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This article establishes a connection between the arithmetic properties of linear group representations and their structural properties, particularly focusing on representations that factor through monomial representations. This connection is then applied to characterize the ambiguity of weighted finite automata (WFA) over a field.

Abstract

**Bibliographic Information:**Puch, A., & Smertnig, D. (2024). Factoring through monomial representations: arithmetic characterizations and ambiguity of weighted automata.*arXiv preprint arXiv:2410.03444v1*.**Research Objective:**This paper aims to characterize linear group representations that factor through monomial representations using arithmetic properties. The authors further apply these findings to analyze the ambiguity of weighted finite automata.**Methodology:**The authors utilize results on unit equations from Diophantine number theory, specifically leveraging theorems by Evertse, van der Poorten, Schlickewei, Derksen, and Masser. They employ these tools to analyze the structure of linear groups and their representations.**Key Findings:**- The research establishes an equivalence between locally B'ezivin linear groups and groups admitting faithful monomial representations, under certain conditions.
- It demonstrates that linear groups with locally finitely generated spectrum are virtually solvable.
- The study reveals a correspondence between the ambiguity of invertible weighted finite automata and the arithmetic properties of their minimal linear representations.

**Main Conclusions:**- The paper provides a novel characterization of linear group representations that factor through monomial representations based on arithmetic properties.
- It offers insights into the structure of linear groups with specific arithmetic constraints.
- The results have significant implications for understanding the complexity of weighted finite automata, particularly regarding their ambiguity.

**Significance:**This research contributes significantly to representation theory and its applications to theoretical computer science, particularly in analyzing the complexity of weighted finite automata.**Limitations and Future Research:**The study primarily focuses on invertible weighted finite automata. Further research could explore extending these results to non-invertible WFAs and investigate the potential for more efficient algorithms for deciding ambiguity in specific cases.

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by Daniel Smert... at **arxiv.org** 10-07-2024

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Extending the results on the ambiguity of weighted finite automata (WFA) to semirings is a challenging but potentially fruitful endeavor. Here's a breakdown of the challenges and possibilities:
Challenges:
Loss of Invertibility: A key aspect of the presented results is the focus on invertible WFA, where the transition matrices are invertible over the field K. This invertibility is heavily utilized in leveraging the theory of linear groups. Semirings, in general, lack invertibility, making a direct translation of the techniques difficult.
Weaker Arithmetic Structure: Semirings have a weaker arithmetic structure compared to fields. Concepts like eigenvalues, eigenvectors, and the unit equation, which are central to the proofs for fields, might not have direct analogs or might behave differently in the context of semirings.
Role of Zero Divisors: The presence of zero divisors in semirings introduces additional complexities. The behavior of matrix representations and the notion of ambiguity itself might need careful re-evaluation in the presence of zero divisors.
Possibilities:
Restricting to Specific Semirings: One possible approach is to focus on specific classes of semirings that possess some additional structure, such as division semirings (where non-zero elements have multiplicative inverses) or tropical semirings. These structures might offer a more amenable setting for extending some of the results.
Adapting the Techniques: Instead of direct translation, adapting the techniques used for fields to the semiring setting might be necessary. This could involve developing semiring analogs of key concepts or exploring alternative proof strategies.
New Characterizations: The challenges posed by semirings might necessitate developing entirely new characterizations of ambiguity classes. These characterizations might rely on different algebraic or combinatorial properties of semirings and their matrix representations.
In summary, extending the ambiguity results to semirings is a non-trivial problem that requires careful consideration of the structural differences between fields and semirings. However, by focusing on specific semiring classes, adapting existing techniques, or developing new characterizations, it might be possible to obtain meaningful extensions.

The computational complexity of decision problems related to WFA ambiguity can indeed be influenced by the representation of the field K. Here's a breakdown of how different representations might affect complexity:
1. Number Fields:
Explicit Algebraic Number Fields: If K is a number field represented explicitly, meaning we have a concrete polynomial whose root generates K, then the decision problems mentioned (like M-ambiguity, finite ambiguity, polynomial ambiguity) are decidable. This is because we can leverage algorithms for computing the Zariski closure of matrix groups over number fields.
Implicit Representations: If K is represented implicitly, for example, as the field of real numbers with computable coefficients, the complexity might increase. Decidability might still hold, but the algorithms could become significantly more involved.
2. Finite Fields:
Standard Representation: When K is a finite field represented in the standard way (e.g., as integers modulo a prime), the decision problems are typically simpler than for number fields. The finite nature of the field allows for more straightforward algorithms.
3. Function Fields:
Complexity Depends on Representation: The complexity for function fields (fields of rational functions) depends heavily on how the field is represented. Explicit representations might lead to decidable problems, while implicit representations could make the problems significantly harder.
Factors Influencing Complexity:
Field Arithmetic: The complexity of performing arithmetic operations in K directly affects the overall complexity. For example, computing inverses or testing for roots of unity can be more expensive in some representations.
Zariski Closure Computation: The complexity of computing the Zariski closure of matrix groups, a crucial step in the decision algorithms, is highly sensitive to the field representation.
Representation Size: The size of the representation of K (e.g., the degree of the defining polynomial for number fields) can also play a role.
In essence, while the presented results provide a theoretical framework for deciding ambiguity properties, the practical complexity hinges significantly on the chosen representation of the field K. Choosing a representation that allows for efficient arithmetic and Zariski closure computation is crucial for practical algorithms.

Yes, the findings on the structure of linear groups, particularly those related to monomial representations and finitely generated spectra, have the potential to be applied to other areas like coding theory and cryptography. Here are some potential avenues:
Coding Theory:
Code Construction: Monomial matrices are closely related to permutation codes, which are used in error-correcting codes. The characterizations of groups with monomial representations could potentially lead to new constructions of codes with desirable properties.
Decoding Algorithms: Understanding the structure of groups related to the generator matrices of codes could lead to more efficient decoding algorithms. For example, the block-triangular representations might be exploited to simplify decoding procedures.
Code Equivalence: Determining whether two codes are equivalent (related by a permutation of coordinates) is a fundamental problem in coding theory. The results on group representations might offer new tools for tackling code equivalence problems.
Cryptography:
Cryptosystems Based on Linear Groups: Several cryptographic schemes rely on the hardness of problems related to linear groups, such as the Conjugacy Search Problem or the Discrete Logarithm Problem. The structural insights gained from the presented findings could potentially be used to analyze the security of such cryptosystems or even design new ones.
Pseudorandom Generators: Linear groups with specific properties, like those with large girth (shortest cycle in their Cayley graphs), are used in constructing pseudorandom generators. The characterizations of groups with finitely generated spectra might be relevant in this context.
Lattice-Based Cryptography: Lattices, which can be viewed as discrete subgroups of vector spaces, play a crucial role in lattice-based cryptography. The techniques used to analyze linear groups might offer insights into the structure of lattices used in cryptographic applications.
Beyond Coding and Cryptography:
Computational Group Theory: The results contribute to the field of computational group theory, providing new algorithms for deciding properties of linear groups. These algorithms could have applications in other areas where linear groups arise.
Representation Theory: The findings deepen our understanding of the representation theory of groups, particularly regarding monomial and block-triangular representations. This knowledge could have implications in other areas of mathematics where representation theory is used.
In conclusion, the structural insights into linear groups presented in the context of WFA ambiguity have the potential to be valuable tools in other fields. Their applications in coding theory, cryptography, and beyond are worth exploring further.

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