On composition and Right Distributive Law for formal power series of multiple variables

TL;DR

This paper establishes conditions for composing formal power series with multiple variables and clarifies the Right Distributive Law, extending it from single-variable to multivariable cases.

Contribution

It provides a necessary and sufficient condition for composition of multivariable formal power series and generalizes the Right Distributive Law to multiple variables.

Findings

Derived a condition for composition existence in multivariable series.

Clarified and extended the Right Distributive Law to multivariable formal power series.

Removed ambiguities in the law's application to multiple variables.

Abstract

In the first part of the paper we prove a necessary and sufficient condition for the existence of the composition of formal power series in the case when the outer series is a series of one variable while the inner one is a series of multiple variables. The aim of the second part is to remove ambiguities connected with the Right Distributive Law for formal power series of one variable as well as to provide analogues of that law in the multivariable case.