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Sushma Palimar

Publications and source records attributed to Sushma Palimar.

8 recordsLinked to original sources
arXiv2023

Iterates of prime producing polynomials and their Galois groups

Let $\F_q$ be a finite field of characteristic $p>0$. We prove that, given $F(t,x)\in \F_q[t][x]$ an irreducible separable monic polynomial in the variable $x$ and a generic monic polynomial $ϕ(t)$ in the variable $t$, the polynomial $F(t,ϕ)$ is a prime producing polynomial over large finite fields under suitable irreducible specialization. We also prove that $F(t,ϕ)$ satisfies Odoni's conjecture, namely the arboreal Galois representation associated to $F(t,ϕ)$ is surjective.

math.NT↗
arXiv2023

Iterates of polynomials over $\F_q(t)$ and their Galois groups

A conjecture of Odoni stated over Hilbertian fields $K$ of characteristic zero asserts that for every positive integer $d$, there exists a polynomial $f\in K[x]$ of degree $d$ such that for every positive integer $n$, each iterate $f^{\circ n}$ of $f$ is irreducible and the Galois group of the splitting field of $f^{\circ n}$ is isomorphic to $[S_d]^{n}$, the $n$ folded iterated wreath product of the symmetric group $S_{d}$. We prove an analogue this conjecture over $\F_q(t)$, the field of rational functions in $t$ over a finite field $\F_q$ of characteristic $p>0$. We present some examples and see that most polynomials in $\F_q[t][x]$ satisfy these conditions.

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arXiv2019

Prime polynomial values of quadratic functions in short intervals

In this paper we establish the function field analogue of Bateman-Horn conjecture in short interval in the limit of a large finite field. Hence we start with counting prime polynomials generated by primitive quadratic functions in short intervals. To this end we further work out on function field analogs of cancellation of Mobius sums and its correlations(Chowla type sums) and confirm that square root cancellation in Mobius sums is equivalent to square root cancellation in Chowla type sums.

math.NT↗
arXiv2014

Gaussian Mersenne Primes of the form $x^2+dy^2$

In this paper we study Gaussian ring $\Z[i]$ with a focus on representing Gaussian Mersenne primes $G_p$ in the form $x^2+7y^2$. Interestingly when such a form exists, one can observe that, $x\equiv \pm 1\pmod{8}$ and $y\equiv 0\pmod{8}$. To prove this property of Gaussian Mersenne primes, we show that Gaussian Mersenne primes splits completely in the cyclic quartic unramified extension of $\Q(\sqrt{-14})$ and have a trivial Artin symbol in this extension. We generalize this result for $d\equiv 7\pmod{24}$. We also attempt to give an alternate proof using Artin's reciprocity law, which was earlier given by H. W. Lenstra and P. Stevenhagen to prove a similar property on ordinary Mersenne Primes.

math.NT↗
arXiv2014

Hensel's Lemma, Backward Dynamics and p-adic Approximations

The problem of backward dynamics over the ring of p-adic integers is studied. It is shown that Inverse Limit Theory provides the right framework. Backward iterations of a polynomial with p-adic integer coefficients are constructed by solving congruences modulo powers of p, which inturn are solved by Hensel's lifting lemma.

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arXiv2012

Mersenne Primes in Real Quadratic Fields

The concept of Mersenne primes is studied in real quadratic fields of class number 1. Computational results are given. The field $Q(\sqrt{2})$ is studied in detail with a focus on representing Mersenne primes in the form $x^{2}+7y^{2}$. It is also proved that $x$ is divisible by 8 and $y\equiv \pm3\pmod{8}$ generalizing the result of F Lemmermeyer, first proved in \cite{LS} using Artin's Reciprocity law.

math.NT↗