Ask Your Question

Groebner Basis with transcendental coefficients?

asked 2019-04-23 11:31:27 +0100

Losty gravatar image

updated 2019-05-27 10:48:39 +0100

slelievre gravatar image

I would like to compute on Sage a Groebner basis generated by a set of polynomials containing coefficients like $\exp(-2/10)$ without rounding the coefficients, for example:

$$f_1 = e^{-1/2} x^2 - e^{-3/50} y x$$

$$f_2 = e^{-3/20} x + x y^2$$

Is there a simple way to do that? I tried to create an extension of QQbar including the transcendental coefficients, but I could not successfully do it. Any help would be greatly appreciated!

edit retag flag offensive close merge delete



Why not define a new variable z (instead of $\exp-1/50$), then work algebraically... ?!

dan_fulea gravatar imagedan_fulea ( 2019-04-23 17:35:32 +0100 )edit

1 Answer

Sort by ยป oldest newest most voted

answered 2019-05-26 21:59:36 +0100

tmonteil gravatar image

You can define your field as a fraction field as follows:

sage: S.<em100> = QQbar[]
....: K = S.fraction_field()
sage: K
Fraction Field of Univariate Polynomial Ring in em100 over Algebraic Field

Here, i used the name em100 to model the transcendent number $e^{-100}$

Then, you can do:

sage: R.<x,y> = K[]
sage: f1 = em100^50*x^2-em100^6*y*x
sage: f2 = em100^15*x + x*y^2
sage: I = R.ideal([f1,f2])

sage: I
Ideal (em100^50*x^2 + (-em100^6)*x*y, x*y^2 + em100^15*x) of Multivariate Polynomial Ring in x, y over Fraction Field of Univariate Polynomial Ring in em100 over Algebraic Field

sage: I.groebner_basis()
[x*y^2 + em100^15*x, x^2 + ((-1)/em100^44)*x*y]
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


Asked: 2019-04-23 11:31:27 +0100

Seen: 276 times

Last updated: May 27 '19