Sums of two powers of linear forms (Q2781460)

Let \(Z(n,m,X)\) stand for the number of those forms \(F(\vec x)\) of degree \(m\) in \(n\) variables \(\vec x:= (x_1,\dots, x_n)\) with integral coefficients, which can be written as a sum NEWLINE\[NEWLINEF(\vec x)= L_1(\vec x)^m+ L_2(\vec x)^m,NEWLINE\]NEWLINE of two arbitrary linear forms \(L_1(\vec x)\) and \(L_2(\vec x)\) with algebraic coefficients, and such that \(H(F)\leq X\), where \(H(F)\) denotes the maximum of the coefficients of \(F\). Assuming \(m\geq 3\) and \(n> 4m\), the author proves that \(X^{2n/m}\ll Z(n,m,X)\ll X^{2n/m}\) with the constants, implicit in \(\ll\), depending at most on \(n,m\). This result is a generalization of the author's previous estimate of \(Z(3,m,X)\) proved in [the author, J. Number Theory 73, 472-517 (1998; Zbl 0926.11026)].