Hilbert's tenth problem via additive combinatorics (2412.01768v2)

Abstract: For all infinite rings $R$ that are finitely generated over $\mathbb{Z}$, we show that Hilbert's tenth problem has a negative answer. This is accomplished by constructing elliptic curves $E$ without rank growth in certain quadratic extensions $L/K$. To achieve such a result unconditionally, our key innovation is to use elliptic curves $E$ with full rational $2$-torsion which allows us to combine techniques from additive combinatorics with $2$-descent.

• The paper introduces a novel integration of 2-descent methods with additive combinatorics to control 2-Selmer ranks.
• The study constructs elliptic curves that avoid rank growth in quadratic extensions without assuming the finiteness of the Tate–Shafarevich group.
• The authors generalize prior conjectures by proving that ℤ is Diophantine over the ring of integers in any number field.

Insights on Hilbert’s Tenth Problem via Additive Combinatorics

The paper "Hilbert's tenth problem via additive combinatorics" by Peter Koymans and Carlo Pagano presents significant advancements in the understanding of Hilbert's Tenth Problem (HTP) through methods in additive combinatorics. Hilbert's Tenth Problem, as originally posited, asked for a general algorithm to determine the solvability of Diophantine equations over the integers. The problem was resolved in the negative via the collaborative efforts leading to the MRDP theorem. The paper under discussion seeks to provide a negative answer to this problem for every infinite ring finitely generated over the integers.

Main Contributions

1. Elliptic Curves without Rank Growth: The authors construct elliptic curves $E$ that exhibit no rank growth in certain quadratic extensions $L/K$. Crucially, they manage this without assuming the finiteness of the Tate–Shafarevich group, which is a notable departure from prior reliance on such assumptions by prominent researchers like Mazur and Rubin.
2. Integration of Additive Combinatorics: A pivotal innovation in this paper is the integration of techniques from additive combinatorics with $2$-descent on elliptic curves. This method allows for the control of $2$-Selmer ranks, ultimately informing the solvability conditions of the HTP over these infinite rings.
3. Generalization to Number Fields: The authors extend the results from rings of integers to more general number fields, showing that the integers $\mathbb{Z}$ are Diophantine over the ring of integers of any number field, thus generalizing Denef and Lipshitz’s conjectures.

Numerical Results

Key to the paper’s methodology is leveraging the properties of elliptic curves with full rational $2$-torsion. This facilitates the manipulation of $2$-Selmer groups through additive combinatorial techniques. Unlike polynomial twists, which are typically hard to control, especially beyond linear cases, the authors employ prime element selection facilitated by Green–Tao-type theorems to achieve the necessary rank reductions and Selmer rank control.

Implications and Speculations

Practical Implications: Practically, the outcome of this paper suggests that determining the solvability of polynomial equations in many cases extends beyond integer coefficients to broader classes of number-theoretic structures without resorting to additional conjectural frameworks.

Theoretical Framework: Theoretically, the formulation offers robust new techniques for engaging with classic number-theoretic problems. By combining deep insights from disparate areas such as additive combinatorics with traditional number theory, the authors propose a pathway that may address other unsolved conjectures, such as Skorobogatov's conjecture, without dependence on the finiteness of Sha.

Future Developments: As computational techniques in understanding elliptic curves evolve, the approaches outlined might help elucidate further intricacies of $L$-function zeros or directions in arithmetic geometry with implications for cryptographic applications. The potential discovery of elliptic curves with rank growth over every number field appears plausible with these methodologies.

In conclusion, the integration of additive combinatorics into the paper of Hilbert's Tenth Problem opens substantial new avenues for research, leading to both practical advances in algorithmic number theory and theoretical expansions regarding the nature of Diophantine equations over complex number fields. The framework presented by Koymans and Pagano bears promising tools and results that may influence future studies in related domains.