arXiv subjects

Srijonee Shabnam Chaudhury

Publications and source records attributed to Srijonee Shabnam Chaudhury.

4 recordsLinked to original sources

arXiv2024

Sums of squares of integer-multiple of an integral element on real bi-quadratic fields

For any given positive integer $m$ we construct certain totally positive algebraic integers $α$ of a real bi-quadratic field $K$ and obtain some necessary conditions for which $mα$ can not be represented as sum of integral squares. We show this for integers lie in quadratic subfields of $K$ and for integers which are in $K$ but not in any quadratic subfield of $K$. We provide examples in tabular form for each cases to corroborate the results.

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arXiv2024

A Matrix Analogue of Schur-Siegel-Smyth Trace Problem

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.

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arXiv2021

Sums of integral squares in complex bi-quadratic fields and in CM fields

Let $K$ be a complex bi-quadratic field with ring of integers $\mathcal{O}_{K}$. For $K = \mathbb{Q}(\sqrt{-m}$, $\sqrt{n}$), where $ m \equiv 3 \pmod 4 $ and $ n \equiv 1 \pmod 4$, we prove that every algebraic integer can be written as sum of integral squares. Using this, we prove that for any complex bi-quadratic field $K$, every element of $4\mathcal{O}_K$ can be written as sum of five integral squares. In addition, we show that the Pythagoras number of ring of integers of any CM field is at most five. Moreover, we give two classes of complex bi-quadratic fields for which $p(\mathcal{O}_{K})= 3$ and $p(4\mathcal{O}_{K})=3 $ respectively. Here, $p(\mathcal{O}_{K})$ is the Pythagoras number of ring of integers of $K$ and $p(4\mathcal{O}_{K})$ is the smallest positive integer $t$ such that every element of $4\mathcal{O}_{K}$ can be written as sum of $t$ integral squares.

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arXiv2020

Sums of Integral Squares In Certain Complex Bi-quadratic Fields

Let K be an algebraic number field and O_K be its ring of integers. Let S_K be the set of elements in O_K which are sums of squares in O_K and s(O_K) the minimal number of squares necessary to represent -1in O_K. Let g( S_K ) be the smallest positive integer t such that every element in S_K is a sum of t squares in O_K. Here K is generated over field of rational number by square root of m and -n , where m congruent 3 mod 4 and n congruent 1 mod 4 are two distinct positive square free integers, we prove that $ S_K= O_K. We also prove that g(O_ K) less or equals to s(O_K)+1 or s(O_K)+2. Applying this, we shows that if s(O_K)=2, then g(O_K)=3. This work is continuation of a recent study initiated by Zhang and Ji .

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