Prove that for any two distinct points of an irreducible curve there exists a rational function that

Savanah Boone

Savanah Boone

Answered question

2022-07-01

Prove that for any two distinct points of an irreducible curve there exists a rational function that is regular at both, and takes the value 0 at one and 1 at the other.
I think I can construct such a function, for example, u ( x , y ) = ( x a ) 2 + ( y b ) 2 for given two points ( a , b ) and ( c , d ). However, this doesn't work for general algebraically closed field, for example, the case of ( c , d ) = ( a + i , b + 1 ). Hence now I have no clue. Could you give me a hint for this problem?

Answer & Explanation

Zachery Conway

Zachery Conway

Beginner2022-07-02Added 7 answers

Perhaps this is a bit late, but here's what I first thought:
Suppose a c , b d, and characteristic is not 2. Then let u ( x , y ) := x a 2 ( c a ) + y b 2 ( d b ) . Then clearly u ( a , b ) = 0 and u ( c , d ) = 1 2 + 1 2 = 1

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