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.Sat, 23 Jul 2022 16:31:08 +0200Degree of a rational map and the corresponding map between function fieldshttps://ask.sagemath.org/question/63229/degree-of-a-rational-map-and-the-corresponding-map-between-function-fields/Let $X$ and $Y$ be two curves defined over $\mathbb{F}_q $ and $f:X \rightarrow Y$ be a separable rational map. Then there is field embedding
$$
f^\ast : \mathbb{F}_q (Y) \rightarrow \mathbb{F}_q (X)
$$ defined by
$f^\ast(\alpha) = \alpha \circ f$.
The degree of $f$ is then defined to be $[\mathbb{F}_q (X) : f^\ast(\mathbb{F}_q (Y))]$.
If I take two curves $X$ and $Y$ in sagemath over some $\mathbb{F}_q $ in sagemath, is there any way to automatically get the map $f^\ast$ and degree of $f$?Wed, 13 Jul 2022 08:56:42 +0200https://ask.sagemath.org/question/63229/degree-of-a-rational-map-and-the-corresponding-map-between-function-fields/Answer by dan_fulea for <p>Let $X$ and $Y$ be two curves defined over $\mathbb{F}_q $ and $f:X \rightarrow Y$ be a separable rational map. Then there is field embedding
$$
f^\ast : \mathbb{F}_q (Y) \rightarrow \mathbb{F}_q (X)
$$ defined by
$f^\ast(\alpha) = \alpha \circ f$.
The degree of $f$ is then defined to be $[\mathbb{F}_q (X) : f^\ast(\mathbb{F}_q (Y))]$.
If I take two curves $X$ and $Y$ in sagemath over some $\mathbb{F}_q $ in sagemath, is there any way to automatically get the map $f^\ast$ and degree of $f$?</p>
https://ask.sagemath.org/question/63229/degree-of-a-rational-map-and-the-corresponding-map-between-function-fields/?answer=63246#post-id-63246Here is some sample code to have a concrete situation:
F.<a> = GF(25)
P2_XYZ.<X,Y,Z> = ProjectiveSpace(F, 2)
P2_STU.<S,T,U> = ProjectiveSpace(F, 2)
eqC = -(a + 1)*Y^2*Z - X^3 + Z^3
eqD = T^2*U - S^3 + U^3
C = Curve(eqC)
D = Curve(eqD)
s, t, u = D.coordinate_ring().gens() # not really needed below
x, y, z = C.coordinate_ring().gens()
f = C.Hom(D)( [x^5, a*y^5, z^5] ) # so S -> X^5, T -> a*Y^5, U -> Z^5 in the dual polynomial world
Then we can ask for the degree:
sage: f.degree()
5
The related function fields are:
sage: C.function_field()
Function field in Z defined by Z^3 + ((4*a + 4)*Y^2)*Z + 4
sage: D.function_field()
Function field in U defined by U^3 + T^2*U + 4
We can ask for the image of $f$:
sage: f.image()
Closed subscheme of Projective Space of dimension 2 over Finite Field in a of size 5^2 defined by:
S^3 - T^2*U - U^3
Thu, 14 Jul 2022 04:55:12 +0200https://ask.sagemath.org/question/63229/degree-of-a-rational-map-and-the-corresponding-map-between-function-fields/?answer=63246#post-id-63246Comment by Dodul for <p>Here is some sample code to have a concrete situation: </p>
<pre><code>F.<a> = GF(25)
P2_XYZ.<X,Y,Z> = ProjectiveSpace(F, 2)
P2_STU.<S,T,U> = ProjectiveSpace(F, 2)
eqC = -(a + 1)*Y^2*Z - X^3 + Z^3
eqD = T^2*U - S^3 + U^3
C = Curve(eqC)
D = Curve(eqD)
s, t, u = D.coordinate_ring().gens() # not really needed below
x, y, z = C.coordinate_ring().gens()
f = C.Hom(D)( [x^5, a*y^5, z^5] ) # so S -> X^5, T -> a*Y^5, U -> Z^5 in the dual polynomial world
</code></pre>
<p>Then we can ask for the degree:</p>
<pre><code>sage: f.degree()
5
</code></pre>
<p>The related function fields are:</p>
<pre><code>sage: C.function_field()
Function field in Z defined by Z^3 + ((4*a + 4)*Y^2)*Z + 4
sage: D.function_field()
Function field in U defined by U^3 + T^2*U + 4
</code></pre>
<p>We can ask for the image of $f$:</p>
<pre><code>sage: f.image()
Closed subscheme of Projective Space of dimension 2 over Finite Field in a of size 5^2 defined by:
S^3 - T^2*U - U^3
</code></pre>
https://ask.sagemath.org/question/63229/degree-of-a-rational-map-and-the-corresponding-map-between-function-fields/?comment=63358#post-id-63358Thank you very much.Sat, 23 Jul 2022 16:31:08 +0200https://ask.sagemath.org/question/63229/degree-of-a-rational-map-and-the-corresponding-map-between-function-fields/?comment=63358#post-id-63358