On a functional equation of Swiatak on groups

On a functional equation on groups with an involution related to quadratic polynomials in two variables

“…The following result in [1] follows from Theorem 3.1 with σx = x −1 for all x ∈ G. C is a bihomomorphism and a, b : G → C are arbitrary functions. The converse is also true.…”

Factoring biquadratic maps on modules

“…In 1970, H. Swiatak [4] proposed a functional equation which is both a generalized parallelogram law and a generalized d'Alembert equation. In the course of solving this equation, in [2] the following result was proved. Suppose G is a group and R is a unique factorization domain, with char R Φ 2.…”

“…More precisely, suppose B is a Symmetrie map and is a homogeneous polynomial of degree n in each variable. Is (1) then equivalent to (2) with A homogenous of degree nl In the present paper, we treat this question for n = 2 and Brought to you by | provisional account Unauthenticated Download Date | 6/26/15 8:05 AM…”

“…Suppose G is a group and R is a unique factorization domain, with char R Φ 2. Then a Symmetrie biadditive map B: G x G -> R satisfies (1) B (x,y} 2 if and only if B has a factorization (2) B(x,y) = where α is a nonzero constant and A: G -» R is additive. It is natural to ask whether (1) and (2) are equivalent for other types of polynomials.…”

“…Then a Symmetrie biadditive map B: G x G -> R satisfies (1) B (x,y} 2 if and only if B has a factorization (2) B(x,y) = where α is a nonzero constant and A: G -» R is additive. It is natural to ask whether (1) and (2) are equivalent for other types of polynomials. More precisely, suppose B is a Symmetrie map and is a homogeneous polynomial of degree n in each variable.…”