Qbar means different things to different people (or to the same people at different times).
Carl's view is to have Qbar fixed as a subfield of C so that an element of Qbar is represented as a (necessarily approximate) complex value together with its minimal polynomial. Number theorists and algebraists would like a more general viewpoint, to allow p-adics to enter the picture for example. If you have just one algebraic number a with minimal polynomial f(x) then the algebraist can be happy working with the abstract field K=Q[x]/(f(x)); this can be embedded into any larger field L (such as C) in which f has a root alpha just by mapping x to alpha; and if there are several roots then there will be that many different embeddings (into the same L). It gets more complicated if you now introduce a second algebraic number b with minimum polynomial g(x), defineing a second abstract field K'=Q[x]/(g(x)). How do add a to b, since they live in entirely different worlds? The algebraists approach is that this only makes sense if you have field L into which both K and K' embed and you do arithmetic in L. Taken to the limit (literally) one arrives at Qbar as the direct limit of all fields like K, which in abstract terms is obtained by taking the disjoint union of all the K=Q[x]/(f(x)) together with embeddings between these, and identifying two elements if they have the same image in a field containing both. Our discussion about how to make sense of symbolic expressions like sqrt(2)+sqrt(3)-sqrt(6) is relevant here. To make sense of this expression you need not only the individual fields Q(sqrt(a)) for a=2, 3, 6; you need a single field L containing sqrts of all three numbers, together with those embeddings, i.e. a labelling of *which* root of x^2-2 in this big field is the one which you want the abstract sqrt(2) to. One solution mentioned previously takes L=R (reals) and "labels" the positive roots. But, as we have seen that simple solution breaks down quickly for more complicated algebraic numbers. Excuse the ramblings: but I always find that computational ambiguities like this are best understood by treating the mathematics seriously! John On 9/25/07, cwitty <[EMAIL PROTECTED]> wrote: > > On Sep 25, 11:24 am, Bill Hart <[EMAIL PROTECTED]> wrote: > > Actually, to make it work, it might have to switch between polar > > coordinates and rectangular coordinates, always ensuring the point you > > are talking about is inside the region, regardless of whether it is a > > polar rectangle or a right rectangle. > > > > Clearly I don't know anything about complex interval arithmetic if > > such a thing exists. Is there a reference I can read. Shame you aren't > > going to be at SAGE days 5 Carl. Are you going to number 6. > > > > Bill. > > I know very little about complex interval arithmetic myself...just > what I learned from Googling for "complex interval arithmetic" and > spending a couple of hours skimming papers I found on the Web. > > Some implementations use "intervals" represented as circles in the > complex plane; others use right rectangles. I don't remember any that > use polar rectangles, but they might exist. In any case, if you want > the tightest possible bounds, there's some fairly tricky math. > > Fortunately, for implementing Qbar, I'm pretty sure we wouldn't need > an implementation of complex interval arithmetic that was carefully > optimized to get the tightest possible bounds. I believe (although I > haven't gone through the math to check) that the "extra looseness" of > the simple algorithms is mostly a problem if the original bounds were > loose relative to the number. However, if Qbar were implemented > similarly to my algebraic reals package, we start with very tight > bounds, and the result of a precision error is to make the bounds much > much tighter. So I think the simplest possible implementation (a > complex interval is a pair of real intervals, and arithmetic > operations are implemented textbook fashion using these real > intervals) would probably suffice. > > No, I'm not planning to go to SD6. (After all, computer algebra is > only a hobby for me!) > > Carl > > > > > -- John Cremona --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
