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Mathematics > Number Theory

arXiv:2210.00086 (math)
[Submitted on 30 Sep 2022 (v1), last revised 23 Mar 2023 (this version, v2)]

Title:The Diophantine problem for systems of algebraic equations with exponents

Authors:Richard Mandel, Alexander Ushakov
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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.
Subjects: Number Theory (math.NT)
MSC classes: 11Y16, 68Q15, 68W30
Cite as: arXiv:2210.00086 [math.NT]
  (or arXiv:2210.00086v2 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.2210.00086

Submission history

From: Alexander Ushakov [view email]
[v1] Fri, 30 Sep 2022 20:41:13 UTC (17 KB)
[v2] Thu, 23 Mar 2023 15:14:07 UTC (20 KB)
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