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Orientable quadratic equations in wreath products

Alexander Ushakov, Chloe Weiers

TL;DR

The paper addresses the solvability of orientable quadratic equations over wreath products $A\wr B$ of finitely generated abelian groups. It reduces the Diophantine problem to a central combinatorial object, the quotient-sum problem ${\mathsf{QSP}}$, and analyzes its complexity across finite/infinite $B$, genus $g$, and the rank of $B$. The main findings establish ${\mathbf{NP}}$-membership for the general uniform setting, with ${\mathbf{NP}}$-hardness when $|B|=\infty$, and polynomial-time decidability in several parameter regimes (notably large genus relative to rank$(B)$ and bounded $m$); the rank of $A$ largely does not affect hardness, though allowing $A$ to be part of the input can induce hardness even for small $B$. The results map a detailed boundary between tractable and intractable cases and connect algebraic structure with computational complexity through notions like clusters and quotient reduction. This advances understanding of Diophantine problems in wreath products and offers a framework for analyzing orientable equations in broader group classes, with implications for algorithmic group theory and related lattice-based techniques.

Abstract

In this paper we study the complexity of solving orientable quadratic equations in wreath products $A\wr B$ of finitely generated abelian groups. We give a classification of cases (depending on genus and other characteristics of a given equation) when the problem is computationally hard or feasible.

Orientable quadratic equations in wreath products

TL;DR

The paper addresses the solvability of orientable quadratic equations over wreath products of finitely generated abelian groups. It reduces the Diophantine problem to a central combinatorial object, the quotient-sum problem , and analyzes its complexity across finite/infinite , genus , and the rank of . The main findings establish -membership for the general uniform setting, with -hardness when , and polynomial-time decidability in several parameter regimes (notably large genus relative to rank and bounded ); the rank of largely does not affect hardness, though allowing to be part of the input can induce hardness even for small . The results map a detailed boundary between tractable and intractable cases and connect algebraic structure with computational complexity through notions like clusters and quotient reduction. This advances understanding of Diophantine problems in wreath products and offers a framework for analyzing orientable equations in broader group classes, with implications for algorithmic group theory and related lattice-based techniques.

Abstract

In this paper we study the complexity of solving orientable quadratic equations in wreath products of finitely generated abelian groups. We give a classification of cases (depending on genus and other characteristics of a given equation) when the problem is computationally hard or feasible.

Paper Structure

This paper contains 35 sections, 43 theorems, 90 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Classification of quadratic equations
  3. Previous results
  4. Our results
  5. Model of computation and representation of basic objects
  6. Outline
  7. Preliminaries
  8. Wreath products of abelian groups
  9. $A^B$ as a group ring
  10. Some properties of $A^B$
  11. Several useful formulae
  12. Cayley graph
  13. The rank of an abelian group
  14. Lattices
  15. The $3$-partition problem
  16. ...and 20 more sections

Key Result

Lemma 2.1

$\tau(f)=a_1\cdot b_1+\dots+a_k\cdot b_k \ \ \ \Leftrightarrow\ \ \ \tau(f^\delta)=a_1\cdot \delta^{-1}b_1+\dots+a_k\cdot \delta^{-1}b_k$ for every $f\in A^B$ and $\delta\in B$.

Figures (4)

  • Figure 1: The function component of the element $c_y$ in ${\mathbb{Z}}_2\wr{\mathbb{Z}}$.
  • Figure 2: The function component for the element $c$ in ${\mathbb{Z}}_2\wr{\mathbb{Z}}$ consists of $k$ clusters of $L$ lit lamps, each cluster separated by a single unlit lamp.
  • Figure 3: Schematic picture for the lamp configuration in $c_y$ defined by \ref{['eq:mid-h']}, where $h=3$.
  • Figure 4: Schematic picture for the lamp configuration in $c$ defined by \ref{['eq:mid-h']}, where $h=3$.

Theorems & Definitions (87)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • ...and 77 more