Let p 1 </msub> , . . . , p n </msub> be n

Emery Boone

Emery Boone

Answered question

2022-05-26

Let p 1 , . . . , p n be n distinct points in C , and let q 1 , . . . , q n be n distinct points in C . Is there a rational function f C ( x ) such that f ( p i ) = q i for each 1 i n? Thinking of C ( x ) as a monoid under composition, this question asks whether C ( x ) acts n-transitively on C .
More importantly, if this is false, then is there a nice description of the set of pairs ( ( a 1 , . . . , a n ) , ( b 1 , . . . , b n ) ) C n × C n for which there is a rational function f C ( x ) such that f ( a i ) = b i ?
A rational function is determined by its preimages counting multiplicities at 3 distinct points, so I suspect this is false.

Answer & Explanation

Harper Heath

Harper Heath

Beginner2022-05-27Added 9 answers

By Lagrange interpolation, you don't even need rational functions — polynomials suffice.

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