[flint-devel] Updates 25/7/2008

[<D.C...@wa...>]

Hi,

Another update as to how I've got on this week.

* Here are some papers I've come across. The first two are both recent
papers which look interesting but are heavy going;

http://www.ams.org/mcom/2008-77-262/S0025-5718-07-02027-3/S0025-5718-07-02027-3.pdf
   -- An algorithm for Computing p-adic Polylogarithms

http://www.ams.org/mcom/0000-000-00/S0025-5718-08-02091-7/S0025-5718-08-02091-7.pdf
    --  A p-adic algorithm to compute the Hilbert Class Polynomial


Heres another paper on polynomial factorisation over Q_p;

http://citeseer.ist.psu.edu/291135.html -- Factoring Polynomials over
P-adic Fields


And heres the paper which relates to what we were discussing the other day;

http://www.emis.de/journals/JTNB/2006-3/article07.pdf   -- Constructing
class fields over local fields


* I've been looking at how PARI calculates the p-adic logarithm. To
calculate log(x) when x is real, it uses the formula log(x) ~=
Pi/(2agm(1,4/s) - m*log2    where s = x*2^m. It says that p-adic values
for x are accepted, but I don't know and haven't been able to tell yet by
playing around if this means that the above formula is still used or if a
Taylor series expansion is used or something else, which I hope to find
out.



* I havent been able to find many algorithms as of yet in algebraic number
theory which make use of p-adics apart from the stuff we discussed the
other day on abelian Galois groups etc., but I could just be missing (lots
of) things so I'll be looking into that again over the weekend and next
week. It seems so far though that apart from basic arithmetic, FLINT will
probably need to be able to deal with actual p-adic fields themselves and
not just their elements when things are built into FLINT which use the
p-adics.


* I'll probably email you early on next week with any progress that I make
on the function interface, but really I need a good list of algorithms
that use p-adics before I can do this. It looks like when we create a
p-adic we will want to store at least an integer, but sometimes maybe
other things like the valuation.


* I think I'm going to look a little closer into ramified extensions
during my
remaining weeks. Class field theory looks interesting from what I've been
reading so I'd like to look into that a little also.


* When I start coming to the end of my project I have to write a short
report for the URSS . I was also going to write a separate report
detailing everything Ive done during my project for yourself, basically
give an overview of what I've found, put together all the papers I've
found useful and interesting into an organised list, put general research
I've done in there, combine parts of the update emails I've been sending
and include the (now updated) PARI functionality list into it as well. Is
there anything else that I should put into it?


Thanks,

Daniel Ellam