Find all polynomials P(x) which have the property P[F(x)]=F[P(x)], P(0)=0 where F(x) is a given function with the property F(x)>x for all x>=0.

Madilyn Quinn 2022-10-18 Answered
Find all polynomials P ( x ) which have the property
P [ F ( x ) ] = F [ P ( x ) ] , P ( 0 ) = 0
where F ( x ) is a given function with the property F ( x ) > x for all x 0
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Answers (1)

veirenca77
Answered 2022-10-19 Author has 9 answers
Denote x 0 = 0 and x n + 1 = F ( x n ) for every n 0. Since F ( x ) > x for every x 0 x n + 1 > x n for every n 0. Note that P ( x 0 ) = x 0 and
P ( x n + 1 ) = P ( F ( x n ) ) = F ( P ( x n ) ) , n 0.
Then by induction, it is easy to see that
(1) P ( x n ) = x n , n 0.
(1) implies that the polynomial P ( x ) x has infinitely many roots, i.e. P ( x ) x
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