2020/09/21 by Kathuria, Leetika, Raka, Madhu
#11H31 #11H46 #11J20 #11J37 #52C15 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2009.09992
Let \mathbbL be a lattice in n-dimensional Euclidean space ℝn reduced in the sense of Korkine and Zolotareff and having a basis of the form ~(A1,0,0,⋯ ,0), ~(a2,1,A2,0,⋯,0),⋯, (an,1,an,2,⋯,an,n-1,An). A famous conjecture of Woods in Geometry of Numbers asserts that if A1A2⋯ An = 1 and Ai≤ A1 for each i then any closed sphere in ℝn of radius √(n/4) contains a point of \mathbbL. 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.