On Bounds of Extension Degrees for Similarity of Integral Matrices over Number Fields

Abstract: It is well-known that if $n\times n$ integral matrices $A$ and $B$ of a number field $K$ are similar over all completions of the ring of integers of $K$, then $A$ and $B$ are similar over the ring of integers of a finite extension of $K$. We prove that there is no uniform bound of the degree of extension of $K$ valid for all $n\times n$ matrices. On the other hand, we provide a upper bound of the degree of extension of $K$ for a given separable characteristic polynomial.

• The paper demonstrates that no uniform bound exists for field extension degrees in establishing similarity, as the minimal degree can grow arbitrarily large.
• It employs module-theoretic and class group techniques to explicitly construct sequences of matrix pairs with increasing extension degree complexity.
• For matrices with a fixed separable characteristic polynomial, the paper provides effective upper bounds via class field theory, connecting arithmetic invariants to matrix similarity.

Bounds on Extension Degrees for Similarity of Integral Matrices over Number Fields

Introduction and Problem Statement

The paper "On Bounds of Extension Degrees for Similarity of Integral Matrices over Number Fields" (2606.21628) investigates the minimal degree of field extensions required to establish similarity of integral matrices defined over the ring of integers of a given number field $K$. The classical local-global principle, proven by Guralnick, asserts that if two $n \times n$ integral matrices $M$, $N$ over the ring of integers $\mathcal{O}_K$ are similar in each local completion $\mathcal{O}_{K_v}$, then they are similar over the ring of integers of some finite extension $L/K$. However, the constructive and quantitative aspects of this field extension, specifically whether its degree can be uniformly bounded for all such matrices, have remained unresolved.

Non-Existence of Uniform Bounds

The central result of the paper is the negative answer to the uniformity question. The authors construct explicit sequences of integral matrix pairs $(M_n, N_n)$, each similar over all local completions, but with the minimal degree $C_K(M_n, N_n)$ of extensions required for similarity tending to infinity with $n$. The construction employs non-derogatory matrices and leverages module-theoretic and class group techniques, particularly focusing on the structure of locally free rank-one modules (line bundles) over the centralizer ring $n \times n$0 of $n \times n$1.

For two such matrices, the obstruction to global similarity is measured by the class $n \times n$2 in the class group $n \times n$3 of locally free rank-one $n \times n$4-modules. The paper establishes the lower bound

$n \times n$5

and shows this order can be made arbitrarily large, by explicit construction over $n \times n$6. These constructions, based on carefully chosen parameters (e.g., using primitive root modulo large primes for the $n \times n$7 case), demonstrate that, for fixed $n \times n$8, no uniform bound exists that works for all integral matrices.

Effective Upper Bounds for Separable Characteristic Polynomials

Despite the absence of a uniform bound for arbitrary matrices, the paper provides an effective and explicit upper bound for the extension degree under the restriction that the characteristic polynomial is separable and fixed. The key insight is to relate the set of matrices similar to $n \times n$9 over all local completions to a torsor over the multi-norm torus associated with the characteristic polynomial, thereby reducing the question to the existence of integral points on certain group schemes.

Employing recent advances (notably Wei and Xu [WX12], [WX13]), the authors utilize criteria for the existence of integral points and construct the necessary field extensions via class field theory. More precisely, for fixed monic separable $M$0, there exists a constant $M$1 such that

$M$2

where $M$3 is the $M$4-fold Hilbert class field, and $M$5 is an explicit open subgroup defined in terms of the roots of $M$6 and their discriminant. For non-split polynomials, the bound incorporates the degree of the splitting field. This result is effective, providing an explicit recipe to compute the required extension degree for a given characteristic polynomial.

Optimality and Special Cases

The bounds obtained match known cases and extend classical results. When the base ring is a principal ideal domain and matrices are diagonal with pairwise distinct entries, the extension degree is $M$7, confirming the local-global principle in these cases (see Corollary and Example in the paper). For certain constructions over $M$8 with primitive roots modulo primes, both lower and upper bounds coincide, establishing optimality.

Implications and Future Directions

The results have notable theoretical implications, sharpening the understanding of the local-global principle for similarity of integral matrices and elucidating the obstructions in terms of arithmetic invariants. The absence of uniform bounds underscores the complexity arising from the arithmetic of the underlying ring and its class group. The effective bounds for fixed characteristic polynomials create a bridge to computational number theory and arithmetic geometry, facilitating explicit algorithms for verifying similarity and constructing extension fields.

Potential avenues for further research include extending the methods to more general algebraic group actions, exploring finer invariants of module structure beyond locally free rank-one modules, and generalizations to non-separable or inseparable cases. Moreover, the explicit nature of the bounds may enable quantitative studies in the context of integral representations and applications to algebraic structures conducive to field extensions in computational contexts.

Conclusion

The paper rigorously answers the question regarding bounds on extension degrees for similarity of integral matrices over number fields, demonstrating that no uniform bound exists in general but providing explicit, optimal bounds when the characteristic polynomial is fixed and separable. The results blend matrix theory, class field theory, and algebraic geometry, yielding both negative and positive answers depending on the structural constraints imposed, and paving the way for further exploration at the intersection of arithmetic algebra and representation theory.