The Diophantine problem in iterated wreath products of free abelian groups is undecidable

Olga Kharlampovich; Alexei Miasnikov

doi:10.1142/S0218196726400096

The Diophantine problem in iterated wreath products of free abelian groups is undecidable

Olga Kharlampovich
, Alexei Miasnikov

Department of Mathematical Sciences

City University of New York

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we prove that the Diophantine problem in iterated restricted wreath products G of arbitrary non-trivial free abelian groups A₁,…,A_k, k > 1 of finite ranks is undecidable, i.e., there is no algorithm that given a finite system of group equations with coefficients in G decides whether or not the system has a solution in G.

Original language	English
Journal	International Journal of Algebra and Computation
DOIs	https://doi.org/10.1142/S0218196726400096
State	Accepted/In press - 2026

Keywords

Diophantine problem
equations
wreath product of abelian groups

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10.1142/S0218196726400096

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title = "The Diophantine problem in iterated wreath products of free abelian groups is undecidable",

abstract = "In this paper we prove that the Diophantine problem in iterated restricted wreath products G of arbitrary non-trivial free abelian groups A1,…,Ak, k > 1 of finite ranks is undecidable, i.e., there is no algorithm that given a finite system of group equations with coefficients in G decides whether or not the system has a solution in G.",

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AU - Miasnikov, Alexei

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N2 - In this paper we prove that the Diophantine problem in iterated restricted wreath products G of arbitrary non-trivial free abelian groups A1,…,Ak, k > 1 of finite ranks is undecidable, i.e., there is no algorithm that given a finite system of group equations with coefficients in G decides whether or not the system has a solution in G.

AB - In this paper we prove that the Diophantine problem in iterated restricted wreath products G of arbitrary non-trivial free abelian groups A1,…,Ak, k > 1 of finite ranks is undecidable, i.e., there is no algorithm that given a finite system of group equations with coefficients in G decides whether or not the system has a solution in G.

KW - Diophantine problem

KW - equations

KW - wreath product of abelian groups

UR - https://www.scopus.com/pages/publications/105039310728

UR - https://www.scopus.com/pages/publications/105039310728#tab=citedBy

U2 - 10.1142/S0218196726400096

DO - 10.1142/S0218196726400096

M3 - Article

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SN - 0218-1967

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

ER -

The Diophantine problem in iterated wreath products of free abelian groups is undecidable

Abstract

Keywords

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