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.Wed, 27 Apr 2016 21:00:21 +0200How to construct random divisor once the set of rational points over the Jacobian of a hyperelliptic curve is created?https://ask.sagemath.org/question/33232/how-to-construct-random-divisor-once-the-set-of-rational-points-over-the-jacobian-of-a-hyperelliptic-curve-is-created/ I know how to deliberately create a divisor with knowledge of the mumford coordinates, but is there a way to generate a divisor randomly inside of X = J(FF) and extract its mumford coordinates as polynomials over FF?
Where divisors D = X([a,b]) are elements of X = J(FF),
a and b are the mumford polynomials, and
J is the Jacobian of a hyperelliptic curve.
Thanks!Tue, 26 Apr 2016 19:06:01 +0200https://ask.sagemath.org/question/33232/how-to-construct-random-divisor-once-the-set-of-rational-points-over-the-jacobian-of-a-hyperelliptic-curve-is-created/Answer by nbruin for <p>I know how to deliberately create a divisor with knowledge of the mumford coordinates, but is there a way to generate a divisor randomly inside of X = J(FF) and extract its mumford coordinates as polynomials over FF?</p>
<p>Where divisors D = X([a,b]) are elements of X = J(FF),
a and b are the mumford polynomials, and
J is the Jacobian of a hyperelliptic curve.</p>
<p>Thanks!</p>
https://ask.sagemath.org/question/33232/how-to-construct-random-divisor-once-the-set-of-rational-points-over-the-jacobian-of-a-hyperelliptic-curve-is-created/?answer=33247#post-id-33247I assume FF is a finite field. One approach is to randomly choose the polynomial $a$ and then try and solve for a polynomial $b$. The condition you have to meet is that for every irreducible factor $x_i$ of $a$, a root $r_i$ of $a_i$ is an $x$-coordinate of a point on the hyperelliptic curve defined over $FF(r_i)$, say with $y$-coordinate $y_i$. So the probability you choose a polynomial $a$ that works is $(1/2)^m$, where $m$ is the number of irreducible factors of $m$.
Recovering $b$ is now a matter of interpolating the points $(x_i,y_i)$ and their conjugates. Make sure to properly randomize the sign choice in the square roots that you end up taking.Wed, 27 Apr 2016 21:00:21 +0200https://ask.sagemath.org/question/33232/how-to-construct-random-divisor-once-the-set-of-rational-points-over-the-jacobian-of-a-hyperelliptic-curve-is-created/?answer=33247#post-id-33247