Tagged Questions

A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.

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Symmetric, commuting matrices in $\mathrm{SL}(3,\mathbb{Z})$

Can someone evaluate whether the following is true: Given two symmetric, commuting matrices $M_1,M_2 \in \mathrm{SL}(3,\mathbb{Z})$ (integer entries with determinant 1), where $M_2$ is not ...
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Eigenvalue counting function for the Dirichlet and Neumann Laplacians on a rectangle

I'm doing some research on the Eigenvalue counting function for the Dirichlet and Neumann Laplacians on a rectangle and I was wondering if anyone knew why we consider the integer lattice points within ...
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Does the determinant give you the index over $\mathcal{O}_k$ as well as over $\mathbb{Z}$?

It is a standard fact that if $M$ is a nonsingular $n\times n$ integer matrix, the index of the $\mathbb{Z}$-span of its columns as an abelian group in $\mathbb{Z}^n$ is $|\det M|$. What happens if we ...
I have some trouble understanding the concept of lattice reduction. As I understand, an integer lattice $$\{ A k : k \in \mathbb{Z}^n \} \subset \mathbb{Z}^n$$ is defined by a regular matrix \$A \in \...