> > I have not been able to get the following behaviour out of maxima
> > directly, but the following code illustrates that maxima's solve
> > forgets about some solutions:
>
> > def realzeros(f,g):
> > t = polygen(QQbar)
> > P = parent(f)
> > R = f.resultant(g,P.0)
> > yvals = [yval[0] for yval in R(0,t).roots() if yval[0] in RR]
> > result = []
> > for yval in yvals:
> > xpol = GCD(f(t,yval),g(t,yval))
> > result += [(xval[0],yval) for xval in xpol.roots() if xval[0]
> > in RR]
> > return result
> > var("x,y")
> > f=(x^2-y^2+y^4)*(x^2+y^2-1)
> > g=diff(f,x)+2*diff(f,y)
> > temp=solve([f,g],x,y)
> > S1 = [ (RR(p[0]),RR(p[1])) for p in [(x.subs(s[0]),y.subs(s[1])) for s
> > in temp] if p[0] in RR and p[1] in RR]
> > S2 = realzeros(f.polynomial(QQ),g.polynomial(QQ))
> > implicit_plot(g,(x,-2,2),(y,-2,2))+implicit_plot(f,(x,-2,2),(y,-2,2))
> > +points(S1,color="red",pointsize=40)+points(S2,color="blue",pointsize=20)
>
> > This code produces a picture that clearly shows that maxima's solve
> > misses some of the intersection points that a straightforward
> > resultant-and-roots-of-univariate-polynomials does find.
>
> > Perhaps someone more expert in using maxima can get this problem to
> > happen directly in maxima so that it can be reported upstream?
>
> I'm too lazy to see what roots you actually want :) but here is the
> appropriate Maxima session:
>
> (%i15) f:(x^2-y^2+y^4)*(x^2+y^2-1);
> 2 2 4 2 2
> (%o15) (y + x - 1) (y - y + x )
> (%i16) g:diff(f,x)+2*diff(f,y);
> 4 2 2 2 2 3
> 4 2 2
> (%o16) 2 (2 y (y - y + x ) + (y + x - 1) (4 y - 2 y)) + 2 x (y -
> y + x )
>
> 2 2
> + 2 x (y
> + x - 1)
> (%i17) solve([f,g],[x,y]);
> 2 1 2 1
> (%o17) [[x = - -------, y = -------], [x = -------, y = - -------],
> sqrt(5) sqrt(5) sqrt(5) sqrt(5)
> [x = 0, y = - 1], [x = 0, y = 1], [x =
> 0, y = 0]]
>
> So it gets five intersection points, including three on the x-axis and
> two other conjugate ones on the unit circle. Which ones is it
> missing? (to_poly_solve gets the same roots, by the way, in the Sage
> Maxima.)
to_poly_solve([f,g],[x,y],use_grobner=true) finds nine roots.
HTH, Andrej
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