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Find all polynomials $P$ such that

$P(x^2+1)=P(x)^2+1$

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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. Also, many find the use of imperative ("Find", "Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. –  Michael Albanese Jan 6 '13 at 4:42
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Is $P$ a real polynomial? –  Frank Science Jan 6 '13 at 6:12
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$P(x)=x$ works. Don't know if there are others. Nice question. –  coffeemath Jan 6 '13 at 6:53
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@Coffeemath Let $\alpha$ satisfy $\alpha=\alpha^2+1$ then $p(x)=\alpha$ for all $x$ works as well. –  Amr Jan 6 '13 at 7:14
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I found the problem here: imomath.com/index.php?options=346&lmm=0 The solution is the same as miracle173's. –  sdcvvc Jan 7 '13 at 13:56

6 Answers 6

One solution is $P(x) = x^2 + 1$:

$$ P(x^2 + 1) = (x^2 + 1)^2 + 1 = P(x)^2 + 1$$

Another solution is $P(x) = x^4 + 2x^2 + 2$:

$$ \begin{align} P(x^2 + 1) &= (x^2 + 1)^4 + 2(x^2 + 1)^2 + 2 \\&= x^8 + 4 x^6 + 8 x^4 + 8 x^2 + 5 \\&= (x^4 + 2x^2 + 2)^2 + 1 \\&= P(x)^2 + 1 \end{align}$$

How did I find these? Turns out it's obvious after rewriting the equation! Let $Q(x) = x^2+1$. Then the equation is

$$ P(Q(x)) = Q(P(x)) $$

or more succinctly, $P \circ Q = Q \circ P$. It's now clear that $P=Q$ is one solution, $P = Q \circ Q$ is another solution, $P = Q \circ Q \circ Q$ is yet another, and so forth.

Note that $P(x) = x$ is in this family as well, as the identity function is the empty repeated composition (just like the empty sum is $0$ and the empty product is $1$).

This argument is easily adapted to show the set of solutions is a monoid under composition.

I haven't worked out the complete solution though. Alas I don't know much about the monoid of polynomials under composition.


This answer to a similar question cites a 1922 theorem of Ritt that contains a characterization of polynomials that commute under composition; in particular, we can conclude the repeated compositions of $Q$ are indeed the entire solution space, excluding nonconstant polynomials. The remaining solutions are thus the two functions $P(x) = \beta$ where $\beta$ is a solution to $\beta = Q(\beta)$.

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Aha. Nice observation. –  Amr Jan 6 '13 at 7:47
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Interesting that you did not start with $P(x) = x$. –  user17762 Jan 6 '13 at 8:37
    
@Marvis There is also another solution, Let α satisfy $α=α^2+1$ then $p(x)=α$ for all x works as well. –  Amr Jan 6 '13 at 8:47
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I suspect that all non-constant solutions are compositional powers of $x^2+1$. I also suspect that asc.tuwien.ac.at/funkana/woracek/papers/ppol.pdf is relevant here. –  Robert Israel Jan 6 '13 at 10:35

A full characterization of solutions which is quite short and sweet.

Plugging in $x = \pm a$, we see that $P(x) = \pm P(-x)$ for each $x$. Clearly one sign holds infinitely often. By looking at the roots of the polynomial $P(x) - P(-x)$ or $P(x) + P(-x)$, we easily see that $P(x)$ is either odd or even. If $P(x)$ is odd, then note that $P(0) = 0$. Then $P(1) = 1$, $P(2) = 2$, $P(5) = 5$, etc. and by induction we can get infinitely many values $a$ such that $P(a) = a$. By looking at the roots of the polynomial $P(x) - x$ we see that $P(x) = x$ for all $x$ then.

Now suppose $P$ is even. This means $P(x) = Q(x^2)$ for some polynomial $Q$. Let $R(x) = x^2 + 1$. Remark there clearly exists a polynomial $S(x)$ such that $P(x) = S(x^2+1)$, just shift $Q$. Now remark that $P \circ R = R \circ P$. Hence $S \circ R \circ R = R \circ S \circ R \implies S \circ R = R \circ S$, so $S$ is a solution to the functional equation. By an easy induction on degree, it follows the only even solutions to the equations are iterations of $R$, so we are done. (EDIT : Also there is the constant solution $P(x) = c$ where $c = c^2 + 1$, because the induction starts on degree 1 which I forgot)

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Wow. Great answer! –  Jesse Madnick Jan 8 '13 at 7:21
    
(two constant solutions) –  Hurkyl Jan 8 '13 at 7:30
    
the first part i exactly the same as in my solution. The second parts avoids using the sqareroot. that is nice. –  miracle173 Jan 8 '13 at 7:45

There is another method (besides that posted here) to prove that solutions have the structure $\sum_{k=0}^n\,a_{2k}x^{2k}$ without constructing formulas to

calculate the coefficients as in my other answer.

We have $$P(x)^2-P(-x)^2=(P(x^2+1)-1)-(P((-x)^2+1)-1)=0$$

At least one of $P(x)-P(-x)$ and $P(x)+P(-x)$ must have infinitely many zeros and therefore must be identical to $0$.

Asume that $P(-x)=-P(x)$. This means that $P$ is an odd function and $P(0)=0$. Let us define $$Q(x)=x^2+1$$ then $P(Q^n(0)=Q^n(0)$ and so $$ P(x)-x=0, \quad x=0,\,Q(0),\,Q^2(0),\ldots$$ for infintely many $x$, so $P(x)=x$.

$P$ must be an even polynomial if not $P(x)=x$. But the even polynomials are exactly the polynomial that contain only even powers of x.

The polynomials $Q^n$ are even polynomials that satisfy the functional equation. They have a degree of $2^n$

In contrast to my other answer I was not able to prove with this method that there is at most one polynomial of a certain degree that satisfy the functional equation.


Proof
a polynomial is even $\Longleftrightarrow$ the polynomial contains only even powers of $x$

The even polynomial $P(x)$ can be expressed as sum $f(x)+g(x)$ were $f(x)$ is a polynomial that only contains even powers of $x$ and $g(x)$ is a polynomial that contains only odd powers of $x$. We have $$P(x)-f(x)=g(x)$$ The left side is an even polynomial, the right side an odd one, so $P(x)=f(x)$. So polynomials that contain only even powers of $x$ are exactly the even polynomials.

EDIT:

The following completes the proofs:

Lemma $Q(x)=x^2+1$,$P(Q(x)=Q(P(x))$ and $P(x)$ is even. Then there is a polynomial $T(x)$ with $P=T \circ Q = Q \circ T$

So a polynomial $P$ of degree $n$ can be reduced to a polynomial with degree $\frac{n}{2}$ that also satisfies the functional equation. This process can repeated until we

arrive at a polynomial with odd degree. But the only polynomial with odd degree that satisfies the functional equation is $T(x)=x$.

Therefore the functional equation $$Q \circ P = P \circ Q $$ has exactly the following polynomial solutions:

$$ -\frac{\sqrt{3}i-1}{2} \\ \frac{\sqrt{3}i+1}{2} \\ x \\ Q(x)\\ Q^2(x) \\ Q^3(x) \\ \ldots $$

Proof of the Lemma
We define $R(x)=\sqrt[+]{x-1}$, then $$R(Q(x))=Q(R(x))=x,\quad \forall x>1$$ and $$R(x)^2=x-1, \quad \forall x>1$$ We have $$ Q \circ P = P \circ Q$$ and therefore $$ Q \circ P \circ R = P \circ Q \circ R = P = P \circ R \circ Q , \quad \forall x>1$$ For $x> 1$ the polynomial $$T(x)=P(R(x))=\sum_{k=0}^{n}a_{2n}(x-1)^{n}$$ satisfies $$P=T \circ Q = Q \circ T$$ But if a polynomial equation is satisfied for infinite many $x$ it is satisfied for all $x$.

The coefficients of the polynomials can be efficiently calculated by the formulae $(3)$ and $(4)$ of my other post

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Are you sure you meant that degree of $P$ is reduced from $n$ to $n-2$, not $n/2$? –  sdcvvc Jan 7 '13 at 14:48
    
@ sdcvvc: thank you, fixed. –  miracle173 Jan 7 '13 at 15:03
    
Ah! I had thought to look at the inverse of $Q$ and its iterates, but it hadn't clicked that $P \circ Q^{-1}$ is a polynomial! Well done. –  Hurkyl Jan 8 '13 at 6:23

From @lab bhattacharjee post we see that the formulae for $P(1+x^2)$ and $P(x)^2+1$ are

$$ \begin{eqnarray} P(1+x^2)&=&\sum_{k=0}^{n}{\left(\sum_{i=0}^{n-k}{a_{n-i}\,{{n-i}\choose{k}}} \right)\,x^{2\,k}} \tag{1} \\ P(x)^2+1&=&\sum_{k=0}^{n}{\left(\sum_{i=0}^{k}{a_{n-i}\,a_{n-k+i}}\right)\,x^{ 2\,n-k}} +\sum_{k=0}^{n-1}{\left(\sum_{i=0}^{k}{a_{i}\,a_{k-i}} \right)\,x^{k}}+1 \tag{2} \\ \end{eqnarray} $$

The coefficients of the monomials with odd exponents in $(1)$ are $0$. For the coefficients of the exponent $2n$ we get the equation

$$ a_{n}^2=a_{n}$$ for $n>0$ and therefore $$a_{n}=1$$ If $a_{n}=0$ then it would be $degree(g)<n$. For $n=0$ one gets the equation $a_{0}^2+1=a_{0}$ instead.

The remaining $a_i$ can calculated successively. After calculation $a_n \ldots a_{n-(k-1)}$ for even $k=2m$ the value $a_{n-k}$ can be calculated by the coefficient

of $x^{2n-2m}$

$$ \sum_{i=0}^{2m}{a_{n-i}\,a_{n-2m+i}}= \sum_{i=0}^{m}{a_{n-i}\,{{n-i}\choose{n-m}}} $$

which gives

$$ a_{n-2m}=\frac{\sum_{i=0}^{m}{a_{n-i}\,{{n-i}\choose{n-m}}}-\sum_{i=1}^{2m-1}{a_{n-i}\,a_{n-2m+i}}}{2a_n} \tag{3} $$

indexes of the $a$ on the right hand side of this equation are all larger than $2n-2m$ and therefore already known. For odd $k=2m+1$ we got the equation

$$ \sum_{i=0}^{2m+1}{a_{n-i}\,a_{n-2m-1+i}}=0 $$

and therefore

$$ a_{n-2m-1}=-\frac{\sum_{i=1}^{2m}{a_{n-i}\,a_{n-2m-1+i}}}{2a_n} \tag{4} $$

This shows that $a_{n-2m-1}$ must be $0$. This can be proven by inductions starting with $m=0$ whch gives $a_{n-1}=0$ from $(4)$. In equation $(4)$ one factor $a_j$ of

each summand has an odd index because 2m-1 is odd and so $a_i=0$ and also the whole summand by induction. So the whole sum is $0$. So $0=a_{n-1}=a_{n-3}=...$. For $n$ odd this means especially $a_0=0$.

With the formulae $(3)$ and $(4)$ we can calculate the sequence $a_n, a_{n-1}, \ldots, a_0$ from the coefficients of $x^{2n}, \ldots, x^{n}$. But it is possible that

$a_i$ calculated do not fullfil the equations for the coefficients of $x^{0}, \ldots, x^{n-1}$. These coefficients give raise to the following equations

$$ \sum_{i=0}^{2m}{a_{i}\,a_{2m-i}}=\sum_{i=0}^{n-m}{a_{n-i}\,{{n-i}\choose{m}}} \tag{5} $$

or $$ a_0^2+1=\sum_{i=0}^{n} a_i $$

if $k=m=0$

$$ \sum_{i=0}^{2m+1}{a_{i}\,a_{2m+1-i}}=0 \tag{6} $$

So we can only condlude that there is at most one polynomial for each degree. The polynomials from the the post of @Hurkyl shows that there exists polynomial with $degree(P)=2^n$ for each $n$.

From @Hurkyl We take the definition of the polynomial

$$Q(x)=x^2+1$$

and we know that the problem can be restated as

Find all polynomial $P$ that $$P(Q(x))=Q(P(x))$$ or $$ P \circ Q = Q \circ P $$

therefore

$$ P \circ Q^n = Q^n \circ P$$

and @Hurkyl showed that all

$$P=Q^n$$ are solutions

I checked for $degree(P)$ from $0$ to $16$ that there are only the following polynomials:

$$ -\frac{\sqrt{3}i-1}{2} \\ \frac{\sqrt{3}i+1}{2} \\ x \\ x^2+1 \\ x^4+2x^2+2 \\ x^8+4x^6+8x^4+8x^2+5 \\ x^{16}+8x^{14}+32x^{12}+80x^{10}+138x^8+168x^6+144x^4+80x^2+26 \\ $$

Besides the constatn solutions the only solutions that exists where the solutions @Hurkyl found.

Let $P$ be a polynomial that fullfills the equation $P * Q = Q *P$. We showed that $a_0$=0 if $degree(P)$ is odd. Therefore $P(0) =0$ and also

$$P(Q^n(0))=Q^n(P(0))=Q^n(0)$$

So $Q^n(0), n=0,1,2,3,\ldots$ is a strictly increasing and therefore infinite sequence with $P(x)=x$ and therefore $P(x)-x=0$. But if a polynomial $P(x)-x$ is $0$ for

infinite many values $x$ the the polynomial es equal to $0$. So $P=id$. THis means that $x$ is the only polynomial with odd degree that satisfies our functional equation.

The remaining problem: Show that $degree(P)=2^n$ if $degree(P)$ is even.

Edit:
The solution fo the remaining problem is now already included in my other post

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addeda proof that there are no solutions with odd degree –  miracle173 Jan 7 '13 at 0:50

Let $P(y)=\sum_{0\le r\le n}a_ry^r$

So, $$P(1+x^2)=\sum_{0\le r\le n}a_r(1+x^2)^r=a_0+a_1(1+x^2)+a_2(1+\binom 21x^2+x^4)+\cdots +a_{n-1}(1+\binom {n-1}1x^2+\binom {n-1}2x^4+\cdots+\binom {n-1}{n-2}x^{2(n-2)}+\binom {n-1}{n-1}x^{2(n-1)}) +a_n(1+\binom n1x^2+\binom n2x^4+\cdots+\binom n{n-1}x^{2(n-1)}+\binom n nx^{2n})$$

$$=x^{2n}a_n+x^{2n-2}(a_n\binom n{n-1}+a_{n-1})+x^{2n-4}(a_n\binom n{n-2}+a_{n-1}\binom {n-1}{n-2}+a_{n-2})+x^2(a_n\binom n1+a_{n-1}\binom{n-1}1+\cdots+a_2\binom21+a_1)+\sum_{0\le r\le n}a_r$$

and $$\{P(x)\}^2+1=\{\sum_{0\le r\le n}a_rx^r\}^2+1$$ $$=a_n^2x^{2n}+x^{2n-1}2a_na_{n-1} +x^{2n-2}(a_{n-1}^2+2a_na_{n-2})+x^{2n-3}2(a_na_{n-3}+a_{n-1}a_{n-2}) +x^{2n-4}(a_{n-2}^2+2a_na_{n-4}+2a_{n-1}a_{n-3})+\cdots+x^2(a_1^2+2a_0a_2)+\sum_{0\le r\le n}a_r^2+1$$

Comparing the coefficients of the different powers of $x$

$r=n\implies a_n=a_n^2\implies a_n=1$ as $a_n\ne0$

$r=n-1\implies 2a_na_{n-1}=0\implies a_{n-1}=0$

$r=n-2\implies a_n\binom n{n-1}+a_{n-1}=a_{n-1}^2+2a_na_{n-2}\implies a_{n-2}=\frac n2$

$r=n-3\implies 2(a_na_{n-3}+a_{n-1}a_{n-2})=0\implies a_{n-3}=0$

$r=n-4\implies a_n\binom n{n-2}+a_{n-1}\binom {n-1}{n-2}+a_{n-2}=a_{n-2}^2+2a_na_{n-4}+2a_{n-1}a_{n-3}\implies a_{n-4}=\frac {n^2}8=\frac1{2!}\left(\frac n2\right)^2$

$r=n-5\implies 2(a_na_{n-5}+a_{n-1}a_{n-4}+a_{n-2}a_{n-3})=0\implies a_{n-5}=0$

$r=n-6\implies 2(a_na_{n-6}+a_{n-1}a_{n-5}+a_{n-2}a_{n-4})+a_{n-3}^2=a_n\binom n{n-3}+a_{n-1}\binom{n-1}{n-3}+a_{n-2}\binom{n-2}{n-3}+a_{n-3}\implies a_{n-6}=\frac1{3!}\left(\frac n2\right)^3-\frac n3$

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What about the rest of the cofficents ? –  Amr Jan 6 '13 at 8:19
    
@Amr, I was just trying to find the pattern, observe that $n=2$ gives us Hurkyl's solution. –  lab bhattacharjee Jan 6 '13 at 8:36

It depends on whether the coefficients of the polynomial come from a field (or ring) of characteristic $0$. [ http://en.wikipedia.org/wiki/Characteristic_%28algebra%29 ]

There are extra solutions in characteristic $p$, such as $P(x) = x^p$. It could be a hard problem to determine whether solutions exist that are not generated by compositions of $x^p$ and $x^2+1$. The general problem of determining all commuting pairs of polynomials over a finite field is well-known and unsolved.

In characteristic $0$, solutions of $P(Q(x))=Q(P(x))$ are classified. The case $Q(x)=x^2+1$ does not fall into one of the families of nontrivial solutions (it is not linearly conjugate to $x^2$ or to the degree 2 Chebyshev polynomial), so that the only possibililty is $P(x) = Q^{\circ \hskip0.7pt n}$ for some $n$.

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