MHB Proving that $P(P(x))=x$ Has No Real Solution

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The discussion revolves around proving that if a polynomial \( P(x) \) with real coefficients has no real solution for \( P(x) = x \), then it also has no real solution for \( P(P(x)) = x \). Participants express confusion regarding the notation, specifically whether \( X \) and \( x \) are distinct variables or a typographical error. Anemone acknowledges the mistake, clarifying that \( P(x) = x \) was intended. The conversation highlights the importance of precise notation in mathematical arguments. Ultimately, the proof hinges on the initial condition of \( P(x) \) lacking real solutions.
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Let $P(X)$ be a polynomial with real coefficients for which the equation $P(X)=x$ has no real solution.

Prove that the equation $P(P(x))=x$ has no real solution either.
 
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Consider the polynomial $f(x) = P(x) - x$. If it has no real roots, then clearly either $f(x) > 0$ for all $x \in \mathbb{R}$, or $f(x) < 0$ for all $x \in \mathbb{R}$, since if it assumes both positive and negative values then it must assume the value zero at some point, being continuous. As a consequence, if $P(x) = x$ has no real solutions, then either $P(x) > x$ for all $x \in \mathbb{R}$ or $P(x) < x$ for all $x \in \mathbb{R}$.

Assume that $P(x) > x$ for all $x \in \mathbb{R}$, then for any $x \in \mathbb{R}$ it follows that $P(P(x)) > P(x) > x$ since $P(x) \in \mathbb{R}$, hence $P(P(x)) \ne x$ for all $x \in \mathbb{R}$ so $P(P(x)) = x$ has no real solutions either. The case for $P(x) < x$ is proved similarly.
 
anemone said:
Let $P(X)$ be a polynomial with real coefficients for which the equation $P(X)=x$ has no real solution.

Prove that the equation $P(P(x))=x$ has no real solution either.

There is no real X for which

$P(X) = x$

for $P(P(x)) = x$ , P(x) cannot be real so complex

$P(x)$ with real coefficient cannot be complex with real $x$

so $P(P(x)) =x $ does not have real solution
 
Thanks to Bacterius and kaliprasad for the solution and participating in this challenge of mine.

Here is another solution:

Let $Q(x)=P(x)-x$. Then $Q(x)$ is a polynomial that never vanishes. We argue that it must always have the same sign.

Supose if possible that $Q(a)<0<Q(b)$ for some reals $a$ and $b$. Since $Q(x)$ being a polynomial is continuous, the Intermediate Value Theorem applies and there must be a number $c$ between $a$ and $b$ for which $Q(c)=0$, yielding a contradiction.

Thus, either $Q(x)>0$ for all $x$ or else $Q(x)<0$ for all $x$. Then

$P(P(x))-x=P(P(x))-P(x)+P(x)-x=Q(P(x))+Q(x)$ for all real $x$.

Since $Q$ never changes sign, both $Q(x)$ and $Q(P(x))$ have the same sign (either positive or negative) so their sum cannot vanish. Hence $P(P(x))\ne x$ for any real $x$.
 
kaliprasad said:
There is no real X for which

$P(X) = x$

for $P(P(x)) = x$ , P(x) cannot be real so complex

$P(x)$ with real coefficient cannot be complex with real $x$

so $P(P(x)) =x $ does not have real solution

I don't follow this argument. Anemone, did you mean $P(x) = x$ in the problem statement? Is $X$ just a typo for $x$ or are $X$ and $x$ distinct?
 
Bacterius said:
I don't follow this argument. Anemone, did you mean $P(x) = x$ in the problem statement? Is $X$ just a typo for $x$ or are $X$ and $x$ distinct?

Ops...yes, I meant $P(x) = x$. Sorry for the typo mistake...
 
Bacterius said:
I don't follow this argument. Anemone, did you mean $P(x) = x$ in the problem statement? Is $X$ just a typo for $x$ or are $X$ and $x$ distinct?

I took x and X different. if this was a typo then my argument does not hold.
 
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