论文标题
关于Minkowski和Woods的猜想，$ n = 10 $
On conjectures of Minkowski and Woods for $n=10$
论文作者
论文摘要
令$ \ mathbb {l} $为$ n $ dimensional Euclidean Space $ \ Mathbb {R}^n $在Korkine和 Zolotareff和具有$〜（a_1,0,0，\ cdots $ $，0），$〜$（a_ {2,1}，a_2,0，\ cdots，0），\ cdots，$ $（a_ {a_ {n，1}，a_ {n，1}，a_ {n，2}，a_ {n，2}，$ cdots $ n n n n n n n n n n n n n n n n n n n n n n n n n n，a_1,0,0，\ cdots $ $，0）的基础的基础。著名的森林猜想 在数字的几何形状中断言，如果$ a_1a_2 \ cdots a_n = 1 $和$ a_i \ leq a_1 $对于每个$ i $，则在$ \ mathbb {r}^n $ radius $ \ sqrt {n/4} $中的任何封闭的球体中的任何闭合球都包含了$ \ m m i \ l} $ c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. cand}} a}。 （2005年），伍兹对固定的$ n $的猜想的真相意味着Minkowski的长期经典猜想是$ n $ n $ nothomesous linear formal of $ n $的$ n $非均匀线性形式。在较早的论文`proc。印度学院。科学。 （数学科学）卷。 126，2016，501-548'我们以$ n = 9 $的形式证明了伍兹的猜想。在本文中，我们证明了伍兹的猜想，因此Minkowski的猜想是$ 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 $n=10$.
