The equation \(F(x)+M(x)G(1/x)=0\) and homogeneous biadditive forms (Q1090873)

The paper deals with the functional equation \(F(x)+M(x)G(1/x)=0\) defined on a commutative field k of characteristic \(\neq 2\). Here F, G are additive maps and M is multiplicative on k. This equation is studied very extensively and then applied to a complete description of all biadditive forms \(T: V\times V\to k\) which are functionally homogeneous. The main result is contained in the following theorem: If F, G, \(M: k\to k\) satisfy the considered equation, then they have one of the following three exclusive representations: (i) For some morphism \(\phi\) : \(k\to k\), \(\phi\neq 0\), some \(\phi\)-derivation D, and a constant \(b\in k\) with (D,d)\(\neq (0,0)\), we have \(F=D+b\phi\), \(G=D-b\phi\), \(M=\phi^ 2\), (ii) For some morphisms \(\phi\),\(\psi\) : \(k\mapsto k\), \(\phi\),\(\psi\neq 0\) and \(\phi\neq \psi\), and some constants \(c_ 1,c_ 2\leq k\), with \((c_ 1,c_ 2)\neq (0,0)\), \(F=c_ 1\phi -c_ 2\psi\), \(G=c_ 2\phi -c_ 1\psi\), \(M=\phi \psi\) (iii) For some constant \(c\in k\), with \(a=\sqrt{c}\not\in k\), some embedding \(\phi\) : \(k\to k(a)\) with \(\phi\neq {\bar \phi}\), and some nonzero constant \(\lambda\in k(a)\), \(F=\lambda \phi +\overline{\lambda \phi}\), \(G=-{\bar \lambda}\phi -\lambda {\bar \phi}\), \(M=\phi {\bar \phi}\). The converse is also true. This theorem is applied to the description of all biadditive forms, which are M-homogeneous (i.e. \(f(\lambda u)=M(\lambda)f(u)\) for all \(\lambda\in k\), \(u\in U\) for the functional \(f: U\mapsto k)\), generalizing the Halperin problem (raised by I. Halperin in 1963 in Paris) and unifying results of \textit{J. A. Baker} [Glas. Mat., III. Ser. 3(23), 215-229 (1968; Zbl 0159.201)]; \textit{A. M. Gleason} [Am. Math. Mon. 73, 1049-1056 (1966; Zbl 0144.020)]; \textit{S. Kurepa} [Period. Math.-Phys. Astron., II. Ser. 19, 23-26 (1964; Zbl 0134.326) and ibid. 20, 79-92 (1965; Zbl 0147.352)].