Question 118
A $(n + 1)$-tuple $(h_1, h_2, \dots, h_{n+1})$ where for every $1 \le i \le n + 1$, $h_i(x_1, x_2, \dots, x_n)$ is an $n$-variable polynomial with real coefficients is called **good** if the following condition holds.
For any $n$ functions $f_1, f_2, \dots, f_n : \mathbb{R} \to \mathbb{R}$, if for all $1 \le i \le n + 1$, $P_i(x) = h_i(f_1(x), f_2(x), \dots, f_n(x))$ is a polynomial in variable $x$, then $f_1(x), f_2(x), \dots, f_n(x)$ are polynomials.
a) Prove that for all positive integers $n$, there exists a **good** $(n+1)$-tuple $(h_1, h_2, \dots, h_{n+1})$ such that the degree of each $h_i$ is more than one.
b) Prove that there does not exist any integer $n \gt 1$ for which there is a **good** $(n+1)$-tuple $(h_1, h_2, \dots, h_{n+1})$ such that all $h_i$ are symmetric polynomials.
For any $n$ functions $f_1, f_2, \dots, f_n : \mathbb{R} \to \mathbb{R}$, if for all $1 \le i \le n + 1$, $P_i(x) = h_i(f_1(x), f_2(x), \dots, f_n(x))$ is a polynomial in variable $x$, then $f_1(x), f_2(x), \dots, f_n(x)$ are polynomials.
a) Prove that for all positive integers $n$, there exists a **good** $(n+1)$-tuple $(h_1, h_2, \dots, h_{n+1})$ such that the degree of each $h_i$ is more than one.
b) Prove that there does not exist any integer $n \gt 1$ for which there is a **good** $(n+1)$-tuple $(h_1, h_2, \dots, h_{n+1})$ such that all $h_i$ are symmetric polynomials.