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arXiv · 2510.18794

Undecidability on Diophantine equations over $\mathbb Z[i]$ with $20$ unknowns

Yuri Matiyasevich·Zhi-Wei Sun
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Abstract

It is known that Hilbert's Tenth Problem over the Gaussian ring $\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}$ is undecidable. In this paper we obtain the following further result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{20})=0$ (with integer coefficients) is solvable over $\mathbb Z[i]$. This improves the previous record involving $52$ variables.

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Yuri Matiyasevich, Zhi-Wei Sun. 2025-10-21. Undecidability on Diophantine equations over $\mathbb Z[i]$ with $20$ unknowns. https://arxiv.org/abs/2510.18794

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