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2017 Iran TST2 day2 p4

Source: 2017 Iran TST second exam day2 p4

April 24, 2017
algebrapolynomialIranIranian TSTfunction

Problem Statement

A n+1n+1-tuple (h1,h2,⋯ ,hn+1)\left(h_1,h_2, \cdots, h_{n+1}\right) where hi(x1,x2,⋯ ,xn)h_i\left(x_1,x_2, \cdots , x_n\right) are nn variable polynomials with real coefficients is called good if the following condition holds: For any nn functions f1,f2,⋯ ,fn:R→Rf_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R if for all 1≤i≤n+11 \le i \le n+1, Pi(x)=hi(f1(x),f2(x),⋯ ,fn(x))P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right) is a polynomial with variable xx, then f1(x),f2(x),⋯ ,fn(x)f_1(x),f_2(x), \cdots, f_n(x) are polynomials.
a)a) Prove that for all positive integers nn, there exists a good n+1n+1-tuple (h1,h2,⋯ ,hn+1)\left(h_1,h_2, \cdots, h_{n+1}\right) such that the degree of all hih_i is more than 11.
b)b) Prove that there doesn't exist any integer n>1n>1 that for which there is a good n+1n+1-tuple (h1,h2,⋯ ,hn+1)\left(h_1,h_2, \cdots, h_{n+1}\right) such that all hih_i are symmetric polynomials.
Proposed by Alireza Shavali
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