On an estimation of the number of solutions of certain ``chipped'' systems of equations (Q1909783)

The author considers the conjecture that for distinct natural numbers \(\alpha_1,\dots, \alpha_n\), the number of solutions of the system \[ x^{\alpha_j}_1+\cdots+ x^{\alpha_j}_{n+1}= y^{\alpha_j}_1+\cdots+ y^{\alpha_j}_{n+ 1},\quad j= 1,\dots, n \] with \(1\leq x_i\), \(y_i\leq P\) does not exceed \(c(n, \varepsilon) P^{n+ 1+ \varepsilon}\) for any \(\varepsilon> 0\). This is established for all \(n\) in the case \(\alpha_j= 2j- 1\), and also generally for \(n= 2\) (the case \(n= 1\) goes back to Hardy and Littlewood). The key observation in the argument is that the symmetric polynomial \(\prod_{1\leq i< j\leq n+ 1} (x_i+ x_j)\) can be writen as a polynomial of the power sums \(s_{2j- 1}\) \((j= 1,\dots, n)\) with rational coefficients.