Timeline for Is the square of the covering radius of an integral lattice/quadratic form always rational?
Current License: CC BY-SA 3.0
9 events
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| Jul 4, 2011 at 4:17 | comment | added | Noam D. Elkies | Thanks; this trick does work, but it produces $$ $$ very wide spacing between the lines! | |
| Jul 4, 2011 at 2:39 | comment | added | Will Jagy | One can force a line break by putting in a blank displayed formula, two dollar signs, then a blank space, then two more dollar signs. $$ $$ and then whatever. About denominators, dimension is no larger than 10, and as Pete knows I have been searching far an a priori proof of class number one. | |
| Jul 4, 2011 at 2:37 | comment | added | David E Speyer | Sadly, there is no way to force line breaks. And thanks for including the geometric discussion in your answer. | |
| Jul 4, 2011 at 2:35 | history | edited | David E Speyer | CC BY-SA 3.0 |
added 127 characters in body
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| Jul 4, 2011 at 2:34 | comment | added | David E Speyer | I just found my error, $2^{n+1}$ is right. Forgot that the matrix entries $\langle v_i, v_j \rangle$ could be half integers, so the adjoint matrix can get denominators from that as well of from the determinant. Editing now. | |
| Jul 4, 2011 at 2:30 | comment | added | Noam D. Elkies | You beat me to this by a few minutes... And only because I included the connections with the formulas for the circumradius of a triangle and a tetrahedron :-) \\ $2^{n+1}$ vs. 4: could be we're both right but your argument gives a better bound. For $n=2$ I can reclaim a factor of 2 because $M_0$, being square of odd order, has even determinant. \\ Trivial typo: "immediately obviously" → "immediately obvious". Most likely there's at least one such typo in my own answer. \\ [Is there a way to force line breaks in comments here?] | |
| Jul 4, 2011 at 2:29 | comment | added | Will Jagy | Thanks, David. Usually for conjectures I have a vast amount of numerical evidence, plus I have learned the value of being able to test "random" cases rather than hand-picked. In this case, I have under 100 quadratic forms which have a quite definite non-random property (Pete's). It did really jump out at me about the denominators, I wrote my own code for the task in C++ and watched as certain checks were performed. If true, the part about the denominators is very useful. Pete asked me how i did it, in turn I want to know how Magma does it... | |
| Jul 4, 2011 at 2:26 | comment | added | David E Speyer | Noam Elkies has just posted an extremely similar answer which has $2^{n+1}$ where I had $4$. I'm not sure which of us has the correct power of $2$. | |
| Jul 4, 2011 at 2:19 | history | answered | David E Speyer | CC BY-SA 3.0 |