Construct local ring of function field variety
Hello sage community,
I want to localize a variety's field at a certain point. First I construct the field of fractions K(V) for a coordinate ring K[V]
K.<x, y> = GF(11)[]
S = K.quotient(y^2 - x^3 - 4*x).fraction_field()Now I wish to construct the ideal ⟨x−2⟩, but unforunately 1/(x−2)∈S so I=S.
sage: I = S.ideal(x - 2)
sage: I
Principal ideal (1) of Fraction Field of Quotient of Multivariate Polynomial Ring in x, y over Finite Field of size 11 by the ideal (-x^3 + y^2 - 4*x)How can I exclude all denominators from the fraction field corresponding to x=2,y=4?
That is S is all f/g∈K[V]:g(2,4)≠0.
Thank you
I do not understand what it means to exclude denominators. Even if g1(2,4)≠0 and g2(2,4)≠0, we still may have g1(2,4)+g2(2,4)=0. That is, such polynomials do not form an ideal.
Maybe I should have excluded the part about the ideal. I just want to construct the local ring, which is K[V]P aka the coordinate ring localized at a certain point. This means excluding all denominators which are a zero at P.