Ask Your Question
1

Values of Hilbert function

asked 2011-01-19 19:34:33 -0500

niceq gravatar image

Is there a built in sage command to get values of the Hilbert function of an ideal? Of course for for large enough integers the value of the Hilbert function is equal to the value of the Hilbert polynomial, and there is a command to in sage for the Hilbert polynomial. But I need the values of the Hilbert function for small integers. Also, I could use the command for Hilbert series but the output is a rational function that I have to expand in a power series in order to see the values of the Hilbert function. Is there a built in command to give me the value directly?

edit retag flag offensive close merge delete

1 answer

Sort by » oldest newest most voted
0

answered 2014-10-22 14:46:05 -0500

FrédéricC gravatar image

Well, there is no built-in function, but you can make your own:

sage: rng = QQ['x, y']         
sage: rng.inject_variables()
Defining x, y
sage: I = rng.ideal([x**3, y*x**2, y**10*x])
sage: I.hilbert_series()
(t^11 + t^3 - t^2 - t - 1)/(t - 1)
sage: def hilbert_coeffs(ideal):
....:     t = PowerSeriesRing(QQ, 't').gen()
....:     return ideal.hilbert_series()(t).coefficients()
....: 
sage: 
sage: hilbert_coeffs(I)
[1, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1]
edit flag offensive delete link more

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: 2011-01-19 19:34:33 -0500

Seen: 321 times

Last updated: Oct 22 '14