Linearisation, splitting property and homotopy algebras

In this paper, we study the formal linearisation problem for vector fields in the framework of graded coalgebras. We prove that a formal vector field is linearisable if and only if it satisfies a splitting property, by providing an explicit recursive construction of the isomorphism that linearises it. This criterion yields a streamlined proof of Basto-Gon\c{c}alves' theorem on admissible resonant vector fields. We also establish a corresponding splitting criterion for morphisms of formal manifolds, proving that a morphism is linearisable if and only if it satisfies this property. Furthermore, we obtain an elementary and explicit proof of Bandiera's characterisation of linearisable (equivalently, homotopy abelian) $L_\infty[1]$ algebras. Finally, we extend this framework to $A_\infty[1]$ algebras, showing that their linearisability is similarly characterised by an analogous splitting property.