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Jan
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awarded | Yearling |
Sep
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awarded | Nice Question |
Sep
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answered | Computing minimal polynomials using LLL |
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accepted | Computing minimal polynomials using LLL |
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revised |
Computing minimal polynomials using LLL
Better tags. |
May
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revised |
Question about the Weeks Manifold
Added reference to paper |
May
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Question about the Weeks Manifold
I should add that Regina is able to do this calculation for any triangulation of a closed or cusped manifold. The algorithm is at least exponential in the number of tetrahedra but is still practical even when this number becomes quite large. For example in 2009 Burton, Rubinstein and Tillmann used it to show that the Seifert-Weber dodecahedral space is not Haken using a triangulation with 23 tetrahedra. See arxiv.org/abs/0909.4625 |
May
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answered | Question about the Weeks Manifold |
May
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Computing minimal polynomials using LLL
@JoeSilverman Yes but shouldn't the bounds on the degree and height of $\alpha$ mean that $a$ determines a unique algebraic number, and so a unique polynomial? Perhaps a more basic question would be: is it true that $|a_i| \leq N$ (or some similar bound)? |
May
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Computing minimal polynomials using LLL
@JoeSilverman: I should also add that the PARI documentation for algdep says that "the polynomial which is obtained is not necessarily the "correct" one". However its code requests a 0.99---LLL Reduced Basis using ZM_lll. With $\delta=0.99$ being so close to 1, we get "better guarantees for the basis (in principle smaller basis vectors) but longer running times". |
May
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Computing minimal polynomials using LLL
@JoeSilverman: Thanks for the suggestion. I've only had a brief chance to look at PARI but the relevant code appears to be "lindep2" in ./basemath/bibli1.c (the precision to work to (equiv. the choice of $N$) is chosen here not in algdep0). It appears that if the binary expansion of $a$ has $i$ bits before the decimal and $j$ bits afterwards then PARI uses $N = 10^{32+i}$ if $j = 0$ and $N = 10^{0.8 j}$ otherwise (or a user given precision). |
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Computing minimal polynomials using LLL
@Igor: I guess I have two questions: 1) What exactly does Cohen mean by "subtle"? 2) Have there been any further developments in how to choose $N$ in the 20 years since Cohen's book? |
May
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asked | Computing minimal polynomials using LLL |
Mar
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awarded | Enlightened |
Mar
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awarded | Nice Answer |
Mar
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answered | 2-bridge knots in the Rolfsen's table |
Mar
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awarded | Nice Answer |
Jan
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awarded | Yearling |
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awarded | Necromancer |
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comment |
Fantastic properties of Z/2Z
So I guess that the correct reference should be Gromov's "Asymptotic invariants of infinite groups" but it is ~300 pages and I haven't read it. Danny gave a talk at Cornell and in the first ~15 mins he covers the density model and gives a sketch of this result. It's available here: cornell.edu/video/danny-calegari-random-groups-diamonds-glass |