Ask Your Question
0

One point from infinity variety

asked 2020-07-14 03:00:35 -0600

philipp7 gravatar image

Given an ideal, I want to compute a non-zero point in the variety of that ideal. This can be done with .variety() if the ideal has 0-dimension. How can I get a non-zero point in the variety if the ideal is not 0-dimensional?

Example:

R.<x,y> = PolynomialRing(QQ)
I = R.ideal([x^2-y^2])
I.variety()

yields an error. How can I get a point like (1,1) which lies in the variety?

Thanks!

edit retag flag offensive close merge delete

Comments

Cut by hyperplanes until the dimension is zero.

FrédéricC gravatar imageFrédéricC ( 2020-07-14 04:43:55 -0600 )edit

Thank you! Two problems come into my mind with that approach: 1. If I intersect with hyperplanes I do not necessarily reduce the dimension by 1. Say I start with dimension 3, I might get the entire ring (so no solution at all) just by adding one hyperplane. 2. I am searching for elements in the variety which are (say) rational. By intersecting with a hyperplane I might get rid of possible rational solutions. However, I suppose mathematically this problem should be quite hard, as it is already very difficult to find rational points on some "easy" curves.

philipp7 gravatar imagephilipp7 ( 2020-07-14 06:41:08 -0600 )edit

1 answer

Sort by » oldest newest most voted
0

answered 2020-07-14 06:02:39 -0600

Emmanuel Charpentier gravatar image

updated 2020-07-14 06:03:27 -0600

I tink that this is along the lines provided by FrédéricC :

sage: RR.<x,y>=PolynomialRing(QQ) ## R has some use already
sage: J=RR.ideal([x^2-y^2]) ## Similarly, I has an interesting use
sage: [J.random_element(u) for u in (1..5)]
[0,
 -1/2*x^2 + 1/2*y^2,
 -1/39*x^3 - 2/3*x^2*y + 1/39*x*y^2 + 2/3*y^3,
 -x^3*y + x*y^3 - x^3 + x*y^2 + x^2 - y^2,
 -4*x^2*y^3 + 4*y^5 + x^3 + 7/4*x^2*y - x*y^2 - 7/4*y^3 - x^2 + y^2]

HTH,

edit flag offensive delete link more

Comments

Thank you. I am probably missing something here. How do I get a random point in the variety from some random polynomials in the ideal?

philipp7 gravatar imagephilipp7 ( 2020-07-14 06:43:47 -0600 )edit

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower

Stats

Asked: 2020-07-14 03:00:35 -0600

Seen: 44 times

Last updated: Jul 14 '20