Undecidability of Moment Membership for Commutative and Non-commutative Polynomial Rings

Core Concepts

The moment membership problem is undecidable for commutative and non-commutative polynomial rings over the integers, where the positivity cone is defined by polynomials with non-negative coefficients.

Abstract

The paper studies the moment membership problem, which asks whether all moments of a given matrix belong to a fixed set of positive elements. The authors show that this problem is decidable in several cases, including for orthogonal, unitary, and matrices with a unique dominant eigenvalue or only real eigenvalues.
However, the main result is that the moment membership problem is undecidable for two important classes of polynomial rings:

Commutative polynomials Z[x1, ..., xd]: The authors prove that for sufficiently large matrix size s and number of variables d, the problem is undecidable when the positivity cone is defined by polynomials with non-negative coefficients.

Non-commutative polynomials Z⟨z1, ..., zd⟩: The authors show that the moment membership problem is undecidable in this case as well, even when restricting to linear matrix polynomials.

As a byproduct, the authors prove a free version of Pólya's Theorem, showing that a non-commutative polynomial has non-negative coefficients if and only if it is entrywise non-negative on the set of entrywise non-negative matrices.

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Key Insights Distilled From

Positive Moments Forever: Undecidable and Decidable Cases

by Gemma De les... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2404.15053.pdf

Deeper Inquiries

What are some potential applications of the undecidability results for the moment membership problem in commutative and non-commutative polynomial rings

The undecidability results for the moment membership problem in commutative and non-commutative polynomial rings have significant implications in various fields. In commutative polynomial rings, where the cone of positive polynomials is considered, the undecidability indicates limitations in determining the positivity of matrix moments. This has implications in areas such as optimization, control theory, and signal processing, where positivity constraints play a crucial role. For example, in optimization problems involving polynomial optimization or semidefinite programming, the undecidability results highlight the complexity of ensuring positivity in the presence of polynomial constraints.
In non-commutative polynomial rings, the undecidability results have implications in areas like quantum information theory and quantum computing. The inability to determine the positivity of matrix moments in this context can impact the analysis of quantum systems and the design of quantum algorithms. Understanding the undecidability of the moment membership problem in non-commutative settings is essential for addressing challenges in quantum information processing and quantum error correction.

Can the undecidability results be extended to other cones of positive polynomials, such as the cone of sum-of-squares polynomials

The undecidability results for the moment membership problem in commutative and non-commutative polynomial rings can potentially be extended to other cones of positive polynomials, such as the cone of sum-of-squares (SOS) polynomials. The undecidability in these contexts indicates fundamental limitations in determining positivity for more general classes of polynomials. Extending the undecidability results to the SOS cone would highlight the complexity of verifying positivity for polynomials that can be expressed as sums of squares of other polynomials.
The extension to the SOS cone would have implications in polynomial optimization, control systems, and machine learning, where SOS polynomials are commonly used to represent convex constraints. Understanding the undecidability of the moment membership problem in the SOS cone would provide insights into the computational complexity of verifying positivity in polynomial optimization and related areas.

Are there any decidable cases of the moment membership problem that were not covered in this paper, or any other open problems in this area

While the paper covers various decidable cases of the moment membership problem, there may be additional decidable instances that were not explicitly addressed. One potential area for further exploration is the decidability of the moment membership problem for specific classes of structured matrices, such as Toeplitz matrices or Hankel matrices. Investigating the decidability of the moment membership problem for structured matrices could provide insights into the complexity of positivity verification for structured sequences of matrix moments.
Another open problem in this area is the decidability of the moment membership problem for matrices with specific spectral properties, such as matrices with prescribed eigenvalue distributions or matrices with structured eigenvectors. Understanding the decidability of the moment membership problem for matrices with spectral constraints could have implications in spectral graph theory, quantum mechanics, and signal processing, where spectral properties play a crucial role in system analysis and design.