A characterization of quadratic polynomials by functional equations and a Diophantine problem (Q734103)

The author considers the functional equation \[ f(x)+ f(y)+ f(z)= f(ax+ by+ cz)+ f(bx+ cy+ az)+ f(cx+ ay+ bz)\quad (x,y,z\in\mathbb R) \] and shows that, if \((a,b,c)\) is a nontrivial solution to the system \(a+b+c=1\), \(a^2+ b^2+ c^2= 1\), then \(f: \mathbb R\to\mathbb R\) is a continuous (or locally integrable) solution of the functional equation, iff \(f\) is a quadratic polynomial (i.e., a polynomial with grade \(\leq 2\)). By considering rational solutions to \(a+b+c= 1\), \(a^2+ b^2+ c^2= 1\), the author is conducted to a very interesting study of the system of Diophantine equations \(x+ y+ z= m\), \(x^2+ y^2+ z^2= m^2\), where \(m\) is a positive integer. By combining methods of algebraic number theory and analytic number theory, he proves e.g. that for the number \(R(m)\) of solutions of this system one has the asymptotic result \(\sum_{n\leq t} R(n)\sim{2\sqrt{3}\over \pi}\,t\log t\) as \(t\to\infty\).