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.

Question

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.

Harper Heath · Accepted Answer

By Lagrange interpolation, you don&#039;t even need rational functions — polynomials suffice.

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

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