Lagrange inversion in infinitely many variables

Abstract

This paper is concerned with the problem of finding an effective definition for the compositional inverse of formal power series in several variables, with special regard to series in infinitely many variables. The problem of defining analogs for several variables of the Lagrange inversion formula has been open for a long time (see, e.g., [3, 53). So far, all indications point to the fact that an explicit Lagrange inversion formula in infinitely many variables does not exist in the complete generality that we find for series in finitely many variables [4]. Our main result in this paper is a Lagrange inversion formula for sets of power series in infinitely many variables c~r(xr, x2 ,... ), c(~(x~, x2 ,...) ,..., where the formal power series X,(X,, x,,...) does not depend on the variables Xl, x27.7 x, , * This situation occurs in several instances, most notably in the composition for series in infinitely many variables which has come to be called plethysm. We recall that the plethysm of a series /?(x,, x2,...) with a series u(x,, x2 ,...) is defined as follows. Set c(,(x,, x2 ,...) := a(~,,, x2,, ,... ); the plethysm of p with a is then the series /?(a,, a,,...). There are other applications as well. As is well known, a Lagrange inversion formula requires the understanding of the embedding of the ring of formal power series into a ring of formal Laurent series (which turns out to be a field for series with coefficients in a field). This is an old and thorny problem, which also has not received to this day a satisfactory answer. There are at least two possible definitions of formal Laurent series in many variables [2, 81: they can be required to be either series such that the set of their monomials (which are always of finite length) has an infimum, or series for which the set,of weights of their monomials has a minimum; in this second case it is necessary to add a finiteness condition on monomials