Determinant of integer lattice basis of $L=\{(x_1,\ldots,x_n): a_1x_1+\cdots+a_nx_n=0\}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T07:01:44Z http://mathoverflow.net/feeds/question/103152 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103152/determinant-of-integer-lattice-basis-of-l-x-1-ldots-x-n-a-1x-1-cdotsa-n Determinant of integer lattice basis of $L=\{(x_1,\ldots,x_n): a_1x_1+\cdots+a_nx_n=0\}$ Victor Wang 2012-07-26T03:44:45Z 2012-07-26T03:44:45Z <blockquote> <p><strong>Question:</strong> Suppose <code>$\{v_1,\ldots,v_{n-1}\}$</code> is an integer basis for the lattice <code>$$L=\{(x_1,\ldots,x_n)\in\mathbb{Z}^n: a_1x_1+\cdots+a_nx_n=0\},$$</code> where the $a_i$ are fixed nonzero integers. Is the volume <code>$V(P)=\det(L)$</code> (see <a href="http://numbertheoryreadinggroup.wordpress.com/2008/04/24/geometry-of-numbers-lecture-2-determinant-of-the-lattice-and-the-fundamental-parallelepiped-lee/" rel="nofollow">this</a> for a proof that they are equal) of its fundamental parallelotope <code>$P=\{t_1v_1+\cdots+t_{n-1}v_{n-1} \mid t_i\in[0,1)\}$</code> necessarily equal to <code>$$\frac{\sqrt{a_1^2+\cdots+a_n^2}}{\gcd(a_1,\ldots,a_n)}?$$</code></p> </blockquote> <p>I used the case $n=3$ along with <a href="http://en.wikipedia.org/wiki/Minkowski%27s_theorem" rel="nofollow">Minkowski's theorem</a> (in the <a href="http://en.wikipedia.org/wiki/Geometry_of_numbers" rel="nofollow">geometry of numbers</a>) to solve the following <a href="http://www.math.u-szeged.hu/~mmaroti/schweitzer/schweitzer-2000-eng.pdf" rel="nofollow">Miklos problem</a> from 2000:</p> <blockquote> <p>Let <code>$a&lt;b&lt;c$</code> be positive integers. Prove that there exist integers <code>$x,y,z$</code>, not all zero, such that <code>$ax+by+cz=0$</code> and <code>$\max(|x|,|y|,|z|)\le 1+\frac{2}{\sqrt3}\sqrt{c}$</code>, and show that the constant <code>$\frac{2}{\sqrt3}$</code> cannot be improved.</p> </blockquote> <p>However, I was only able to find a brute force proof for this special case (see lemma 1 in my AoPS post <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2471417#p2471417" rel="nofollow">here</a>), and I'm not sure if it's as easy for larger values of $n$.</p> <p>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!</p>