Our aim is to describe the solutions of the functional equation ∑i=1naif(bix+ciy)=0\documentclass[12pt]{minimal}
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\begin{document}$${\sum^{n}_{i=1} a_if(b_ix + c_iy) =0}$$\end{document}, where ai,bi,ci∈C\documentclass[12pt]{minimal}
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\begin{document}$${a_i,b_i,c_i \in \mathbb{C}}$$\end{document}, and the unknown function f is defined on the field K=Q(b1,…,bn,c1.…,cn)\documentclass[12pt]{minimal}
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\begin{document}$${K = \mathbb{Q} (b_1,\ldots, b_n, c_1.\ldots,c_n )}$$\end{document}. Since the set of solutions constitutes a variety on the discrete multiplicative group K* of the field K, our approach is to apply spectral synthesis on K* and on its powers. We prove that spectral synthesis holds in every variety on K* which consists of functions additive on K with respect to addition. As an application we show that the set S1 of additive solutions of the equation is spanned by S1∩D\documentclass[12pt]{minimal}
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\begin{document}$${S_1 \cap \mathcal{D}}$$\end{document}, where D\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{D}}$$\end{document} is the set of functions ϕ∘D\documentclass[12pt]{minimal}
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\begin{document}$${\phi \circ D}$$\end{document}, where ϕ\documentclass[12pt]{minimal}
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\begin{document}$${\phi}$$\end{document} is a field automorphism of C\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{C}}$$\end{document} and D is a differential operator on K. We prove that if V is a variety on the Abelian group (K*)k under multiplication, and every function F∈V\documentclass[12pt]{minimal}
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\begin{document}$${F \in V}$$\end{document} is k-additive on Kk with respect to addition, then spectral synthesis holds in V. From this we infer that, under some mild conditions on the equation, the set S of all solutions is spanned by S∩A\documentclass[12pt]{minimal}
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\begin{document}$${S\cap \mathcal{A}}$$\end{document}, where A\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{A}}$$\end{document} is the algebra generated by D\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{D}}$$\end{document}. This implies that if S is translation invariant with respect to addition, then spectral synthesis holds in S considered as a variety on the additive group of K. We give several applications, and describe the set of solutions of equations having some special properties (e.g. having algebraic coefficients etc.).
