For a field k , let V be an affine variety over k . Denote by k ( V )

Erin Lozano

Erin Lozano

Answered question

2022-06-20

For a field k, let V be an affine variety over k. Denote by k ( V ) the function field of V, containing all rational functions r : V A k 1 . My question is, if a rational function f k ( V ) has a pole at p V, is there an expression f = g h where g , h k [ V ] are regular functions, and g ( p ) 0, h ( p ) = 0?
When V A k 1 , this is clear, since if we have f = g h where g ( p ) = h ( p ) = 0, we can simply reduce the expression of g and h and eliminate the factor ( x p ) until we get f = g h such that g ( p ) 0, h ( p ) = 0. But when g , h are multivariate functions, I wonder how to get such a reduced expression?

Answer & Explanation

Jake Mcpherson

Jake Mcpherson

Beginner2022-06-21Added 23 answers

This is a community wiki answer consisting of the answer from the comments in order to mark this question as answered.
Consider the case when V = A 2 with co-ordinate functions x, y and p defined by x = y = 0. Then look at the rational function r = x / y, which you can not write as g / h with g ( p ) 0.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?