An absolute bound for generalized Diophantine tuples over polynomial rings

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 $n=s<sup>2$ 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.

• The paper establishes an absolute bound, showing that for k ≥ 18 any generalized Diophantine tuple in F[x] has at most 6 elements (5 when n is nonsquare and k is even), except for a specific affine exception.
• The authors employ determinant analysis, the Mason–Stothers theorem, and the Combinatorial Nullstellensatz to derive structural restrictions on polynomial Diophantine tuples.
• The work provides a uniform, unconditional bound in contrast to the integer case, demonstrating the effectiveness of function field techniques in resolving classical Diophantine problems.

Absolute Bounds for Generalized Diophantine Tuples over Polynomial Rings

Introduction and Background

The study of Diophantine $m$-tuples, originally sets of positive integers such that the product of any two distinct elements increased by $1$ is a perfect square, has undergone substantial generalization. This extends both in the arithmetic type (replacing $+1$ and the square condition with an arbitrary $n$ and the $k$-th power, respectively) and in the underlying ring (notably to polynomial rings over fields). The classical case over $\mathbb{N}$, and subsequently over $\mathbb{Z}$, has culminated in the absolute determination that no Diophantine quintuple exists. However, the behavior in function fields, particularly polynomial rings over algebraically closed fields, has resisted a general unconditional resolution outside small-degree or special cases.

This paper by Tsang and Yip ("An absolute bound for generalized Diophantine tuples over polynomial rings" (2607.01165)) addresses the polynomial ring setting: For subsets $A\subset F[x]$, where $F$ is an algebraically closed field of characteristic $0$, and for nonzero $1$0, the set $1$1 is a generalized Diophantine tuple with property $1$2 if $1$3 is a $1$4-th power in $1$5 for all distinct $1$6. The central objective is to provide absolute bounds on $1$7—that is, bounds independent of $1$8 and its degree—for sufficiently large exponents $1$9.

Main Results

The primary result is as follows: For $+1$0, any generalized Diophantine tuple $+1$1 with property $+1$2 satisfies

$+1$3

with a unique exception: if $+1$4 is itself a $+1$5-th power and $+1$6, in which case no finite bound on $+1$7 is possible, as $+1$8 may be infinite. Moreover, if $+1$9 is nonsquare and $n$0 is even, the stronger bound $n$1 holds.

This is an absolute bound: for all nonzero $n$2, regardless of degree, all sufficiently large exponents $n$3 obviate any dependence of $n$4 on the arithmetical complexity of $n$5. The simultaneous occurrence of absoluteness and generality is particularly notable, given that over $n$6 only conditional (on the Bombieri–Lang and Lander–Parkin–Selfridge conjectures) absolute boundedness is known.

An explicit statement:

• For $n$7, $n$8 for any $n$9 with property $k$0, unless $k$1 is itself a $k$2-th power and $k$3.
• If $k$4 is nonsquare and $k$5 is even, $k$6.

In addition, a conditional improvement for the integer case is recorded: assuming the Lander–Parkin–Selfridge conjecture, for all $k$7 and $k$8, one has $k$9 (unconditionally, only $\mathbb{N}$0 is known).

Techniques and Novel Approaches

The proof develops an overview of algebraic, combinatorial, and valuation-theoretic methods adapted for the polynomial ring context. Several pivotal ideas underlie the argument.

• Determinant and Cross-Ratio Analysis: Generalized Diophantine tuples naturally result in algebraic relations among products $\mathbb{N}$1; by casting these as entries in low-rank matrices, determinant conditions are derived whose vanishing encodes structural restrictions.
• Mason–Stothers (ABC) Theorems in Function Fields: Quantitative results of Mason–Stothers type and their refinements (notably those of Vaserstein–Wheland) are used to bound $\mathbb{N}$2 in equations involving sums of $\mathbb{N}$3-th powers of polynomials, leveraging the function-field analogy to the number field $\mathbb{N}$4-conjecture.
• Combinatorial Nullstellensatz: To treat cases where $\mathbb{N}$5 is locally “affine” or contained in translates of $\mathbb{N}$6-lines in $\mathbb{N}$7, the Combinatorial Nullstellensatz isolates configurations which must exhibit a certain algebraic independence.
• Selection and Extension Lemmas: The proof demonstrates that large generalized Diophantine tuples necessarily contain "forbidden" $\mathbb{N}$8 or $\mathbb{N}$9 subconfigurations, except in the explicitly described exceptional family. Detailed analysis of cross-ratio conditions is conducted both by valuation-theoretic partition arguments (for nonsquare $\mathbb{Z}$0) and geometric arguments (for square $\mathbb{Z}$1 not in special position).

A crucial step is the exclusion of $\mathbb{Z}$2 configurations (Proposition 18): for $\mathbb{Z}$3, there do not exist six distinct elements $\mathbb{Z}$4 such that all $\mathbb{Z}$5 and a certain cross-ratio is nonconstant. The threshold $\mathbb{Z}$6 arises from function field $\mathbb{Z}$7 estimates in special symmetric six-term $\mathbb{Z}$8-th power relations. It is observed that improvements to these exponential thresholds (possibly to $\mathbb{Z}$9) would directly improve the main theorem.

Numerical Thresholds and Exceptional Behavior

The main bounds are explicitly sharp outside the exceptional affine family ($A\subset F[x]$0 for $A\subset F[x]$1 a $A\subset F[x]$2-th power), where no finite bound is possible. The restriction $A\subset F[x]$3 corresponds to key divisibility barriers in the Mason–Stothers and Vaserstein–Wheland theorems for symmetric six-term sums. For $A\subset F[x]$4 below this threshold, uniform rigidity fails due to the existence of exceptional algebraic relations (especially for small values of $A\subset F[x]$5, e.g., quadratic, cubic).

In the integer setting, assuming the Lander–Parkin–Selfridge conjecture (asserting that $A\subset F[x]$6 nontrivial $A\subset F[x]$7-th power sums over $A\subset F[x]$8 are impossible for $A\subset F[x]$9 sufficiently large), the maximal size of a generalized Diophantine tuple for $F$0 is at most 5, substantially strengthening prior conditional results.

Implications and Perspectives

From an arithmetic geometry perspective, the result evidences that, in polynomial rings over algebraically closed characteristic zero fields, generalized Diophantine phenomena obey a form of uniform boundedness for large exponents. This stands in contrast to the integer case, where such boundedness is not unconditional, and highlights subtle distinctions in the function-field/integer analogy.

Practically, this provides comprehensive control over the possible structure of polynomial Diophantine tuples beyond earlier work limited to small $F$1 or $F$2. The explicit exclusion of affine lines in the exceptional family (where $F$3 is a $F$4-th power and $F$5) classifies all possible "large" polynomial Diophantine tuples.

Theoretically, the combination of determinant, valuation, and combinatorial techniques provides a flexible toolkit for related problems in polynomial Diophantine analysis, suggesting further applications to function field analogs of classical number-theoretic results.

It is noteworthy that further sharpening the function field $F$6-type estimates for sums of $F$7-th powers (specifically, in highly symmetric configurations) would lower the threshold on $F$8 in the main absolute bound. Thus, advances in function field sum-of-powers or zero estimates would immediately propagate to stronger uniform results for generalized Diophantine tuples.

Conclusion

This work establishes, unconditionally, that for $F$9 the cardinality of generalized Diophantine tuples over polynomial rings in characteristic zero is absolutely bounded by 6 (or 5 in the nonsquare, even $0$0 case), outside a single explicit exceptional family. The results are achieved by integrating determinant techniques, function field $0$1 theorems, and fine combinatorial selection arguments, and mark a significant advance in the uniformity of Diophantine phenomena in function fields. The methods and results set a framework for addressing analogous questions in more general algebraic settings and highlight promising directions for future improvements connected to better function-field analogues of number-theoretic conjectures.