A Matrix Analogue of Schur-Siegel-Smyth Trace Problem
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866929218587197440 |
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| author | Chaudhury, Srijonee Shabnam |
| author_facet | Chaudhury, Srijonee Shabnam |
| contents | Let $\mathcal{S}$ be the set of all positive-definite, symmetrizable integer matrices with non-zero upper and lower diagonal and $\mathcal{T}$ to be the set of all positive-definite real symmetric matrices with nonzero upper diagonal such that all non-zero entries are square-roots of some positive integers and the matrices satisfy a certain cycle condition.
In this paper, for any $n \times n$ matrix $A \in \mathcal{S} \cup \mathcal{T}$ and any $k \in \mathbb{N}$ we find a general lower bound for $Tr_{2^k}(A)$, i.e, the sum of $2^k$-th power of eigenvalues of $A$, which depends on $n$ as well as some other variables. In particular, we obtain the best possible lower bound for $Tr_2(A) $ that is $6n - 5$. As a strong outcome of this result we show that the smallest limit point of $\overline{Tr_2(A)} = \frac{Tr_2(A)}{n}$ is $6$. This is a solution of an analogue of ``Schur - Siegel - Smyth trace problem" for characteristic polynomials of matrices in $\mathcal{S} \cup \mathcal{T}$. We also obtain a lower bound of smallest limit point of $\overline{Tr_{2^k}(A)}$ for any positive integer $k > 1$ and for the same set of matrices. Furthermore, we exhibit that the famous results of Smyth on density of absolute trace measure and absolute trace-2 measure of totally positive integers are also true for the set of symmetric integer connected positive definite matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_12871 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Matrix Analogue of Schur-Siegel-Smyth Trace Problem Chaudhury, Srijonee Shabnam Number Theory Combinatorics 15A18 15B36 15B57 11C08 Let $\mathcal{S}$ be the set of all positive-definite, symmetrizable integer matrices with non-zero upper and lower diagonal and $\mathcal{T}$ to be the set of all positive-definite real symmetric matrices with nonzero upper diagonal such that all non-zero entries are square-roots of some positive integers and the matrices satisfy a certain cycle condition. In this paper, for any $n \times n$ matrix $A \in \mathcal{S} \cup \mathcal{T}$ and any $k \in \mathbb{N}$ we find a general lower bound for $Tr_{2^k}(A)$, i.e, the sum of $2^k$-th power of eigenvalues of $A$, which depends on $n$ as well as some other variables. In particular, we obtain the best possible lower bound for $Tr_2(A) $ that is $6n - 5$. As a strong outcome of this result we show that the smallest limit point of $\overline{Tr_2(A)} = \frac{Tr_2(A)}{n}$ is $6$. This is a solution of an analogue of ``Schur - Siegel - Smyth trace problem" for characteristic polynomials of matrices in $\mathcal{S} \cup \mathcal{T}$. We also obtain a lower bound of smallest limit point of $\overline{Tr_{2^k}(A)}$ for any positive integer $k > 1$ and for the same set of matrices. Furthermore, we exhibit that the famous results of Smyth on density of absolute trace measure and absolute trace-2 measure of totally positive integers are also true for the set of symmetric integer connected positive definite matrices. |
| title | A Matrix Analogue of Schur-Siegel-Smyth Trace Problem |
| topic | Number Theory Combinatorics 15A18 15B36 15B57 11C08 |
| url | https://arxiv.org/abs/2206.12871 |