Inversion problem, Legendre transform and inviscid Burgers’ equations

Abstract

Let F ( z ) = z - H ( z ) with order o ( H ( z ) ) ⩾ 1 be a formal map from C n to C n and G ( z ) the formal inverse map of F ( z ) . We first study the deformation F t ( z ) = z - tH ( z ) of F ( z ) and its formal inverse G t ( z ) = z + tN t ( z ) . (Note that G t = 1 ( z ) = G ( z ) when o ( H ( z ) ) ⩾ 2 .) We show that N t ( z ) is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for G t ( z ) . Secondly, motivated by the gradient reduction obtained by de Bondt and van den Essen (A Reduction of the Jacobian Conjecture to the Symmetric Case, Report No. 0308, University of Nijmegen, June 2003, Proc. of the AMS, to appear) and Meng (Legendre Transform, Hessian Conjecture and Tree Formula, math-ph/0308035) 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, N t ( z ) = ∇ Q t ( z ) for some Q t ( z ) ∈ C [ [ z , t ] ] and the PDE satisfied by N t ( z ) becomes the n-dimensional inviscid Burgers’ equation, from which a recurrent formula for Q t ( z ) also follows. Furthermore, we clarify some close relationships among the inversion problem, the 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 Q t ( 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 the Legendre transforms of the power series f ( z ) with o ( f ( z ) ) ⩾ 2 and det Hes ( f ) ( 0 ) ≠ 0 .