Carl and Robert, I presume that in the system you are proposing, an algebraic number would be specified as a specific root of a polynomial f with coefficients in Qbar. But f would be *represented* by a polynomial whose coefficients were complex intervals. The root alpha of f of interest would be specified by giving a complex interval containing the root alpha and no other roots of f. Is this correct?
Assuming this is correct, here is my question (it always makes one look much smarter than one really is to ask a question instead of proposing an answer): if I specified two algebraic numbers alpha and beta in this way, with corresponding polynomials f and g, how would I construct the approximate polynomial of alpha + beta? Would I use the method of Dedekind, reducing the problem to writing down a determinant? Here is my second question (one looks doubly smart if one has two difficult questions and still no clue): is it true that one can easily tell if two algebraic numbers are *not* equal in this system, but to check if they are equal, one *always* has to drop back to algebraic techniques? Someone seemed to indicate that one could use root refinement to *prove* equality. Note I am not assuming anything about the construction of the polynomials of these algebraic numbers here, nor anything about how these algebraic numbers were constructed. I don't even think I can assume their polynomials have the same degree, since it is hard to give a minimum polynomial without exact coefficients being known. Even to check that two quantities are *not* equal, I may have to refine the approximation of the coefficients of their polynomials, yes? This turns it into a recursive problem; in order to refine a coefficient, I may have to refine its polynomial, etc, right down to QQ itself (assuming we were working in an extension of an extension of an extension ... of QQ). Well, my final question is, why not just define an algebraic number to be a specific root of a polynomial over QQ. One can easily then just use basic linear algebra to get a minimum polynomial over QQ of a+b, a- b, a*b, a/b and nthroot(a). If one never cares what field one is in, then this works just fine, no Galois groups required. Checking equality of algebraic numbers is also trivial. The only problem I see is if one writes QQ[a, b] where a and b have different degrees over Q (especially if those degrees are not coprime). How does interval arithmetic deal with this? I guess if one always demands polynomials over QQ (for which I cannot see any use for interval arithmetic) one can just work in QQ[x, y]/ (f(x), g(y)) where f(a) = 0 and g(b) = 0, etc. But this is precisely Allan Steel's solution! Bill. On 25 Sep, 20:04, "John Cremona" <[EMAIL PROTECTED]> wrote: > 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/ -~----------~----~----~----~------~----~------~--~---
