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

TL;DR

This paper proves that it is impossible to decide whether polynomial equations with 20 unknowns over the Gaussian integers have solutions, extending the known undecidability from higher variable counts.

Contribution

The paper improves the known undecidability result for Diophantine equations over z[i] from 52 to 20 variables, advancing the understanding of Hilbert's Tenth Problem in this domain.

Findings

No algorithm can decide solvability of polynomial equations with 20 unknowns over z[i].

Undecidability holds for equations with fewer variables than previously established.

The result narrows the gap towards the minimal number of variables for undecidability.

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.