If phcpack does not report any 'failure' solutions in
classified_solution_dicts(), then it should have found all the roots.
But the groebner methods are preferable if you are interested in exact
solutions.
-Marshall
On Jul 16, 11:29 am, Doug wrote:
> > The last element of that Groebner basis is
> The last element of that Groebner basis is a univariate polynomial in
> x2, so it is relatively easy to analyze its solutions. There is one
> triple root at x2=0, and then four others which are presumably the
> four you are thinking of. Once you have a value for x2, you can
> substitute it int
The last element of that Groebner basis is a univariate polynomial in
x2, so it is relatively easy to analyze its solutions. There is one
triple root at x2=0, and then four others which are presumably the
four you are thinking of. Once you have a value for x2, you can
substitute it into the othe
I confess I'm not a mathematician (I'm an economist) and it's been
almost 25 years since I took a basic course in abstract algebra. But
this is interesting. From the wikipedia page on Grobner bases, it
seems I should be able to compute solutions to the system based on the
Grobner basis, but I ca
Does this help any?
sage: R = PolynomialRing(QQ, 2, 'x1,x2', order='lp')
sage: x1,x2 = R.gens()
sage: f1 = 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)
sage: f2 = 1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2)
sage: I = (f1,f2)*R; I
Ideal (1/2*x1^2*x2^2 + x1^2*x2 + 1/2*x1^2 +
