ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 01 Apr 2021 05:08:59 +0200Why am I getting a type error when I attempt to take the projective closure of this intersection?https://ask.sagemath.org/question/56448/why-am-i-getting-a-type-error-when-i-attempt-to-take-the-projective-closure-of-this-intersection/I am attempting to take the projective closure of the intersection of the following affine polynomials (not the intersection of the closure!):
$$y^3-y-x^2= 0 $$
$$w^3-w+y^7-y^5-x^4y^3+x^4y = 0.$$
This affine intersection is a curve of dimension one. Unfortunately, I haven't been able to enter this lovely affine curve defined by this intersection into sage. When I attempt as follows:
A.<x,y,w> = AffineSpace(QQ, 3)
P.<u,v,t,s>=ProjectiveSpace(QQ,3)
C = Curve([y^3-y-x^2, w^3-w+y^7-y^5-x^4*y^3+x^4*y], A)
D=C.projective_closure(1,P)
I get an error at the definition of C due to the second polynomial:
TypeError: F (=[-x^4*y^3 + y^7 + x^4*y - y^5 + w^3 - w]) must be a list or tuple of polynomials of the coordinate ring of A (=Affine Space of dimension 3 over Finite Field of size 3)
I am so confused because this is absolutely in the coordinate ring of $A$. Why am I getting this type error? How can I enter this affine intersection into sage, so that I may take its closure?
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As an aside, I can enter the intersection of the closure, which is not what I want, as follows:
x,y,z,w = GF(3)['x,y,z,w'].gens()
C = Curve([y^3-y*z^2-x^2*z, w^3*z^4-w*z^6+y^7-y^5*z^2-x^4*y^3+x^4*y*z^2]); C
The intersection of the closure has an extra irreducible component $[x: 0: w: 0]$,
masseygirlThu, 01 Apr 2021 05:08:59 +0200https://ask.sagemath.org/question/56448/Map between projective curves defined in an extension fieldhttps://ask.sagemath.org/question/38263/map-between-projective-curves-defined-in-an-extension-field/For example, suppose I have the following 2 projective curves:
k = GF(13)
x,y,z = ProjectiveSpace(k, 2, 'x,y,z').gens()
E = Curve(2*x^2 + 8*y*z + 8*z^2)
W = Curve(x^2 + y*z + z^2)
I like to define a map from E to W that involves $\sqrt 2$ and $\sqrt 8$, which do not exist in k = GF(13), but do in an extension of k:
x = PolynomialRing(k,'x').gen()
K = GF(13**2, 'w', modulus=x^2-2)
w = K.gen()
So $w = \sqrt 2$ and $2w = \sqrt 8$. The map I like to define sends $(x:y:z)$ to $(wx:2wy:2wz)$.
In this particular example, it's obvious that $(wx:2wy:2wz) = (x:2y:2z)$; but it's just a simple example do demonstrate the problem.
Something like this doesn't work:
x,y,z = ProjectiveSpace(k, 2, 'x,y,z').gens() #or ProjectiveSpace(K, 2, 'x,y,z').gens()
E.Hom(W)([w*x, 2*w*y, 2*w*z])
Thank you.RoadFri, 14 Jul 2017 18:08:01 +0200https://ask.sagemath.org/question/38263/