Timeline for Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 6, 2015 at 0:15 | answer | added | Joe Silverman | timeline score: 6 | |
| Feb 6, 2015 at 0:01 | answer | added | Douglas Lind | timeline score: 5 | |
| Feb 5, 2015 at 21:58 | vote | accept | Liam Baker | ||
| Feb 5, 2015 at 21:10 | comment | added | Geoff Robinson | You need to use that fact that all eigenvalues of $DD^{T}$ are totally positive algebraic integers, and if one or more of them is not 1, its algebraic conjugates have mean value at least $1.5$. | |
| Feb 5, 2015 at 20:45 | comment | added | Geoff Robinson | It seems to be Theorem III of "The trace of totally positive and real algebraic integers" by C.L. Siegel, Annals of Maths, 46, 2, (1945), 302-312 | |
| Feb 5, 2015 at 20:38 | comment | added | Liam Baker | @Goeff can you give a reference for that theorem? | |
| Feb 5, 2015 at 20:11 | answer | added | Qiaochu Yuan | timeline score: 7 | |
| Feb 5, 2015 at 19:53 | comment | added | Geoff Robinson | I am not sure of the general case, but I think that if $D \in {\rm GL}(n,\mathbb{Z})$ is not a "signed permutation matrix", ie $DD^{T} \neq I$, a Theorem of Siegel implies that $\rho(DD^{T}) \geq 1.5$. | |
| Feb 5, 2015 at 19:40 | history | asked | Liam Baker | CC BY-SA 3.0 |