# Groebner Basis with transcendental coefficients?

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!

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

( 2019-04-23 17:35:32 +0200 )edit

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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]

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