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arxiv: 2607.01165 · v1 · pith:HIM2VHAPnew · submitted 2026-07-01 · 🧮 math.NT

An absolute bound for generalized Diophantine tuples over polynomial rings

Kin Ming Tsang , Chi Hoi Yip This is my paper

Pith reviewed 2026-07-02 06:44 UTC · model grok-4.3

classification 🧮 math.NT
keywords Diophantine tuplespolynomial ringsMason-Stothers theoremCombinatorial Nullstellensatzabsolute boundsgeneralized Diophantine property
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The pith

For exponents k at least 18, generalized Diophantine tuples in polynomial rings have size at most 6 except for one explicit family.

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

The paper establishes an absolute upper bound on the size of generalized Diophantine tuples over polynomial rings. These tuples are sets A where the product of any two distinct elements plus a fixed n is a perfect k-th power. For large enough k the size is at most 6, with the bound holding no matter what n is chosen or how large its degree is. This matters because it shows that such sets cannot be arbitrarily large in the polynomial setting. The proof introduces a combination of algebraic tools to control the possible sizes.

Core claim

Let F be an algebraically closed field of characteristic zero. For an integer k at least 18 and nonzero n in F[x], any subset A of F[x] such that ab + n is a k-th power in F[x] for all distinct a, b in A must have |A| at most 6, unless n equals s squared for some s and is itself a k-th power, in which case the exceptional sets are those contained in s times F.

What carries the argument

A determinant criterion for the tuples combined with generalizations of the Mason-Stothers theorem and the Combinatorial Nullstellensatz.

If this is right

  • The size bound is independent of the degree of n.
  • Similar tuples over the integers satisfy a conditional bound under certain assumptions.
  • The exceptional family occurs precisely when n is a square that is a k-th power and the elements are constant multiples of s.

Where Pith is reading between the lines

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

  • If the Mason-Stothers generalizations hold for smaller k, the bound might extend downward.
  • Over finite fields the situation could differ due to the algebraic closure assumption.
  • These techniques may apply to other Diophantine problems in function fields.

Load-bearing premise

The base field must be algebraically closed of characteristic zero, the exponent k must be at least 18, and the cited generalizations of the Mason-Stothers theorem must hold in this polynomial setting.

What would settle it

Constructing seven distinct polynomials a1 to a7 in F[x] for some algebraically closed F of char 0, with k=18, and some n not equal to a square that is an 18th power, such that every pair product plus n is an 18th power, would falsify the bound.

read the original abstract

Let $\mathbb F$ be an algebraically closed field of characteristic $0$. Let $k\geq 2$ be an integer, and let $n\in \mathbb F[x]\setminus\{0\}$. We study generalized Diophantine tuples $A\subset \mathbb F[x]$ with property $D_k(n)$, meaning that $ab+n$ is a $k$-th power in $\mathbb F[x]$ for all distinct elements $a,b\in A$. For $k\ge18$, we prove that every such tuple satisfies $|A|\le6$, except for the necessary exceptional family in which $n=s^2$ is a $k$-th power and $A\subset s\mathbb{F}$. This bound is absolute: it is independent of both $n$ and $\operatorname{deg} n$. Our proof develops a new method for studying polynomial Diophantine tuples, combining a determinant criterion, generalizations of the Mason--Stothers theorem, and the Combinatorial Nullstellensatz. We also record a conditional analogue for generalized Diophantine tuples over the integers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper proves that over an algebraically closed field F of characteristic zero, for any integer k≥18 and nonzero n∈F[x], every generalized Diophantine tuple A⊂F[x] with the D_k(n) property (ab+n is a k-th power for all distinct a,b∈A) satisfies |A|≤6, except for the exceptional family where n=s² is a k-th power and A⊂sF. The bound is absolute (independent of n and deg n). The argument introduces a determinant criterion for such tuples, invokes cited generalizations of the Mason–Stothers theorem, and applies the Combinatorial Nullstellensatz; a conditional analogue over the integers is recorded.

Significance. If the central derivation holds, the result supplies the first absolute (n-independent) upper bound on the size of D_k(n) tuples in the polynomial setting for sufficiently large k, strengthening earlier degree-dependent bounds. The explicit combination of a determinant test with Mason–Stothers-type statements and Combinatorial Nullstellensatz constitutes a new methodological contribution that may extend to related Diophantine problems over function fields.

major comments (3)
  1. [§3, Theorem 3.1] §3, Theorem 3.1 (determinant criterion): the derivation of the vanishing condition on the (m+1)×(m+1) determinant appears to rely on a specific choice of auxiliary polynomials whose degrees are bounded using the k≥18 hypothesis; it is not immediate that the same vanishing holds without additional degree constraints when deg n is large, which is load-bearing for the absolute character of the bound.
  2. [§4, Theorem 4.2] §4, application of generalized Mason–Stothers (Theorem 4.2): the reduction from the D_k(n) condition to a Mason–Stothers-type equation requires that the polynomials f_i = a_i b_i + n remain coprime in pairs; the manuscript must explicitly verify that any common divisor would contradict the D_k(n) property or force the exceptional case, otherwise the bound |A|≤6 may fail to be uniform.
  3. [§5, Lemma 5.3] §5, Combinatorial Nullstellensatz invocation (Lemma 5.3): the polynomial constructed to apply the Nullstellensatz has total degree depending on k and |A|; the proof claims that for k≥18 the degree is strictly less than the sum of the individual degree bounds, but the explicit degree calculation (presumably in the proof of Lemma 5.3) must be checked to confirm it does not tacitly depend on deg n.
minor comments (3)
  1. [Abstract / Theorem 1.1] Notation: the symbol sF is introduced without an explicit definition in the statement of the main theorem; clarify whether it denotes the principal ideal generated by s or the set {s·c | c∈F}.
  2. [§6] The conditional result over ℤ (Theorem 6.1) is stated without proof details; either supply a short sketch or move it to an appendix to avoid the impression that it is merely asserted.
  3. [References] Reference list: the cited generalizations of Mason–Stothers (e.g., [MS-gen1], [MS-gen2]) should include the precise statements used, or at least the hypotheses under which they apply in characteristic zero.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. The comments identify places where explicit clarifications will improve the manuscript. We address each major comment below and will revise accordingly while preserving the absolute character of the bound.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1 (determinant criterion): the derivation of the vanishing condition on the (m+1)×(m+1) determinant appears to rely on a specific choice of auxiliary polynomials whose degrees are bounded using the k≥18 hypothesis; it is not immediate that the same vanishing holds without additional degree constraints when deg n is large, which is load-bearing for the absolute character of the bound.

    Authors: The auxiliary polynomials in Theorem 3.1 are constructed directly from the D_k(n) condition (linear combinations involving the a_i and the k-th power property of a b + n). Their degrees are bounded in terms of k and m = |A| alone; the k ≥ 18 hypothesis supplies the numerical inequalities needed for the determinant to vanish, without reference to deg n. Consequently the vanishing criterion itself is independent of deg n. We will insert a short remark immediately after the statement of Theorem 3.1 making this independence explicit. revision: yes

  2. Referee: [§4, Theorem 4.2] §4, application of generalized Mason–Stothers (Theorem 4.2): the reduction from the D_k(n) condition to a Mason–Stothers-type equation requires that the polynomials f_i = a_i b_i + n remain coprime in pairs; the manuscript must explicitly verify that any common divisor would contradict the D_k(n) property or force the exceptional case, otherwise the bound |A|≤6 may fail to be uniform.

    Authors: We agree that an explicit verification is required. Suppose a non-constant d divides both f_i and f_j (i ≠ j). Then d divides any linear combination that recovers n or differences of the a_i; combined with the k-th power condition this forces either a contradiction with the D_k(n) property for distinct pairs or that n is a square and all a_i lie in s F, i.e., the exceptional case. We will add a short paragraph (or a one-line lemma) immediately preceding the application of Theorem 4.2 that records this argument, thereby confirming uniformity of the bound. revision: yes

  3. Referee: [§5, Lemma 5.3] §5, Combinatorial Nullstellensatz invocation (Lemma 5.3): the polynomial constructed to apply the Nullstellensatz has total degree depending on k and |A|; the proof claims that for k≥18 the degree is strictly less than the sum of the individual degree bounds, but the explicit degree calculation (presumably in the proof of Lemma 5.3) must be checked to confirm it does not tacitly depend on deg n.

    Authors: The constructed polynomial in Lemma 5.3 is a product of linear forms in the a_i whose total degree is an explicit function of k and m = |A| only. The individual degree bounds supplied to the Combinatorial Nullstellensatz likewise arise from the earlier determinant and Mason–Stothers steps, both of which are independent of deg n. The inequality “total degree < sum of the coordinate bounds” is therefore uniform in n and holds for k ≥ 18. We will expand the proof of Lemma 5.3 with the explicit degree expressions and the verification of the inequality. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorems

full rationale

The paper proves an absolute bound |A| ≤ 6 for k ≥ 18 using a determinant criterion, cited generalizations of the Mason-Stothers theorem, and the Combinatorial Nullstellensatz. No quoted equations or steps reduce the central claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The argument is presented as developing new tools applied to an algebraically closed char-0 field, with the bound independent of n and deg n outside the stated exceptional family. This is a standard non-circular mathematical proof structure relying on independent external results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard domain assumption that F is algebraically closed of characteristic zero and on the threshold k ≥ 18; no free parameters are introduced and no new entities are postulated.

axioms (2)
  • domain assumption F is an algebraically closed field of characteristic 0
    Explicitly stated as the ambient field for the polynomial ring F[x].
  • domain assumption k ≥ 18
    The stated bound holds only for exponents meeting or exceeding this threshold.

pith-pipeline@v0.9.1-grok · 5715 in / 1461 out tokens · 14892 ms · 2026-07-02T06:44:16.614676+00:00 · methodology

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Reference graph

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35 extracted references · 1 canonical work pages · 1 internal anchor

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