orbits are finite, so higher-degree cases will need another source of starting points.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural testable extension is to treat 'find one sufficiently large integral point, then let the automorphism group produce the rest' as a general strategy for Diophantine equations attached to surfaces with infinite automorphism groups; the exceptional list shows the only bottleneck is producing that first large point.
If the structural automorphism-group results used from the separately circulated preprint are independently verified, the same reduction should apply to all degrees $m,n\ge3$, but the quartic examples suggest the exceptional families can be infinite in higher degree, so a different source of starting points would then be needed.
Because the proof is effective—the finite exceptions are listed and the required large points are explicit—a reader could convert the argument into a terminating procedure that, for any concrete cubic equation of the stated shape, either generates infinitely many solutions or identifies it as one of the exceptional surfaces.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Kollár and Li prove the first fully proven instance of the Mordell–Schinzel conjecture for degrees at least 3: the case m=n=3 for equations xyz = x^3+y^3+1+a2x^2+a1x+b2y^2+b1y (and their general a,b,c reductions). That is a real result; no (m,n) with m,n>=3 had been settled before. The strategy is attractive: instead of using the huge automorphism generator, they work with a 4-surface groupoid and elementary growth estimates to show that any sufficiently large point has infinite orbit. The estimates in Corollaries 10 and 11 are clean and checkable, and the four exceptional surfaces are each supplied with an explicit large integral point.

The soft spots are exactly where the stress-test puts them. The reduction to the finite list in Proposition 12 is load-bearing, and its proof is thin: it cites [KVP24, Thm.29] and [KVP24, 51] to argue that finiteness of the sigma_A,B-orbit implies finiteness of the full Aut-orbit. That step is not proved here and is not self-evident; it requires the groupoid stabilizer to be the cyclic group generated by sigma. If that structural result from the first author's preprint is wrong, the exception list collapses. The computer enumeration behind Proposition 12 is also not certified: a URL is not a certificate, and the protocol is not described. It may be perfectly correct, but for a theorem that reduces to a finite check, the check should be reproducible without trusting the program.

There are also small presentational issues: some apparent sign/label errors and typos, and the paper is candid that it has no direct proof of Remark 13. None of this undermines the central claim, which I think is probably true. But the paper is not self-contained, and the missing pieces are structurally important rather than cosmetic.

Who is this for? Anyone working on Diophantine equations or arithmetic geometry will want to know this result. It deserves a serious referee: send it out, with a request that the authors either prove or independently verify the needed [KVP24] facts and make the finite enumeration reproducible. I would not desk-reject it; I would engage with it, but acceptance should wait until those two pillars are solid.

Referee Report

3 major / 6 minor

Summary. The paper proves that every cubic diophantine equation of the form xyz = ax^3 + by^3 + c + a2 x^2 + a1 x + b2 y^2 + b1 y with a,b nonzero has infinitely many integer solutions. This is the m=n=3 case of the Mordell-Schinzel conjecture. The proof builds on Schinzel's theorem for |abc|>1, reduces the remaining |abc|=1 cases by sign changes to a=b=c=1, and uses an infinite automorphism groupoid of companion surfaces. An elementary growth estimate (Corollary 10) shows that an orbit is infinite once a point exceeds a coefficient-dependent size; Corollary 11 handles all surfaces with coefficient sums at least 7, leaving a finite list that is dispatched by a computer enumeration (Proposition 12) and four explicit large points (Section 14).

Significance. If correct, Theorem 3 settles the cubic case of a conjecture that has been open in this form since Mordell and that Schinzel only resolved for |abc|>1. The overall strategy is attractive: the automorphism groupoid converts an infinite question into a finite verification, and the four large points in Section 14 are immediately checkable by hand. The proof is not fully self-contained, however, because the structural facts about the automorphism group are imported from an arXiv preprint by the first author ([KVP24]) and the finite classification rests on an uncertified computer enumeration. If those inputs are valid, the argument is sound; the gaps identified below are what currently prevent the paper from being a complete proof.

major comments (3)

[§8, Proposition 12] The reduction in Proposition 12 relies on [KVP24, Thm.29] and [KVP24, 51], which are not proved in this paper and are stated as results of an arXiv preprint coauthored by the first author. In particular, the finite-index statement for <sigma_A,B> in Aut_C(S_A,B) is needed to pass from finiteness of all Sigma_A,B-orbits of trivial solutions to finiteness of all Aut(S_A,B)-orbits. If those structural results are flawed or do not apply to the surfaces (4.1), the classification of exceptions in Proposition 12 has no proof. The paper should either include proofs of these facts for m=n=3, or explicitly state and verify them for the finite set of surfaces actually used.
[Proposition 12] The completeness of the finite enumeration underlying Proposition 12 is load-bearing but not supported by the manuscript. The proof only gives a URL and states that a computer program is available; there is no description of the enumeration algorithm, no code listing, no output, and no certificate that all surfaces with |a1|+|a2| <= 6 and |b1|+|b2| <= 6 were checked. The theorem depends on the assertion that (12.1)-(12.4) are the only surfaces with finite Sigma_A,B-orbits for all trivial solutions, so this gap must be closed. Please provide a verifiable certificate, including the program and its output, or an independent computational protocol.
[Lemma 9.2, Corollary 10] Lemma 9.2 is stated in a form that is false as written: the inequality |f(x)| > |x^m| cannot hold for |x| <= -1 + sum |a_i| (for example, f(x)=x^3+x+1 and x=0,m=0 gives 1 > 1, which is false). The condition should presumably be |x| >= 1 + sum |a_i| (or similar), which is exactly what Corollary 10 needs when it applies (9.2). Please correct the statement of Lemma 9.2 and adjust the proof of Corollary 10 accordingly.

minor comments (6)

[Corollary 11] In the proof of Corollary 11, the names sigma_x and sigma_y appear to be interchanged relative to the definitions in (7.3). The displayed computation makes sense if the first step is sigma_y (giving x1 = B(y0)/x0) and the second is sigma_x (giving y2 = A(x1)/y1); please correct the labels or the formulas.
[Notation and typos] The manuscript contains numerous typos: 'stricty' in Corollary 10, missing carets in equations such as 'ax3' in the abstract and Section 1, and 'A(x−1)' in (7.1) should be 'A(x^{-1})'. A careful proofreading pass is needed.
[Remark 7.5] In Remark 7.5, the sentence 'Thus applying sigma_y as in (7.4) we get points with x1 arbitrarily large' is correct because z0 can be chosen arbitrarily, but this should be stated explicitly to make the argument transparent.
[Proposition 12] The statement of Proposition 12 lists four equations and says 'There are only 3 such, up to isomorphism.' Since (12.3) and (12.4) are isomorphic, this is consistent, but a sentence making the count explicit would avoid confusion.
[Section 14] The four points in Section 14 are easily checked to lie on the respective surfaces, but the text should state explicitly that each point satisfies the size condition of Corollary 10 (max{|x0|,|y0|} >= 6), so that the conclusion follows directly.
[References] The reference to [KVP24, 51] uses an unusual numbering; please identify clearly whether '51' refers to an equation, a theorem, or a numbered paragraph in that preprint.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the proof's new content is a growth argument plus finite checks, not an encoding of its inputs.

full rationale

The derivation chain is: reduce to the normalized surfaces (4.1), use Schinzel for |abc|>1, prove the elementary growth criterion Corollary 10 (a norm increase under sigma_x or sigma_y gives an infinite Sigma_A,B-orbit), prove Corollary 11 for coefficient size at least 7, enumerate the remaining finite coefficient range by computer in Proposition 12, and finally verify explicit points with max{|x0|,|y0|} >= 6 in Section 14 for the four displayed exceptions. The only steps not proved inside the paper are structural facts imported from [KVP24, Thm.3, 27, Thm.29, 51] (infinite automorphism group, companion isomorphisms sigma_x and sigma_y, finite index of <sigma_A,B> in Aut_C, and Zariski density) and the completeness of the computer enumeration in Proposition 12. These are reproducibility and correctness gaps, not circular reductions: [KVP24] is co-authored by the first author, but its statements concern the automorphism structure of the surfaces, not the existence of infinitely many integral points, and they are parameter-free external facts rather than fitted values. No equation in the paper is defined in terms of the theorem being proved, and no fitted parameter is relabeled as a prediction. Remark 13 even admits a missing direct proof ('We do not have a direct proof of this'), and Proposition 12's proof consists of a citation plus a URL rather than a certificate; both are unverified support, not self-referential support. Therefore the paper's new contribution is not equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numeric constants are fitted; the coefficients in the theorem are free variables. The proof imports structural results from two prior papers, one of which is an unreviewed preprint by the first author, and relies on a computer search for the finite enumeration. No new physical entities are introduced.

assumptions (5)

domain assumption [KVP24, Thm.3] The surfaces S_{A,B} have an infinite automorphism group over Z with an explicit infinite-order automorphism sigma_A,B.
Used in Section 8 to define the groupoid; this is a cited theorem from a preprint by the first author.
domain assumption [KVP24, Thm.29] For m = n = 3, the cyclic group generated by sigma_A,B has finite index in Aut_C(S_{A,B}).
Used in Section 12 to reduce the analysis to a finite enumeration of exceptions.
domain assumption [KVP24, 51] Every infinite sigma_A,B-orbit is Zariski dense.
Used to infer that an infinite orbit yields infinitely many distinct integral points.
domain assumption [Sch15, Thm.4] For |abc| > 1, equation (1.1) has infinitely many integer solutions.
Used in Section 4 to restrict the proof to a,b,c in {1,-1}; this is a published theorem.
ad hoc to paper The computer enumeration underlying Proposition 12 is correct.
The paper relies on a program at a URL without a certificate or detailed listing of the enumeration algorithm.

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Cite this review

Pith. "Pith review of The Mordell-Schinzel conjecture for cubic diophantine equations." pith.science (2026). https://pith.science/paper/TAL65VVX

@misc{pith2026241212080,
  author       = {Pith},
  title        = {Pith review of: The Mordell-Schinzel conjecture for cubic diophantine equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TAL65VVX}},
  note         = {Machine review of arXiv:2412.12080}
}

abstract

Building on the works of Mordell (1952) and Schinzel (2015), we prove that cubic diophantine equations $xyz=G(x,y)$ have infinitely many solutions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages

[1]

Tim Browning and Florian Wilsch, Integral points on cubic surfaces: heuristics and numerics, https://arxiv.org/abs/2407.16315, 2024

work page Pith review arXiv 2024
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[7]

L. J. Mordell, The congruence ax^3+by^3+c 0\;( mod \,xy) , and integer solutions of cubic equations in three variables , Acta Math. 88 (1952), 77--83. 51852

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[9]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Tim Browning and Florian Wilsch, Integral points on cubic surfaces: heuristics and numerics, https://arxiv.org/abs/2407.16315, 2024

work page Pith review arXiv 2024

[2] [2]

Bogdan Grechuk, Polynomial D iophantine equations---a systematic approach , Springer, Cham, 2024, With contributions by Ashleigh Wilcox. 4823864

work page 2024

[3] [3]

Bogdan Grechuk and Denis Shatrov, Discussion on M athoverflow , https://mathoverflow.net/questions/466362/can-9xy-divide-1x2x3y2, 2024

work page 2024

[4] [4]

6 (1939), 407--427

Ernst Jacobsthal, Zahlentheoretische E igenschaften ganzzahliger P olynome , Compositio Math. 6 (1939), 407--427. 1557037

work page 1939

[5] [5]

J \'a nos Koll \'a r, Log K 3 surfaces with irreducible boundary , 2024, https://arxiv.org/abs/2407.08051

arXiv 2024

[6] [6]

J \'a nos Koll \'a r and David Villalobos-Paz, Cubic surfaces with infinite, discrete automorphism group, 2024, https://arxiv.org/abs/2410.03934

work page Pith review arXiv 2024

[7] [7]

L. J. Mordell, The congruence ax^3+by^3+c 0\;( mod \,xy) , and integer solutions of cubic equations in three variables , Acta Math. 88 (1952), 77--83. 51852

work page 1952

[8] [8]

30, Academic Press, London-New York, 1969

, Diophantine equations, Pure and Applied Mathematics, vol. 30, Academic Press, London-New York, 1969. 249355

work page 1969