Solvability of some entangled Diophantine equations (Q2925721)

Let \(f(x_1,\ldots,x_n)\) be a polynomial with integer coefficients. A Diophantine equation of the type \(f(x_1,\ldots,x_n)=N_f\) is referred to as entangled when \(N_f\) is an integer determined by the coefficents of \(f\). In this paper it is proven that the equation NEWLINE\[NEWLINEaQ(x_1,x_2)+bQ(x_3,x_4)+cQ(x_5,x_6)=abcNEWLINE\]NEWLINE has integral solutions for arbitrary positive integers \(a,b,c\) when \(Q(x,y)\) is one of the norm forms \(x^2+ky^2\) or \(x^2+xy+ky^2\) for \(k\in \{1,2,3\}\). The proof uses the representation theory of Hermitian lattices and unique factorization in the rings of algebraic integers of certain imaginary quadratic fields.