The equation F(x) + M(x)G(1/x) = 0 and homogeneous biadditive forms over fields of characteristic 2

Given a field [Formula: see text] and a multiplicative function [Formula: see text], we classify all additive functions [Formula: see text] and [Formula: see text] that satisfy the functional equation [Formula: see text] for all [Formula: see text]. A functional (form) [Formula: see text] is functionally homogeneous if, for some scalar function [Formula: see text], [Formula: see text] for all [Formula: see text], [Formula: see text]. A simple application of the equation in the title leads to a complete description of all biadditive forms [Formula: see text] which are functionally homogeneous. While the case [Formula: see text] has been thoroughly examined by Ng [The equation [Formula: see text] and homogeneous biadditive forms, Linear Algebra Appl. 93 (1987) 255–279], we focus on the remaining case where [Formula: see text]. Our results address an open problem concerning fields of characteristic 2, posed by Ferreira et al. in their recent work [B. L. M. Ferreira, A. C. Dantas and G. C. Moraes, Functional identity on division algebras, Boll. Unione Mat. Ital. 17 (2023) 667–673].