Igor Schein on Thu, 3 Apr 2003 18:40:12 -0500 |
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Re: polredabs() again |
On Fri, Apr 04, 2003 at 01:27:29AM +0200, Karim BELABAS wrote: > On Thu, 3 Apr 2003, Igor Schein wrote: > > > On Fri, Apr 04, 2003 at 12:18:34AM +0200, Karim BELABAS wrote: > > [deleted] > > > On the other hand, the current specification of polredabs is quite useless. > > > There's no application whatsoever for a "polynomial of absolute smallest > > > T2-norm". It's not even guaranteed to have minimal discriminant, or to > > > yield smallest coefficients. The only one I can see is to give a > > > pseudo-canonical representative for the field (this helps table builders, > > > less isomorphism tests...) > > > > An appropriate observation. Here's an example: > > > > ? p1=x^40-444796*x^30+91848351606*x^20-11536862712826396*x^10+672749994932560009201; > > time = 0 ms. > > ? p2=polredabs(p1) > > time = 10,160 ms. > > %5 = x^40 - 45*x^38 + 1325*x^36 - 32390*x^34 + 714250*x^32 - > > 12409705*x^30 + 186413550*x^28 - 2499178250*x^26 + 29415089025*x^24 - > > 274143015750*x^22 + 2260522643650*x^20 - 16155295928625*x^18 + > > 89060060373750*x^16 - 251196632524250*x^14 + 670266070790625*x^12 - > > 1618446757085125*x^10 + 2840715854153125*x^8 - 389148315762500*x^6 + > > 53298286370000*x^4 - 7270708200000*x^2 + 922368160000 > > > > Anybody in his right mind, I believe, would prefer p1 over p2 as a > > representative for this field. > > I might prefer p2: > > (00:25) gp > poldisc(p2)/poldisc(p1)*1. > time = 70 ms. > %1 = 2.227868806881974253905227025 E-46 I retract the original statement about being in right mind :-) However, I still much prefer p1. If we were to search for representatives with the smallest index though, then I'd like to see an equivalent of Kant's OrdexIndexFormEquation() implemented in pari. OrdexIndexFormEquation() works only for degree<5 polynomials, but maybe there's a room for improvement there. Igor