Kevin Buzzard on Wed, 15 Jul 1998 04:37:22 +0100

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Query about factorpadic

Hi Pari users.

The gp command factorpadic (at least in version 2.0.10) claims
to factor polynomials in Q_p[X], but in the manual page it
states that the precision that we use "must be strictly larger than
the p-adic valuation of the discriminant of (the polynomial)
for the result to make any sense".

I'd be interested in an elaboration of this comment, if anyone
out there knows the nitty-gritty of what factorpadic is doing.
The reason is that I'd like to work out some roots of some
polynomials in Q_p (within the next day or two!). And these
polynomials have huge valuation-of-discriminant and yet
factorpadic is doing just fine. Yet the manual seems to imply
that I should be skeptical about the results.

Here's a concrete example.

? f=x^5 + 449691864*x^4 - 2209450184054433792*x^3 - 736010060393513697870348288*x^2 + 810634763334812972416233648439689216*x + 263222216157060824115203098902237248565018624;

(this is the char poly of the Hecke operator T_2 acting on the space of
cusp forms of level 1 and weight 60, and will be a
polynomial with a discriminant highly divisible by small

? valuation(poldisc(f),2)
%2 = 148

(so I should trust factorpadic(f,2,149) )


? factorpadic(f,2,149)

gives me five 2-adic roots in Q_2, within a couple of seconds.

But, say, 

? factorpadic(f,2,10)

(which according to the documentation should be avoided) gives
me five 2-adic roots in Q_2, defined to a smaller number of
"significant figures", but all of which agree with the
roots that the "honest" computation gave me. Hence on this occasion,
factorpadic(f,2,10) works well.

These aren't special numbers that I've chosen to make things work
out---I've tried this with several polynomials (monic, in Z_2[X])
and several choices of precision and things always seem to work well
(one can even check with Hensel's lemma that factorpadic is getting
good answers in these 'bad' cases). Yet I'm reluctant to take the output of
factorpadic as gospel because of the caveat in the manual.
So does anyone know "how much" factorpadic can be trusted when
the precision is much smaller than the valuation of the poly?

Kevin Buzzard