Finding order of poles

[Naufil Sakran]

Hello. I have been having trouble of finding order of pole of a rational function.

Suppose I have a smooth affine variety X=V(y^2-x^3+x) and I localize it at a rational point P=(1,0). Then how can I find ord_P(f)=max{k, f\in m_p^k and f\notin m_p^{k+1}}.

What I have been trying to do that is one can find this using intersection numbers defined as dim(k[x,y]_P/ (X,g) ) for any g\in k[x,y]_p. But if I localize the rin, the macaulay is not allowing me to find the dimension. 

Please let me know what should I do. Thanks

[Mahrud Sayrafi]

Hi Naufil,

The LocalRings package doesn't work with quotient rings yet (so taking the dimension of a quotient of a local ring won't work) but you can compute the length of the corresponding module. Here's one way:

needsPackage "LocalRings"
S = QQ[x,y]
P = ideal(x-1, y) -- maximal ideal at (1,0)
R = S_P           -- the local ring
f = y^2-x^3+x
M = R^1/(P + f)
length M -- 1

Mahrud

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