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