Quadratic differences that depend on the product of arguments
One of the two main results reads as follows (Theorem 6): If \(\mathbb{K}\) is a field, \(\text{char }\mathbb{K}\neq 2\), which is either finite or \(\mathbb{Q}\) or such that either \(x\) or \(-x\) is a square, for each \(x\in\mathbb{K}\); but \(\mathbb{K}\neq \mathbb{Z}_3\) and \(\mathbb{K}\neq \mathbb{Z}_5\); and if \(G\) is a uniquely both 2- and 3-divisible abelian group then the general solution \(f,g:\mathbb{K}\to G\) of the functional equation \[ Q_f(x,y):= f(x+y)+ f(x-y)- 2f(x)-2f(y)= g(xy) \tag{1} \] (on \(\mathbb{R}^2\)) is of the form \[ f(x)= {\textstyle \frac{1}{12}} A(x^4)+ B(x,x)+\alpha, \quad g(x)= A(x^2)-2\alpha; \qquad x\in\mathbb{R}, \] where \(A:\mathbb{K}\to G\) is additive, \(B:\mathbb{K}\times \mathbb{K}\to G\) is biadditive and \(\alpha\in G\) is an arbitrary constant. The Pexider-type functional equation (1), with five unknown functions, is also solved. The main results follow from the study of those functions \(f\) whose quadratic difference \(Q_f(x,y)\) (cf. (1)) depends on the product of variables \(x\) and \(y\) only. In the study the cases \(\mathbb{K}= \mathbb{Z}_3\), \(\mathbb{K}= \mathbb{Z}_5\) and \(G\) not 3-divisible are also included.
scientific article; zbMATH DE number 1003987