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987 results on '"KUMAR, SURENDER"'

16. On the index of power compositional polynomials

Author

Kaur, Sumandeep, Kumar, Surender, and Remete, László

Subjects
Mathematics - Number Theory, Mathematics - Commutative Algebra, 11R04, 11R09
Abstract

The index of a monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ having a root $\theta$ is the index $[\mathbb{Z}_K:\mathbb{Z}[\theta]]$, where $\mathbb{Z}_K$ is the ring of algebraic integers of the number field $K=\mathbb{Q}(\theta)$. If $[\mathbb{Z}_K:\mathbb{Z}[\theta]]=1$, then $f(x)$ is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial $f(x^k)$ belonging to $\mathbb{Z}[x]$, to be monogenic. As an application of our results, for a polynomial $f(x)=x^d+A\cdot h(x)\in\mathbb{Z}[x],$ with $d>1, \operatorname{deg} h(x)

Published
2024

22. Empirical Band-Gap Correction for LDA-Derived Atomic Effective Pseudopotentials

Author

Kumar, Surender, Bui, Hanh, and Bester, Gabriel

Subjects
Condensed Matter - Materials Science
Abstract

Atomic effective pseudopotentials enable atomistic calculations at the level of accuracy of density functional theory for semiconductor nanostructures with up to fifty thousand atoms. Since they are directly derived from ab-initio calculations performed in the local density approximation (LDA), they inherit the typical underestimated band gaps and effective masses. We propose an empirical correction based on the modification of the non-local part of the pseudopotential and demonstrate good performance for bulk binary materials (InP, ZnS, HgTe, GaAs) and quantum dots (InP, CdSe, GaAs) with diameters ranging from 1.0 nm to 4.45 nm. Additionally, we provide a simple analytic expression to obtain accurate quasiparticle and optical band gaps for InP, CdSe, and GaAs QDs, from standard LDA calculation.

Published
2024

32. DISCRIMINANT and integral basis OF $\mathbb{Q}(\sqrt[12]{a})$

Author

Jakhar, Anuj and Kumar, Surender

Subjects
Mathematics - Number Theory
Abstract

Suppose $m$ be a $12$-th power free integer. Let $K=\mathbb{Q}(\theta)$ be an algebraic number field defined by a complex root $\theta$ of an irreducible polynomial $x^{12}-m$ and $O_K$ be its ring of integers. In this paper, we determine the highest power of $p$ dividing the index of the subgroup $\Z[\theta]$ in $O_K$ and $p$-integral basis of $K$ for each prime $p$. These $p$-integral bases lead to the construction of an integral basis of $K$ which is illustrated with examples. In particular, when $m$ is a square free integer, we provide necessary and sufficient conditions for the set $\{1,\theta,\theta^2,\cdots,\theta^{10},\theta^{11}\}$ to be an integral basis of $K.$

Published
2023

33. On power basis of a class of number fields

Author

Jakhar, Anuj, Kaur, Sumandeep, and Kumar, Surender

Subjects
Mathematics - Number Theory
Abstract

Let $f(x)=x^n+ax^2+bx+c \in \Z[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K=\Q(\theta)$ be an algebraic number field defined by a complex root $\theta$ of $f(x)$. Let $\Z_K$ deonote the ring of algebraic integers of $K$. The aim of this paper is to provide the necessary and sufficient conditions involving only $a,c$ and $n$ for a given prime $p$ to divide the index of the subgroup $\Z[\theta]$ in $\Z_K$. As a consequence, we provide families of monogenic algebraic number fields. Further, when $\Z_K \neq \Z[\theta]$, we determine explicitly the index $[\Z_K : \Z[\theta]]$ in some cases.

Published
2023

34. On common index divisor of the number fields defined by $x^7+ax+b$

Author

Jakhar, Anuj, Kaur, Sumandeep, and Kumar, Surender

Subjects
Mathematics - Number Theory
Abstract

Let $f(x)=x^7+ax+b$ be an irreducible polynomial having integer coefficients and $K=\mathbb{Q}(\theta)$ be an algebraic number field generated by a root $\theta$ of $f(x)$. In the present paper, for every rational prime $p$, our objective is to determine the necessary and sufficient conditions involving only $a,~b$ so that $p$ is a divisor of the index of the field $K$. In particular, we provide sufficient conditions on $a$ and $b$, for which $K$ is non-monogenic. In a special case, we show that if either $8$ divides both $a\pm1$, $b$ or $32$ divides both $a+4$, $b$, then $K$ is non-monogenic. We illustrate our results through examples., Comment: arXiv admin note: text overlap with arXiv:2011.14348

Published
2023

36. Carbon Nanoadsorbents

Author

Yadav, Aruna, Kumar, Surender, Araujo, Paulo, Series Editor, Gomes Sousa Filho, Antonio, Editorial Board Member, Doorn, Stephen K., Editorial Board Member, Franklin, Aaron D., Editorial Board Member, Hartschuh, Achim, Editorial Board Member, Tharini, J., editor, and Thomas, Sabu, editor

Published
2024
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