Re: Bug in bnfcertify?

[James Wanless <james@grok.ltd.uk>]

I don't know if this is relevant - possibly not and it's just a  
regular bug. But it's interesting you mention Minkowski...
When I was reading Weber's "Lehrbuch der Algebra" a year or two ago, I  
discovered a distinctly non-trivial flaw in specifically some of  
Minkowski's reasoning quoted by Weber.
J

On 22 Jul 2013, at 18:38, Stephan Ehlen wrote:

> Hello,
>
> I tried to compute class numbers of Hilbert class fields of several
> imaginary quadratic fields, originally using sage but sage basically
> interfaces pari/gp for this purpose. In one particular example, the
> computation took very long, so I tried to find out what was going on.
> If I perform bnfinit() in pari/gp with the polynomial that defines the
> Hilbert class field in this case, the process terminates after less
> than a minute or so on my laptop. It is the bnfcertify that takes so
> long. Well, in order to see if we end up in an infinite loop or so, I
> turned on debugging and was a bit surprised to see that pari continued
> with the function testprimes beyond the Minkowski bound in this case.
> Looking at the source code only briefly, it seems that the second for
> loop in 'testprimes' in the file buch2.c does never terminate.
>
> Here are the details of the computation:
>
> I take the following number field, which is supposed to have class  
> number one.
>
> L=bnfinit(x^26 - 2*x^25 + 2484*x^24 - 4586*x^23 + 2847822*x^22 -
> 4818170*x^21 + 1995228021*x^20 - 3066971426*x^19 + 953096577112*x^18 -
> 1317359310304*x^17 + 327800081539043*x^16 - 402251522690840*x^15 +
> 83510912485885347*x^14 - 89531620356216582*x^13 +
> 15956451256652162441*x^12 - 14635777566553330924*x^11 +
> 2286606726322811557998*x^10 - 1743936046731088471916*x^9 +
> 242727050765913635497471*x^8 - 147714971678918552212564*x^7 +
> 18551902020333005985565004*x^6 - 8442206085607103221290354*x^5 +
> 966819395778520305239696441*x^4 - 292299434248523177559301894*x^3 +
> 30792768551084507948533502345*x^2 - 4636558762445458519624201052*x +
> 452684584165052348313734803063)
>
> Then I run bnfcertify(L) with default settings...
> The Minkowski bound is about 1.01592210234705e6. Pari continues to
> check for primes beyond that bound in 'testprimes'
> (called from bnfcertify0 in buch3.c) in the second for loop.
>
> Now, if I start gp instead with
>
> gp -p 1060000
>
> then bnfcertify terminates and returns 1 (in this case, it simply does
> not run into the second loop)...
> So this seems to be a bug to me. However, I don't know exactly why the
> second for - loop is built the way it is, so maybe someone who knows
> more about that can help me here.
>
> Best,
> Stephan