One-variable equations over the lamplighter group

TL;DR

This work resolves the decidability of one-variable Diophantine equations over the lamplighter group $L_2=\mathbb{Z}_2\wr\mathbb{Z}$ by reducing ${\mathbf{DP}}_1(L_2)$ to divisibility problems for $\delta$-parametric Laurent polynomials over ${\mathbb{Z}}_2$. It introduces a division-by-$f$ automaton and a Magnus-type embedding to analyze these divisibility questions, and exploits piecewise periodicity of parametric polynomials to obtain a concrete decision procedure with worst-case super-exponential time, while proving that generic instances are solvable in nearly quadratic time. The approach yields a sharp contrast between worst-case and typical computational behavior for equations over wreath products and provides new tools, including geometric $xt$-grid tracing and automaton-based divisibility, for studying Diophantine problems in metabelian groups. Overall, the results advance understanding of equations over wreath products and establish practical methods for assessing solvability in the lamplighter setting. The findings have potential implications for related algebraic problems and cryptographic contexts where one-variable equations over wreath products arise.

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.

Figures (9)

• Figure 1: Tracing the word $w = t^2 a x t^{-1} x^{-2} a$ on the $xt$-grid produces the sets $N_w = \{ x^0 t^0, x^1 t^1 \}$ and $D_w = \{ x^1 t^0, x^2 t^0, x^2 t^1 \}$, corresponding to the polynomials $\mathop{\mathrm{{\contourlength{0.01em}\contour{black}{num}}}}\nolimits(w)$ and $\mathop{\mathrm{{\contourlength{0.01em}\contour{black}{den}}}}\nolimits(w)$, respectively. The left and right diagrams show $N_w$ (red points) and $D_w$ (green points), respectively.
• Figure 2: Substituting $\delta \leftarrow 1$ to the example in Figure \ref{['fig:tracing_example']}, we can visualize the numerator (see left diagram) and denominator (see right diagram) polynomials on $t$-axis. The sets $N_w$ and $D_w$ are $N_w = \{ z^0, z^2 \}$ and $D_w =\{ z^1, z^2, z^3 \}$.
• Figure 3: Tracing the word $w = t^{1-n} x^{n-1} a t^{-1} x^{-n} a$ creates a path from $(0,0)$ to $(1,n)$ and produces the sets $N_w =\{ x^n t^1, x^0 t^0 \}$ and $D_w =\{ x^1 t^0, x^2 t^0 \dots, x^n t^0, x^2 t^1, x^3 t^1, \dots , x^n t^1 \}$.
• Figure 4: $\Gamma_f$ for $f=z^3+z+1$.
• Figure 5: $\Gamma_f$ for $f=z^2+1$ and its normalization which is the Cayley graph of $D_4\simeq {\mathbb{Z}}_4\rtimes {\mathbb{Z}}_2$ with $P_f=8$.

Theorems & Definitions (82)

• Lemma 2.1
• Theorem 2.2: GathenGerhard2003
• Theorem 2.3: GathenGerhard2003
• Theorem 2.4: GathenGerhard2003
• Proposition 2.5
• proof
• Lemma 3.1
• proof
• Lemma 4.1
• proof