|McLaughlin, James on Sat, 20 Dec 2003 22:32:32 +0100|
[Date Prev] [Date Next] [Thread Prev] [Thread Next] [Date Index] [Thread Index]
|Fundamental Units Again|
It seems that I do not know as much about the PARI/GP system as I had thought (as little as that was). Concerning the bnfinit(f,1) command, where f is an irreducible polynomial in Z[x], it says in the manual that "When flag = 1, GP insists on finding the fundamental units exactly, the internal precision being doubled and the computation redone, until the exact results are obtained". I had understood this to me that the output was guaranteed to be a system of fundamental units ( I think it was the words "exactly" and "exact" that threw me off course :-) ). I have been informed that this is not the case and that the output is actually conditional on an assumption that is stronger than GRH. bnfcertify will guarantee the output unconditionally for number fields of low degree over Q fairly easily (up to, say, 11), but using bnfcertify becomes practically impossible as the degree of f goes much beyond this. However, I have also been told that the output of bnfinit(f,1).fu is guaranteed unconditionally to be at least a system of independent units. If this were true, I could work around the difficulty by using known general lower bounds on the regulator of a number field of degree n over Q. I cannot find a statement anywhere in the manual that bnfinit(f,1).fu is at least guaranteed to output a system of independent units (possibly did not look hard enough, but I do not think I missed it). Is this in fact true? Please excuse my ignorance in this matter. Jimmy Mc Laughlin.