One-variable equations over the lamplighter group

TL;DR

This paper proves that the problem of deciding whether a single-variable equation over the lamplighter group is solvable is decidable, introducing an automaton-based method and analyzing typical versus worst-case complexity.

Contribution

It establishes the decidability of single-variable equations over the lamplighter group and develops an automaton-theoretic framework for analyzing related divisibility problems.

Findings

Decidability of single-variable equations over the lamplighter group.

Automaton-theoretic approach to divisibility of parametric Laurent polynomials.

Super-exponential worst-case and nearly quadratic typical-case complexity results.

Abstract

We study one-variable equations over the lamplighter group $\MZ_2 \wr \MZ$. While the decidability of arbitrary equations over $L_2$ remains open, we prove that the Diophantine problem for single equations in one variable is decidable. Our approach reduces the problem to a divisibility question for families of parametric Laurent polynomials over $\MZ_2$, whose coefficients depend linearly on an integer parameter. We develop an automaton-theoretic framework to analyze divisibility of such polynomials, exploiting eventual periodicity phenomena arising from polynomial division over finite fields. This yields an explicit decision procedure, which is super-exponential in the worst case. On the other hand, we show that for a generic class of equations, solvability can be decided in nearly quadratic time. These results establish a sharp contrast between worst-case and typical computational behavior and provide new tools for the study of equations over wreath products.