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

TL;DR

The paper proves undecidability of Hilbert's Tenth Problem over the Gaussian integers $\mathbb Z[i]$ for Diophantine equations with $20$ unknowns, improving the previous bound of $52$ variables. It introduces an auxiliary result over imaginary quadratic fields that links a rational expression $y+\sum x_k/y^k$ to integrality of the $x_k$ via norm estimates, and uses Skolem’s degree-reduction trick to connect to degree $4$ Diophantine equations. The authors extend Matiyasevich–Robinson-style encoding to $\mathbb Z[i]$ to translate a nonrecursive, algorithmically enumerable set into a fixed Diophantine condition in $20$ Gaussian variables. This strengthens undecidability results for HTP in rings of integers of number fields and highlights sharp limits on algorithmic solvability in Diophantine problems over complex quadratic rings.

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.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Remark 3.1
• Lemma 3.4