Faà di Bruno's formula and inversion of power series

Faà di Bruno's formula gives an expression for the derivatives of composition two real-valued functions. In this paper we prove a multivariate and synthesised version in higher dimensions, providing combinatorial chain compositions F ( 1 ) ∘ … m functions l : R N → terms sums over labelled trees. We give several applications formula, including new involution inversion power series. use framework to outline approach studying invertibility polynomial mappings, giving purely restatement Jacobian conjecture. Our methods extend naturally non-commutative case, where free series indeterminates, as tool obtaining