MathDB
Linear form

Source: Baltic Way 2017 Problem 4

November 14, 2017
algebraequationpolynomial

Problem Statement

A linear form in kk variables is an expression of the form P(x1,...,xk)=a1x1+...+akxkP(x_1,...,x_k)=a_1x_1+...+a_kx_k with real constants a1,...,aka_1,...,a_k. Prove that there exist a positive integer nn and linear forms P1,...,PnP_1,...,P_n in 20172017 variables such that the equation x1⋅x2⋅...⋅x2017=P1(x1,...,x2017)2017+...+Pn(x1,...,x2017)2017x_1\cdot x_2\cdot ... \cdot x_{2017}=P_1(x_1,...,x_{2017})^{2017}+...+P_n(x_1,...,x_{2017})^{2017} holds for all real numbers x1,...,x2017x_1,...,x_{2017}.
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