arXiv subjects

Prabhakar Yadav

Publications and source records attributed to Prabhakar Yadav.

6 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, θ, \dots, θ^{mn-1}\} $ forms an integral basis of the number field $ \mathbb{Q}(θ) $, where $ θ$ 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.

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arXiv2026

Monogenic Fields from Polynomial Compositions with Applications

A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[α]$ for some $α\in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ is said to be monogenic if one of its roots generates both the number field and its ring of integers. In this article, we establish the necessary and sufficient conditions for $[\mathbb{Z}_{K_i}:\mathbb{Z}[α_i]]=1$, where $K_i=\mathbb{Q}(α_i)$ and $α_i$ is a root of the composed polynomial $f_i(x^k+b)$ for $i=1,2$. Here, $f_1(x)=x^n+c\sum_{j=1}^{n}(ax)^{n-j}\in\mathbb{Z}[x]$ and $f_2(x)=x^n+c\sum_{j=1}^{n}a^{j-1}x^{n-j}\in\mathbb{Z}[x]$ are irreducible polynomials of degree $n\ge 3$. In addition, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. As an application of our main results, we construct a class of polynomials with non-square-free discriminants. We also analyze the behavior of solutions to certain related differential equations.

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arXiv2026

Arithmetic Aspects of Number Fields Generated by Polynomial Families

Let $f(x)=(x^{k}+c)^{m}-ax^{n}\in\mathbb{Z}[x]$ be an irreducible polynomial over $\mathbb{Q}$, where $k,m,n\in\mathbb{N}$ with $km>n$, and let $K=\mathbb{Q}(θ)$, where $θ$ is a root of $f(x)$. We investigate the arithmetic properties of the number fields that arise from this family. We first obtain an explicit formula for the discriminant of $f(x)$. Using this formula, we establish necessary and sufficient conditions for the monogeneity of $f(x)$, expressed in terms of the prime divisors of $a$ and $c$ and the parameters $k,m,n$. This yields infinite families of monogenic polynomials of arbitrary degree, including families with a non-square-free discriminant. Building on these results, we extend our algebraic characterization to composite polynomials, establishing some explicit conditions for the monogeneity of the composition of $f(x)$ with an arbitrary polynomial $g(x)$. From an analytic point of view, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. We further study non-monogeneity via the field index $i(K)$ and, for each prime $p$, provide sufficient conditions ensuring $ν_p(i(K))=1$, yielding partial progress toward a problem of Narkiewicz. We also highlight a connection with a class of differential equations naturally associated with $f(x)$. As an application, we determine the conditions under which the splitting field of $f(x)$ has a full symmetric Galois group. Several explicit examples illustrate our results.

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arXiv2025

Behaviour of Newton Polygon over polynomial composition

In this paper, we study the structure of Newton polygons for compositions of polynomials over the rationals. We establish sufficient conditions under which the successive vertices of the Newton polygon of the composition $ g(f^n(x)) $ with respect to a prime $ p $ can be explicitly described in terms of the Newton polygon of the polynomial $ g(x) $. Our results provide deeper insights into how the Newton polygon of a polynomial evolves under iteration and composition, with applications to the study of dynamical irreducibility, eventual stability, non-monogenity of tower of number fields, etc.

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arXiv2025

Primitive prime divisors in the forward orbit of a polynomial

For the polynomial $f(z) \in \mathbb{Q}[z]$, we consider the Zsigmondy set $\mathcal{Z}(f,0)$ associated to the numerators of the sequence $\{f^n(0)\}_{n \geq 0}$. In this paper, we provide an upper bound on the largest element of $\mathcal{Z}(f, 0)$. As an application, we show that the largest element of the set $\mathcal{Z}(f,0)$ is bounded above by $6$ when $f(z) = z^d + z^e +c \in \mathbb{Q}[z]$, with $d>e \geq 2$ and $|c|>2$. Furthermore, when $f(z) =z^d+c \in \mathbb{Q}[z]$ with $|f(0)| > 2^{\frac{d}{d-1}}$ and $d >2$, we also deduce a result of Krieger [Int. Math. Res. Not. IMRN, 23 (2013), pp. 5498-5525] as a consequence of our main result.

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arXiv2024

A Study of monogenity of Binomial Composition

Let $θ$ be a root of a monic polynomial $h(x) \in \Z[x]$ of degree $n \geq 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{ 1, θ, θ^2, \ldots, θ^{n-1} \}$ is a basis for the ring $\Z_K$ of integers of $K = \Q(θ)$. In this article, we study about the monogenity of number fields generated by a root of composition of two binomials. We characterise all the primes dividing the index of the subgroup $\Z[θ]$ in $\Z_K$ where $K = \Q(θ)$ with $θ$ having minimal polynomial $F(x) = (x^m-b)^n - a \in \Z[x]$, $m\geq 1$ and $n \geq 2$. As an application, we provide a class of pairs of binomials $f(x)=x^n-a$ and $g(x)=x^m-b$ having the property that both $f(x)$ and $f(g(x))$ are monogenic.

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