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Positive Moments Forever: Undecidable and Decidable Cases

Gemma De les Coves, Joshua Graf, Andreas Klingler, Tim Netzer

TL;DR

The paper investigates the moment positivity problem for matrices, defined by $\\mu_n(A)=\\mathrm{tr}(A^n)$, and its relation to Skolem's problem for linear recurrence sequences. It develops a framework showing that generalized moment sequences correspond to LRS in commutative settings and analyzes decidability across matrix classes and coefficient rings, employing algebraic-group methods for orthogonal/unitary cases and spectral-badius arguments for real-dominant cases. It proves decidability for orthogonal and unitary matrices, and for matrices with a unique dominant eigenvalue or only real eigenvalues, while establishing undecidability for matrices over both commutative and non-commutative polynomial rings; it also derives a free version of Polya's theorem as a byproduct. These results delineate sharp boundaries between decidable positivity problems and undecidable algebraic questions, with implications for simple unitary LRS positivity and tensor-network moment sequences, and they raise further questions about complexity and boundary cases such as rational rings and SOS polynomials.

Abstract

We investigate the generalized moment membership problem for matrices, a formulation equivalent to Skolem's problem for linear recurrence sequences. We show decidability for orthogonal, unitary, and real eigenvalue matrices, and undecidability for matrices over certain commutative and non-commutative polynomial rings. As consequences, we deduce that positivity is decidable for simple unitary linear recurrence sequences and undecidable for linear recurrence sequences over commutative polynomial rings. As a byproduct, we also prove a free version of Polya's theorem.

Positive Moments Forever: Undecidable and Decidable Cases

TL;DR

The paper investigates the moment positivity problem for matrices, defined by , and its relation to Skolem's problem for linear recurrence sequences. It develops a framework showing that generalized moment sequences correspond to LRS in commutative settings and analyzes decidability across matrix classes and coefficient rings, employing algebraic-group methods for orthogonal/unitary cases and spectral-badius arguments for real-dominant cases. It proves decidability for orthogonal and unitary matrices, and for matrices with a unique dominant eigenvalue or only real eigenvalues, while establishing undecidability for matrices over both commutative and non-commutative polynomial rings; it also derives a free version of Polya's theorem as a byproduct. These results delineate sharp boundaries between decidable positivity problems and undecidable algebraic questions, with implications for simple unitary LRS positivity and tensor-network moment sequences, and they raise further questions about complexity and boundary cases such as rational rings and SOS polynomials.

Abstract

We investigate the generalized moment membership problem for matrices, a formulation equivalent to Skolem's problem for linear recurrence sequences. We show decidability for orthogonal, unitary, and real eigenvalue matrices, and undecidability for matrices over certain commutative and non-commutative polynomial rings. As consequences, we deduce that positivity is decidable for simple unitary linear recurrence sequences and undecidable for linear recurrence sequences over commutative polynomial rings. As a byproduct, we also prove a free version of Polya's theorem.
Paper Structure (9 sections, 9 theorems, 37 equations, 1 table)

This paper contains 9 sections, 9 theorems, 37 equations, 1 table.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Problem statement
  4. Relation to the membership problem for linear recurrence sequences
  5. Decidable cases
  6. Known results: small order
  7. Orthogonal and unitary matrices
  8. Matrices with a unique dominant eigenvalue or real eigenvalues
  9. Further generalizations

Key Result

Lemma 4

Let $\mathcal{R}$ be a commutative unital ring, and let $A \in \mathrm{Mat}_s(\mathcal{R})$. Then $\mathopen{}\mathclose{\left(\varphi(A^n)\right)_{n\in\mathbb N}}$ is an LRS of order $s$, for every $\mathcal{R}$-linear map $\varphi\colon \mathrm{Mat}_s(\mathcal{R})\to\mathcal{R}$.

Theorems & Definitions (19)

  • Example 1
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • Lemma 7
  • proof
  • Theorem 8
  • proof
  • ...and 9 more