Groebner Basis with transcendental coefficients?

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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]