Polynomial P ∈ R [ x ] with P ( x ¯ ) ¯ = P ( x ) , ∀ x ∈ C

Question

Polynomial   P  ∈      R    [  x  ] with             P      (              x        ¯            )        ¯    =  P  (  x  )  ,  ∀  x  ∈      C

Jake Landry · Accepted Answer

Let   P  (  x  )  =      ∑          k      =      0        n        a    k        x    k  . The hypothesis applied for x real gives                    ∑                  k          =          0                n                    a        k                    x        k              ¯    =      ∑          k      =      0        n        a    k        x    k    ,hence      ∑          k      =      0        n    (            a      k        ¯    −      a    k    )      x    k    =  0.The LHS is a polynomial with complex entries, which vanishes on the real numbers: it&#039;s the null polynomial, hence       a    k    ∈      R   for   0  ≤  k  ≤  n and   P  ∈      R    [  x  ]

Polynomial P in RR[x] with P(x)=P(x), for all x in CC

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