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Decidability of multiplicative matrix equations and related Diophantine problems

Sebastian Heintze, Armand Noubissie, Robert F. Tichy

TL;DR

The paper advances the decidability of multiplicative matrix equations (knapsack/mortality problems) over algebraic number fields by leveraging effective Diophantine bounds in finitely generated domains. It develops sharp height-based exponents bounds for commuting and certain noncommuting cases (notably Heisenberg-type matrices), and establishes an effective bound for the ABC-identity problem in symmetric $2\times2$ matrices, enabling decidability with explicit solutions. A general decidability framework is then presented, reducing matrix polynomial equations to a finite system of exponential Diophantine equations via eigenvalue analysis and S-unit techniques. The work further provides counting results for multiplicatively dependent tuples of symmetric matrices, including tight upper and lower bounds and constructions from diagonal matrices, with discussions of extensions to number fields. Together, these results yield algorithmic decidability in broad regimes and quantify the complexity of related Diophantine and counting problems, connecting matrix theory, Diophantine approximation, and arithmetic geometry.

Abstract

Some new decidability results for multiplicative matrix equations over algebraic number fields are established. In particular, special instances of the so-called knapsack problem are considered. The proofs are based on effective methods for Diophantine problems in finitely generated domains as presented in the recent book of Evertse and Györy. The focus lies on explicit bounds for the size of the solutions in terms of heights as well as on bounds for the number of solutions. This approach also works for systems of symmetric matrices which do not form a semigroup. In the final section some related counting problems are investigated.

Decidability of multiplicative matrix equations and related Diophantine problems

TL;DR

The paper advances the decidability of multiplicative matrix equations (knapsack/mortality problems) over algebraic number fields by leveraging effective Diophantine bounds in finitely generated domains. It develops sharp height-based exponents bounds for commuting and certain noncommuting cases (notably Heisenberg-type matrices), and establishes an effective bound for the ABC-identity problem in symmetric matrices, enabling decidability with explicit solutions. A general decidability framework is then presented, reducing matrix polynomial equations to a finite system of exponential Diophantine equations via eigenvalue analysis and S-unit techniques. The work further provides counting results for multiplicatively dependent tuples of symmetric matrices, including tight upper and lower bounds and constructions from diagonal matrices, with discussions of extensions to number fields. Together, these results yield algorithmic decidability in broad regimes and quantify the complexity of related Diophantine and counting problems, connecting matrix theory, Diophantine approximation, and arithmetic geometry.

Abstract

Some new decidability results for multiplicative matrix equations over algebraic number fields are established. In particular, special instances of the so-called knapsack problem are considered. The proofs are based on effective methods for Diophantine problems in finitely generated domains as presented in the recent book of Evertse and Györy. The focus lies on explicit bounds for the size of the solutions in terms of heights as well as on bounds for the number of solutions. This approach also works for systems of symmetric matrices which do not form a semigroup. In the final section some related counting problems are investigated.

Paper Structure

This paper contains 7 sections, 21 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Commuting matrices and other special cases
  3. The ABC-identity-problem for symmetric $(2 \times 2)$-matrices
  4. A general decidability problem
  5. Special counting results for multiplicatively dependent tuples
  6. Multiplicatively dependent pairs
  7. Multiplicatively dependent $s$-tuples of symmetric $(2 \times 2)$-matrices

Key Result

Theorem 1

Let $K$ be a number field. Let $A_1$ and $A_2$ be two diagonalizable matrices in $\mathrm{GL}_n(K)$ of bounded height $\mathcal{H}(A_i) \ll H$ for $i \in \left\{ 1,2 \right\}$ which satisfy a multiplicative relation with $k_1 k_2 \neq 0$. Then there are exponents with $\left| k_1 \right|,\left| k_2 \right| \ll \log H$ such that eq:multrel holds.

Theorems & Definitions (43)

  • Theorem 1
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['thm:case2']}
  • Remark 3
  • Lemma 4
  • proof
  • Theorem 5
  • Lemma 6
  • proof : Proof of Theorem \ref{['thm:cases']}
  • ...and 33 more