Timeline for Determinant of integer lattice basis of `$L=\{(x_1,\ldots,x_n): a_1x_1+\cdots+a_nx_n=0\}$`

5 events

when toggle format	what		by	license	comment
Nov 10, 2017 at 0:40	answer	added	Will Jagy		timeline score: 1
Jul 26, 2012 at 4:53	comment	added	Noam D. Elkies		More generally: suppose $L_0$ is a lattice primitively embedded in ${\bf Z}^n$; that is, if $L_0$ contains $mv$ for some nonzero $m\in\bf Z$ and $v \in {\bf Z}^n$ then $L_0$ contains $v$. Let $L \subset {\bf Z}^n$ consist of the integer vectors orthogonal to every vector in $L_0$. Then the fundamental parallelotopes of the lattices $L_0$, $L$ have the same volume. Victor's question is the special case when $L_0$ is the rank-$1$ lattice ${\bf Z}\cdot(a_1,\ldots,a_n)$.
Jul 26, 2012 at 4:48	comment	added	Ilya Bogdanov		Again, wlog gcd is 1. Take a fundamental parallelotope of $L$ and add a vector $v_n$ to obtain a f.p. of ${\mathbb Z}^n$. Then the endpoints of $v_n$ lie in the ``neighboring'' affine hyperspaces parallel to $L$, and the distance between them is exactly $1/\sqrt{a_1^2+\dots+a_n^2}$.
Jul 26, 2012 at 3:53	comment	added	Noam D. Elkies		Without loss of generality the gcd is $1$. Then ${\bf Z}^n$ contains the direct sum $L_1$ of $L$ with ${\bf Z}\cdot(a_1,\ldots,a_n)$ with index $A := a_1^2+\cdots+a_n^2$. Thus a fundamental parallelotope for $L_1$ has volume $A$. But this volume is the product of the corresponding volumes for $L$ and ${\bf Z}\cdot(a_1,\ldots,a_n)$. The latter volume is $A^{1/2}$; hence the former is $A / A^{1/2} = A^{1/2}$, QED.
Jul 26, 2012 at 3:44	history	asked	Victor Wang	CC BY-SA 3.0