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

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 $P(z_1,\dots ,z_{20})=0$ (with integer coefficients) is solvable over $\mathbb{Z}[i]$. This improves the previous record involving $52$ variables.

9 pages