Nils Bruin | 12 Nov 18:58 2012
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Re: analytic_rank() discussed on mathoverflow

On Nov 12, 9:46 am, Dima Pasechnik <dimp... <at> gmail.com> wrote:
> please have a look (and perhaps reply - I am not an expert on this
> stuff):

It has to check that a certain analytic function vanishes to 8th order
at a particular point. That involves proving that all its derivatives
up to 7th order vanish at the point. It's the *proving* that's hard.
However, proving that they're not 0 is straightforward: Just
approximate to sufficient precision.

Thus, if sage gets it wrong and pari and magma get it right then it
looks like sage gets a precision bound wrong somewhere. One would
normally expect that a routine like this will err by *overestimating*
the order of vanishing, since underestimating involves saying "I
cannot really distinguish this approximated value from zero, but I bet
it's nonzero".

That, or sage is correct and we're looking at a counterexample of the
Birch--Swinnerton-Dyer conjecture.

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Gmane