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arXiv · 2412.01768

Hilbert's tenth problem via additive combinatorics

Peter Koymans·Carlo Pagano
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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.

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Peter Koymans, Carlo Pagano. 2024-12-02. Hilbert's tenth problem via additive combinatorics. https://arxiv.org/abs/2412.01768

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