Cauchy type equation in three variables

Let $P(x)$ a polynomial in x with real coefficients such that for all real numbers $x, y, z$ satisfying $xy + yz + zx = 1$,$ P(x) + P(y) + P(z) = P(x + y + z)$. Furthermore, $P(0) = 1$ & $P(1) = 4$. Find $P(2017)$.

This looks to me like the Cauchy functional equation which is why the title. I understand that a Cauchy equation has solutions of the type $f(x)=cx$ but can't figure out the case here with that constraint. My wild guess, however, is that $P(x)$ should be a perfect square. Please help.

Answer

Note that $P(x) = ax^2+bx+a$ solves this problem, as \begin{align*} P(x+y+z)-P(x)-P(y)-P(z) &= a(x+y+z)^2-a(x^2+y^2+z^2)-2a \\ &= 2a(xy+yz+zx-1) \end{align*} Then, we can impose $P(0) = a = 1$ and $P(1) = 2a+b = 4$. The latter implies $b = 2$, so we have $P(x) = x^2+2x+1 = (x+1)^2$. Then we find $P(2017) = 2018^2$.

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To prove that this is the only solution, note that for $$P^*(x, y, z) := P(x+y+z)-P(x)-P(y)-P(z)$$ $P^*(x, y, z)$ must be in the ideal generated by $xy+yz+zx-1$ in $\mathbb{C}[x, y, z]$ as this polynomial is irreducible over $\mathbb{C}$. Therefore, $$P^*(x, y, z) = Q(x, y, z)(xy+yz+zx-1)$$ for some polynomial $Q$. Note that $P^*(x, y, z)$ and $xy+yz+zx-1$ are both symmetric polynomials. In particular, $xy+yz+zx-1 = \frac{1}{2}p_1^2-\frac{1}{2}p_2-1$, where $p_k := x^k+y^k+z^k$. Now, note that for $P(x) = \sum_{k=0}^n a_kx^k$, $$P^*(x, y, z) = \sum_{k=0}^n a_k(p_1^k-p_k)$$ As $Q(x, y, z)$ is the quotient of symmetric polynomials, it must also be a symmetric polynomial. Thus, we write $R^*(p_1, p_2, \ldots, p_n)$ and $R(p_1, p_2, \ldots, p_n)$ to be the polynomial representations of $P^*$ and $Q$ respectively in terms of power sums (which are algebraically independent). After switching to $R$ and $R^*$, our equation in $P^*$ and $Q$ becomes $$R^*(p_1, \ldots, p_n) = \left(\frac{1}{2}p_1^2-\frac{1}{2}p_2-1\right)R(p_1, \ldots, p_n)$$ If $a_n\neq 0$ for some $n\geq 3$, then $R^*(p_1, \ldots, p_n)$ will have a nonzero term $-a_np_n$. However, this term cannot appear on the right-hand side, as if $R$ has a term $a_np_n$, then it would imply that $R^*$ also has terms $-\frac{a_n}{2}p_1^2p_n$ and $-\frac{a_n}{2}p_2p_n$ (which do not appear in the above sum). Thus, $R^*$ must equal $-2a_0+a_2(p_1^2-p_2)$, which implies that $\deg(R) = \deg(R^*)-2 = 0$. Then, $R\equiv c$ for some constant $c$, and $a_0 = a_2 = \frac{c}{2}$. Letting $a = \frac{c}{2}$ and $b = a_1$, we have $$P(x) = ax^2+bx+a$$ as above.