Searcharxiv⌕ Search

arXiv subjects

Saranya G. Nair

Publications and source records attributed to Saranya G. Nair.

3 recordsLinked to original sources
arXiv2026

Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters

For a positive integer $n$, we study the irreducibility of the generalised Laguerre polynomials \[ L_n^{(a)}(x) = \frac{1}{n!}\sum_{j=0}^{n} \binom{n}{j} \frac{(n+a)!}{(j+a)!}(-1)^j x^j, \] where $a \in \mathbb{Z}_{\ge 0}$. Earlier works established that these polynomials are irreducible over $\mathbb{Q}$ for $0 \le a \le 50$. In this paper, we substantially strengthen these results by establishing that $L_n^{(a)}(x)$ are irreducible for all $a \le 1000$. We derive sharper constraints on potential integer roots and make systematic use of the geometry of the associated Newton polygons. This approach allows us to extend the irreducibility range significantly while keeping explicit computations to a minimum.

math.NT↗
arXiv2019

On the Galois group of Generalised Laguerre polynomials II

For real number $α,$ Generalised Laguerre Polynomials (GLP) is a family of polynomials defined by \begin{align*} L_n^{(α)}(x)=(-1)^n\displaystyle\sum_{j=0}^{n}\binom{n+α}{n-j}\frac{(-x)^j}{j!}. \end{align*}These orthogonal polynomials are extensively studied in Numerical Analysis and Mathematical Physics. In 1926, Schur initiated the study of algebraic properties of these polynomials. We consider the Galois group of Generalised Laguerre Polynomials $ L_n^{(\frac{1}{2}+u)}(x^2)$ when $u$ is a negative integer.

math.NT↗
arXiv2019

Irreducibility of extensions of Laguerre Polynomials

For integers $a_0,a_1,\ldots,a_n$ with $|a_0a_n|=1$ and either $α=u$ with $1\leq u \leq 50$ or $α=u+ \frac{1}{2}$ with $1 \leq u \leq 45$, we prove that $ψ_n^{(α)}(x;a_0,a_1,\cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore all the exceptions other than $n=2^{12},α=89/2$ are necessary. The above result with $0\leqα\leq 10$ is due to Filaseta, Finch and Leidy and with $α\in \{-1/2,1/2\}$ due to Schur.

math.NT↗