arXiv subjects

Leetika Kathuria

Publications and source records attributed to Leetika Kathuria.

5 recordsLinked to original sources

arXiv2024

Positive values of non-homogeneous quadratic forms of type (1,4): A conjecture of Bambah, Dumir and Hans-Gill

Let $Q(x_1, \cdots,x_n)$ be a real indefinite quadratic form of the type $(r,s)$, $n=r+s$, signature $σ=r-s$ and determinant $D\neq 0$. Let $Γ_{r,n-r}$ denote the infimum of all numbers $Γ$ such that for any real numbers $c_1, c_2 ,\cdots, c_n$ there exist integers $x_1, x_ 2,\cdots, x_n$ satisfying $$0< Q(x_1+c_1,x_2+c_2,\cdots,x_n+c_n)\leq (Γ|D|)^{1/n}.$$ All the values of $Γ_{r,n-r}$ are known except for $Γ_{1,4}$. Earlier it was shown that $8\leq Γ_{1,4}<12$. It is conjectured that $Γ_{1,4}=8$. Here we shall prove that $Γ_{1,4}=8$, when (i) $c_2 \not \equiv 0 \pmod 1$, (ii) $c_2 \equiv 0 \pmod 1$, $a\geq \frac{1}{2}$, where $a$ is minima of positive definite ternary quadratic forms with determinant $4|D|$, and (iii) in some cases of $c_2 \equiv 0 \pmod 1$, $a< \frac{1}{2}$. We also obtain six critical forms for which the constant 8 is attained. In the remaining cases we prove that $Γ_{1,4}< \frac{32}{3}$.

math.NT↗

arXiv2020

On conjectures of Minkowski and Woods for $n=10$

Let $\mathbb{L}$ be a lattice in $n$-dimensional Euclidean space $\mathbb{R}^n$ reduced in the sense of Korkine and Zolotareff and having a basis of the form $~(A_1,0,0,\cdots$ $,0),$ ~$(a_{2,1},A_2,0,\cdots,0),\cdots,$ $(a_{n,1},a_{n,2},\cdots,a_{n,n-1},A_n)$. A famous conjecture of Woods in Geometry of Numbers asserts that if $A_1A_2\cdots A_n = 1$ and $A_i\leq A_1$ for each $i$ then any closed sphere in $\mathbb{R}^n$ of radius $\sqrt{n/4}$ contains a point of $\mathbb{L}.$ Together with a result of C. T. McMullen (2005), the truth of Woods' Conjecture for a fixed $n$, implies the long standing classical conjecture of Minkowski on product of $n$ non-homogeneous linear forms for that value of $n$. In an earlier paper `Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, 2016, 501-548' we proved Woods' Conjecture for $n=9$. In this paper, we prove Woods' Conjecture and hence Minkowski's Conjecture for $n=10$.

math.NT↗

arXiv2016

$(1-2u^3)$-constacyclic codes and quadratic residue codes over $\mathbb{F}_{p}[u]/\langle u^4-u\rangle$

Let $\mathcal{R}=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}+u^3\mathbb{F}_{p}$ with $u^4=u$ be a finite non-chain ring, where $p$ is a prime congruent to $1$ modulo $3$. In this paper we study $(1-2u^3)$-constacyclic codes over the ring $\mathcal{R}$, their equivalence to cyclic codes and find their Gray images. To illustrate this, examples of $(1-2u^3)$-constacyclic codes of lengths $2^m$ for $p=7$ and of lengths $3^m$ for $p=19$ are given. We also discuss quadratic residue codes over the ring $\mathcal{R}$ and their extensions. A Gray map from $\mathcal{R}$ to $\mathbb{F}_{p}^4$ is defined which preserves self duality and gives self-dual and formally self-dual codes over $\mathbb{F}_{p}$ from extended quadratic residue codes.

math.NT↗

arXiv2015

Refined Estimates on Conjectures of Woods and Minkowski

Let $\wedge$ be a lattice in $\mathbb{R}^n$ reduced in the sense of Korkine and Zolotareff having a basis of the form $(A_1,0,0,\ldots,0),(a_{2,1},A_2,0,\ldots,0)$, $\ldots,(a_{n,1},a_{n,2},\ldots,a_{n,n-1},A_n)$ where $A_1, A_2,\ldots,A_n$ are all positive. A well known conjecture of Woods in Geometry of Numbers asserts that if $A_{1}A_{2}\cdots A_{n}=1$ and $A_{i}\leqslant A_{1}$ for each $i$ then any closed sphere in $\mathbb{R}^n $ of radius $ \sqrt{n}/2$ contains a point of $\wedge$. Woods' Conjecture is known to be true for $n\leq 9$. In this paper we give estimates on the Conjecture of Woods for $10\leq n\leq33$, improving the earlier best known results of Hans-Gill et al. These lead to an improvement, for these values of $n$, to the estimates on the long standing classical conjecture of Minkowski on the product of $n$ non-homogeneous linear forms.

math.NT↗

arXiv2014

On Conjectures of Minkowski and Woods for n=9

Let $\mathbb{R}^n$ be the n-dimensional Euclidean space with $O$ as the origin. Let $\wedge$ be a lattice of determinant $1$ such that there is a sphere $|X|<R$ which contains no point of $\wedge$ other than $O$ and has $n$ linearly independent points of $\wedge$ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in $\mathbb{R}^n $ of radius $ \sqrt{n/4}$ contains a point of $\wedge$. This is known to be true for $n\leq 8$. Here we prove a more general conjecture of Woods for $n=9$ from which this conjecture follows in $\mathbb{R}^9$. Together with a result of C. T. McMullen (2005), the long standing conjecture of Minkowski follows for $n=9$.

math.NT↗