User mebassett - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T05:07:21Z http://mathoverflow.net/feeds/user/8341 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104061/computer-algebra-system-for-polynomial-algebras-over-finite-fields computer algebra system for polynomial algebras over finite fields mebassett 2012-08-05T22:53:05Z 2012-08-20T09:05:27Z <p>Is there a computer algebra system that can do arithmetic over polynomial algebras over finite fields where I can specify the extension?</p> <p>Exempli gratia, if $f(x), g(x) \in \mathbb{F}_p[\mu]/((m(\mu))\large[x\large]$, I'd like the CAS to be able to compute things like $f(x + \mu) + g(x)$ where I specify the polynomial $m(\mu)$.</p> <p>Thanks</p> http://mathoverflow.net/questions/35142/q-lattices-and-commensurability-any-insight-into-the-definition-and-intuition Q-lattices and commensurability, any insight into the definition and intuition? mebassett 2010-08-10T16:42:00Z 2010-08-10T23:24:07Z <p>I've been coming across $\mathbb{Q}$-lattices in $\mathbb{R}^n$ in my reading, and I'm having trouble understanding the definitions. Connes and Marcolli define it as a lattice $\Lambda \in \mathbb{R}^n$ together with a homomorphism $\phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q} \Lambda / \Lambda$. Moreover, two $\mathbb{Q}$-lattices $\Lambda_1$ and $\Lambda_2$ are commensurable iff 1) $\mathbb{Q} \Lambda_1 = \mathbb{Q}\Lambda_2$ and 2) $\phi_1 = \phi_2$ mod $\Lambda_1 + \Lambda_2$. </p> <p>I think I understand condition 1): the lattices must be rational multiples of each other to be commensurable. I don't even understand the notation for condition 2). The best I can gather is that the homomorphism $\phi$ labels which positions in $\mathbb{Q} \Lambda / \Lambda$ come from your more normal "discrete hyper-torus" $\mathbb{Q}^n / \mathbb{Z}^n$. Condition 2) then says that the same points are labelled. Is this anywhere near the right ballpark? Can anyone recommend any literature on the subject? </p> <p>I'm a pretty young mathematician (not even in a PhD program...yet) so please forgive me if this question seems basic. </p> <p>Thanks.</p> http://mathoverflow.net/questions/35142/q-lattices-and-commensurability-any-insight-into-the-definition-and-intuition Comment by mebassett mebassett 2010-08-10T19:34:58Z 2010-08-10T19:34:58Z Connes &amp; Marcolli, Noncommutative Geometry, Quantum Fields, and Motives. Marcolli, Lectures on Arithmetic Noncommutative Geometry. Various other papers from those two. alainconnes.org/docs/Qlattices.pdf is a short one. Thanks for the comments on the notation, very helpful.