On Apr 28, 12:11 am, Ursula Whitcher <urs...@math.hmc.edu> wrote:
> > I'm playing with a family of plane curves with rational coefficients in > the complex projective plane. So rational or complex numbers would be > enough for me to test examples. In a perfect world I'd be able to > specify a family using rational functions of arbitrary constants > (something like a x^2 + b/(a-1) y^2), and compute the projective dual in > terms of those constants. > If you are patient, you could just compute the dual curve using elimination via Groebner bases: Let f(x,y,z) be the homogeneous polynomial defining your curve. The subscheme Z of P^2 x (P^2)^* parameterizing pairs of a point on your curve and a tangent line at that point is cut out by the ideal (f(x,y,z), u - f_x(x,y,z), v - f_y(x,y,z), w - f_z(x,y,z)), where we work with bihomogeneous coordinates [(x,y,z);(u,v,w)] on P^2 x (P^2)^* so that the incidence relation is u*x + v*y + w*z = 0. The dual curve should parameterize tangent lines to your curve. You might take it to be the image of the projection of Z in (P^2)^*, which you can construct by eliminating x, y, and z from the ideal defining Z. In other words (taking f(x,y,z) = x^2 + y^2 - z^2 for example): sage: S.<x,y,z,u,v,w> = PolynomialRing(QQ,6) sage: f = x^2 + y^2 - z^2 sage: I = S.ideal([f, u - f.derivative(x), v - f.derivative(y), w - f.derivative(z)]) sage: I.elimination_ideal([x,y,z]) Ideal (u^2 + v^2 - w^2) of Multivariate Polynomial Ring in x, y, z, u, v, w over Rational Field I tried to get this to work with coefficients Q(a,b) with hopes of accommodating the example f(x,y,z) = x^2 + b/(a-1)*y^2 - z^2, but I had problems getting it to work. I think that Singular can handle such coefficient fields directly, but I didn't know what to tell Sage other than sage: R.<a,b> = PolynomialRing(QQ,2) sage: K = FractionField(R) sage: S.<x,y,z,u,v,w> = PolynomialRing(K,6) It then did not want to do the elimination: sage: f = x^2 + b/(a-1)*y^2 - z^2 sage: I = S.ideal([f, u - f.derivative(x), v - f.derivative(y), w - f.derivative(z)]) sage: I Ideal (x^2 + (b/(a - 1))*y^2 - z^2, (-2)*x + u, ((-2*b)/(a - 1))*y + v, 2*z + w) of Multivariate Polynomial Ring in x, y, z, u, v, w over Fraction Field of Multivariate Polynomial Ring in a, b over Rational Field sage: I.elimination_ideal([x,y,z]) Traceback (most recent call last): ... TypeError: Cannot call Singular function 'eliminate' with ring parameter of type '<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_domain'>' Calculating with suitable discriminants would presumably be speedier. I suppose your choice of method, though, would depend on exactly what you mean by the dual curve. Regards, James Parson -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
