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Mohamed Mooney

Mohamed Mooney

Answered question

2022-06-24

Let k be an algebraic closed field, A n be an affine variety over k, U be open set in A n and f , g k ( x 1 , . . . , x n ) and their denominators is not zero over U.If f(x)=g(x) for all x U, is it true that f=g as rational functions?

Answer & Explanation

podesect

podesect

Beginner2022-06-25Added 20 answers

Yes. By cross multiplying you can reduce this to a question about regular functions. If two regular functions agree on an open subset of a variety (this is because varieties are separated, and any non-empty open set is dense), then they agree everywhere. Use the fact then that I ( A n ) = 0.

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