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arXiv · 2608.03992

On Diophantine equations over the integer rings of quadratic fields

Zhi-Wei Sun
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Abstract

Let $K$ be any quadratic number field, and let $O_K$ be the ring of algebraic integers in $K$. In 1975 J. Denef proved that Hilbert's Tenth Problem over $O_K$ has a negative solution. In this paper we establish the following undecidability result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{16})=0$ (with integer coefficients and 16 unknowns) has solutions over $O_K$. Moreover, when $K$ is a real quadratic field, we show that $15$ unknowns suffice for undecidability.

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Zhi-Wei Sun. 2026-08-04. On Diophantine equations over the integer rings of quadratic fields. https://arxiv.org/abs/2608.03992

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