On exponential diophantine equations over $\mathbb Q$ with few unknowns

TL;DR

This paper proves the undecidability of solving certain exponential Diophantine equations over the rationals, showing no algorithm can determine their solvability in general for equations with up to eight unknowns.

Contribution

It establishes the first undecidability results for exponential Diophantine equations over  with a bounded number of unknowns, specifically up to eight.

Findings

No algorithm can decide solvability of general exponential Diophantine equations over  with eight unknowns.

Proves undecidability results for exponential Diophantine equations over .

Provides three new undecidability theorems in the area.

Abstract

In this paper we obtain three undecidable results for exponential diophantine equations over the field $\mathbb Q$ of rational numbers. For example, we prove that there is no algorithm to decide the solvability of a general exponential diophantine equation $F(x_1,\ldots,x_{8})=0$ over $\mathbb Q$ with eight unknowns.