arXiv subjects

Aakash Choudhary

Publications and source records attributed to Aakash Choudhary.

3 recordsLinked to original sources

arXiv2026

Power Integral Bases in Polynomial Compositions

In this paper, we study the monogeneity of a special class of composed polynomials of the form $ (f \circ g)(x) = (x^m + c)^n + a(x^m + c)^{n-1} + d(x^m + c)^{n-2} + b,$ where $ f(x) = x^n + a x^{n-1} + d x^{n-2} + b \in \mathbb{Z}[x] $ satisfies $ a^2 = 4d $ and $ g(x) = x^m + c \in \mathbb{Z}[x] $. Assuming that $ (f \circ g)(x) $ is irreducible over $ \mathbb{Q} $, we obtain necessary and sufficient conditions on the parameters $ a, b, c, d, m, n $ for the polynomial to be monogenic. These conditions help to identify when the set $ \{1, \theta, \dots, \theta^{mn-1}\} $ forms an integral basis of the number field $ \mathbb{Q}(\theta) $, where $ \theta $ is a root of $ (f \circ g)(x) $. We also provide lower bound for the counting of such monogenic polynomials. Furthermore, we study the behaviour of solutions to certain related differential equations and present a class of polynomials with non-square-free discriminants as an application of the main results.

math.NT↗

arXiv2023

On pairs of $r$-primitive and $k$-normal elements with prescribed traces over finite fields

Given $\mathbb{F}_{q^{n}}$, a field with $q^n$ elements, where $q $ is a prime power and $n$ is positive integer. For $r_1,r_2,m_1,m_2 \in \mathbb{N}$, $k_1,k_2 \in \mathbb{N}\cup \{0\}$, a rational function $F = \frac{F_1}{F_2}$ in $\mathbb{F}_{q}[x]$ with deg($F_i$) $\leq m_i$; $i=1,2,$ satisfying some conditions, and $a,b \in \mathbb{F}_{q}$, we construct a sufficient condition on $(q,n)$ which guarantees the existence of an $r_1$-primitive, $k_1$-normal element $ε\in \mathbb{F}_{q^n}$ such that $F(ε)$ is $r_2$-primitive, $k_2$-normal with $\operatorname{Tr}_{\mathbb{F}_{q^n}/\mathbb{F}_q}(ε) = a$ and $\operatorname{Tr}_{\mathbb{F}_{q^n}/\mathbb{F}_q}(ε^{-1}) = b$. For $m_1=10, \; m_2=11,\; r_1 = 3, \; r_2 = 2, \; k_1=2,\;k_2 = 1$, we establish bounds on $q$, for various $n$, to determine the existence of such elements in $\mathbb{F}_{q^{n}}$. Furthermore, we identify all such pairs $(q,n)$ excluding 10 possible values of $(q,n)$, in fields of characteristics 13.

math.NT↗

arXiv2023

Existence of primitive pairs with two prescribed traces over finite fields

Given $F= \mathbb{F}_{p^{t}}$, a field with $p^t$ elements, where $p $ is a prime power, $t\geq 7$, $n$ are positive integers and $f=f_1/f_2$ is a rational function, where $f_1, f_2$ are relatively prime, irreducible polynomials with $deg(f_1) + deg(f_2) = n $ in $F[x]$. We construct a sufficient condition on $(p,t)$ which guarantees primitive pairing $(ε, f(ε))$ exists in $F$ such that $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(ε) = a$ and $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(ε)) = b$ for any prescribed $a,b \in \mathbb{F}_{p}$. Further, we demonstrate for any positive integer $n$, such a pair definitely exists for large $t$. The scenario when $n = 2$ is handled separately and we verified that such a pair exists for all $(p,t)$ except from possible 71 values of $p$. A result for the case $n=3$ is given as well.

math.NT↗