The Diophantine problem for systems of algebraic equations with exponents

TL;DR

This paper proves that deciding the existence of integer solutions for algebraic equations with exponents is NP-complete, and characterizes the solution set as semilinear, extending understanding of these Diophantine problems.

Contribution

It establishes NP-completeness for the decision problem and describes the structure of all solutions for systems of algebraic equations with exponents.

Findings

Decision problem is NP-complete.

Solution set is semilinear.

Results hold for fixed or input $oldsymbol{ extalpha}$.

Abstract

Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as an \emph{algebraic equation with exponents}. We prove that the problem to decide if a given equation has an integer solution is $\textbf{NP}$-complete, and that the same holds for systems of equations (whether $\alpha$ is fixed or given as part of the input). Furthermore, we describe the set of all solutions for a given system of algebraic equations with exponents and prove that it is semilinear.