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

2012-07-26T03:44:45Z

Question: Suppose $\{v_1,\ldots,v_{n-1}\}$ is an integer basis for the lattice $$L=\{(x_1,\ldots,x_n)\in\mathbb{Z}^n: a_1x_1+\cdots+a_nx_n=0\},$$ where the $a_i$ are fixed nonzero integers. Is the volume $V(P)=\det(L)$ (see this for a proof that they are equal) of its fundamental parallelotope $P=\{t_1v_1+\cdots+t_{n-1}v_{n-1} \mid t_i\in[0,1)\}$ necessarily equal to $$\frac{\sqrt{a_1^2+\cdots+a_n^2}}{\gcd(a_1,\ldots,a_n)}?$$

I used the case $n=3$ along with Minkowski's theorem (in the geometry of numbers) to solve the following Miklos problem from 2000:

Let $a<b<c$ be positive integers. Prove that there exist integers $x,y,z$ , not all zero, such that $ax+by+cz=0$ and $\max(|x|,|y|,|z|)\le 1+\frac{2}{\sqrt3}\sqrt{c}$ , and show that the constant $\frac{2}{\sqrt3}$ cannot be improved.

However, I was only able to find a brute force proof for this special case (see lemma 1 in my AoPS post here), and I'm not sure if it's as easy for larger values of $n$.

But I'm pretty sure this should be true in general (I've tried several cases for $n=4$ and $n=5$), so I would appreciate it if someone could give a (clean?) proof, reference, or counterexample. Thanks!

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

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