20 04 Inversion Problem , Legendre Transform and Inviscid Burgers ’ Equations

Let F (z) = z−H(z) with order o(H(z)) ≥ 1 be a formal map from C to C and G(z) the formal inverse map of F (z). We first study the deformation Ft(z) = z − tH(z) of F (z) and its formal inverse Gt(z) = z + tNt(z). (Note that Gt=1(z) = G(z) when o(H(z)) ≥ 2.) We show that Nt(z) is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for Gt(z). Secondly, motivated by the gradient reduction obtained by M. de Bondt, A. van den Essen [BE1] and G. Meng [M] for the Jacobian conjecture, we consider the formal maps F (z) = z−H(z) satisfying the gradient condition, i.e. H(z) = ∇P (z) for some P (z) ∈ C[[z]] of order o(P (z)) ≥ 2. We show that, under the gradient condition, Nt(z) = ∇Qt(z) for some Qt(z) ∈ C[[z, t]] and the PDE satisfied by Nt(z) becomes the n-dimensional inviscid Burgers’ equation, from which a recurrent formula for Qt(z) also follows. Furthermore, we clarify some close relationships among the inversion problem, Legendre transform and the inviscid Burgers’ equations. In particular the Jacobian conjecture is reduced to a problem on the inviscid Burgers’ equations. Finally, under the gradient condition, we derive a binary rooted tree expansion inversion formula for Qt(z). The recurrent inversion formula and the binary rooted tree expansion inversion formula derived in this paper can also be used as computational algorithms for solutions of certain Cauchy problems of the inviscid Burgers’ equations and Legendre transforms of the power series f(z) of o(f(z)) ≥ 2.