Multivariate polynomial functional equation I’m having some difficulties solving the following functional equation:

Find all polynomials $P(x,y)\in\mathbb{R}[X,Y]$ for which:
  
  
*
  
*$P(x,y)$ is homogeneous (so $\exists n\in\mathbb{N}, \forall x,y,t\in\mathbb{R}: P(tx,ty)=t^n\cdot P(x,y)$).
  
*$\forall a,b,c\in\mathbb{R}: P(a+b,c)+P(b+c,a)+P(c+a,b)=0$
  
*$P(1,0)=1$


My (useful) observations:


*

*$P(x,0)=x^n$

*$P(0,x)=-2x^n$

*$P(2x,x)=0$

*$P(x,x)=\frac{-1}{2}\cdot2^nx^n$

*$P(-x,-x)=-P(x,x)= \frac{1}{2}\cdot2^nx^n$

*$P(y,x)=-P(x,y)-(x+y)^n$

When we write
$$P(x,y) = \sum_{i=0}^na_i\cdot x^iy^{n-i}$$
These observations imply that:


*

*$a_n=1$

*$a_0=-2$

*$\sum_{i=0}^na_i=\frac{-1}{2}\cdot2^n$

*$\sum_{i=0}^n2^ia_i=0$
Also, the fifth observation implies that $n$ is odd.

I’ve noticed that $P(x,y)=x-2y$ satisfies the conditions, but I don’t know how to prove it’s the only solution.
Can someone please give me a hint how to proceed?
 A: As you ask for a hint, here are two hints to help you along. Below is a sketch of a full proof. Let me know when I can 'unhide' all the hidden text to make the answer more legible for future readers.
Hint 1: 

 For all $a,b\in\Bbb{R}$ find $c\in\Bbb{R}$ such that the second identity becomes of the form
 $$P(u,-u)+P(v,-v)+P(w,-w)=0.$$

Hint 2:

 Deduce that if $\deg{P}>1$ then $P$ is divisible by $X+Y$.


Full solution: The polynomials that satisfy the conditions are precisely the polynomials
$$(X-2Y)(X+Y)^n,$$
with $n\in\Bbb{N}$. It is not hard to verify that these polynomials satisfy the conditions. Showing that there are no other solutions is more work. Below is a proof is by induction on the degree.
Observation 1:  The unique solution $P\in\Bbb{R}[X,Y]$ with $\deg P\leq1$ is $P=X-2Y$.
Proof. 
There are no constant solutions, and for $n=1$ setting $P=uX+vY$ shows that
$$(2u+v)(a+b+c)=0,$$
holds for all $a,b,c\in\Bbb{R}$, 
and together with $P(1,0)=1$ this implies $P=X-2Y$.$\hspace{10pt}\square$
Observation 2: If $P\in\Bbb{R}[X,Y]$ satisfies the conditions and $\deg P>1$ then $X+Y$ divides $P$.
Proof. Suppose $P\in\Bbb{R}[X,Y]$ satisfies the conditions and $\deg P>1$. Plugging in $c=-a-b$ shows that for all $a,b\in\Bbb{R}$ we have
$$0=P(a+b,-a-b)+P(-a,a)+P(-b,b)=((a+b)^n+(-a)^n+(-b)^n)P(1,-1),$$
which implies that $P(1,-1)=0$ because $n>1$, and hence that $P(X,-X)=0$.
This means $P$ is divisible $X+Y$.$\hspace{10pt}\square$
Proof of full solution. Now we can prove by induction that for all $n\in\Bbb{N}$ we have

If $P\in\Bbb{R}[X,Y]$ satisfies the conditions and $\deg P=n+1$ then $P=(X-2Y)(X+Y)^n$.

The base case $n=0$ is covered by observation 1. So let $n\in\Bbb{N}$ and suppose that the statement above holds for $n$. 
Suppose $P\in\Bbb{R}[X,Y]$ satisfies the conditions and $\deg P=n+2$. Then $P$ is divisible by $X+Y$ by observation 2, which means there exists $Q\in\Bbb{R}[X,Y]$ such that $P=(X+Y)Q$. Then clearly $\deg Q=n+1$, and we verify that $Q$ also satisfies the conditions:


*

*Because $P$ and $X+Y$ are homogeneous, also $Q$ is homogeneous.

*For all $a,b,c\in\Bbb{R}$ we have
\begin{eqnarray*}
0&=&P(a+b,c)+P(b+c,a)+P(c+a,b)\\
&=&(a+b+c)(Q(a+b,c)+Q(b+c,a)+Q(c+a,b)),
\end{eqnarray*}
which shows that for all $a,b,c\in\Bbb{R}$ with $a+b+c\neq0$ we have
$$Q(a+b,c)+Q(b+c,a)+Q(c+a,b)=0.$$
Because $Q$ is a polynomial, it follows that this holds for all $a,b,c\in\Bbb{R}$.

*Clearly $P(1,0)=1$ implies $Q(1,0)=1$.


This shows that $Q$ satisfies the conditions and $\deg Q=n+1$, so by induction hypothesis 
$$Q=(X-2Y)(X+Y)^n
\qquad\text{ and hence }\qquad
P=(X-2Y)(X+Y)^{n+1},$$
which completes the proof by induction.
A: From $P_n(t x, t y) = t^nP_n(x,y)$ we conclude that $P_n(x,y) = \sum_{k=0}^n a_k x^k y^{n-k}$
now considering
$$
P_n(2x,y) + 2P_n(x+y,x) = 0\Rightarrow \sum_{k=0}^n 2^k a_k x^k y^{n-k}+2\sum_{k=0}^na_k\left(x+y\right)^k x^{n-k}
$$
or
$$
\sum_{k=0}^n 2^k a_k x^k y^{n-k}+2\sum_{k=0}^na_k\left(\sum_{j=0}^k C_j^k x^j y^{k-j}\right) x^{n-k} = \sum_{k=0}^n 2^k a_k x^k y^{n-k}+2\sum_{k=0}^na_k\left(\sum_{j=0}^k C_j^k x^{n-(k-j)} y^{k-j}\right)=0
$$
or making $\nu=k-j$
$$
\sum_{k=0}^n 2^k a_k x^k y^{n-k}+2\sum_{k=0}^na_k\left(\sum_{\nu=k}^0 C_{k-\nu}^k x^{n-\nu} y^{\nu}\right)=0
$$
so to guarantee the polynomial identity we conclude
$$
2^{n-k}a_{n-k}+2 \sum_{j=0}^{n-k}a_{j+k}C_j^{j+k}= 0
$$
with $a_n = 1$ due to $P_n(1,0) = 1$
This is a almost triangular linear system. For $n = 3$ we have
$$
\begin{cases}
2^3a_3+2\left(a_0C_0^0+a_1C_1^1+a_2C_2^2+a_3C_3^3\right)=0\\
2^2a_2+2\left(a_1C_0^1+a_2C_1^2+a_3C_2^3\right)=0\\
2^1a_1+2\left(a_2C_0^2+a_3C_1^3\right)=0\\
2^0a_0+2a_3C_0^3 = 0
\end{cases}
$$
or
$$
\left\{
\begin{array}{rcl}
 2 a_0+2 a_1+2 a_2+10 a_3& = &0 \\
 2 a_1+8 a_2+6 a_3&=&0 \\
 2 a_1+2 a_2+6 a_3&=&0 \\
 a_0+2 a_3&=&0 \\
\end{array}
\right.
$$
and solving we have
$$
a_0 = -2, a_1 = -3, a_2 = 0, a_3 = 1
$$
or
$$
P_3(x,y) = (x-2y)(x+y)^2
$$
and for $n$
$$
P_n(x,y) = (x-2y)(x+y)^{n-1}
$$
as can be easily verified.
