On Jan 19, 4:46 pm, Alex Raichev <[email protected]> wrote:
> Hi everyone:
>
> I'm applying for a grant from the New Zealand government to fund some
> Sage development in the area of computational algebraic and analytic
> geometry. For part of the application i need to report on the 'state
> of the field'. Part of my response to this will be to mention that
> Sage doesn't have an implementation of factorization, ideal, and
> variety calculations over QQbar. To double-check, is this true?
It's essentially true (although as William points out, it would be
straightforward to add some potentially inefficient code for these
tasks).
I have implemented one special case: if you have a zero-dimensional
ideal over the rationals, then you can compute its variety over QQbar:
sage: K.<x, y> = PolynomialRing(QQ, 2, order='lex')
sage: I = Ideal([ x*y - 1, (x-2)^2 + (y-1)^2 - 1])
sage: I = Ideal(I.groebner_basis())
sage: I.variety(ring=QQbar)
[{y: 0.3194484597356763? - 1.633170240915238?*I, x:
0.11535382288068429? + 0.5897428050222055?*I}, {y: 0.3194484597356763?
+ 1.633170240915238?*I, x: 0.11535382288068429? - 0.5897428050222055?
*I}, {y: 0.3611030805286474?, x: 2.769292354238632?}, {y: 1, x: 1}]
> Related to that, do any major computer algebra systems (Singular
> included) have such implementations?
Mathematica supports algebraic numbers; I haven't checked whether they
support factorization, etc. with algebraic coefficients.
Carl
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