Composition and inverse of multivariate functions and algebraic system of equations

TL;DR

This paper explores advanced methods for solving multivariate equations by introducing function promotion, multivariate function composition, and inverse relations, offering new perspectives on expressing solutions and addressing Hilbert's 13th problem.

Contribution

It introduces novel concepts like function promotion, multivariate function composition, and inverse multivariate relations to solve equations more effectively.

Findings

Expressed solutions using inverse multivariate functions when invertible.

Extended inverse functions to multivariate relations for irreversible functions.

Discussed superposition of unary functions related to Hilbert's 13th problem.

Abstract

Every one knows that an equation is equivalent to a multivariate function. Generally speaking, there are more than one unknown x in this multivariate function and it is not easy to reduce the number of unknown x to one. In this paper we achieve this by introducing function promotion which can converse a function of less variables to one of more variables and by introducing multivariate function composition developed from unary function composition. We introduced inverse multivariate functions extended from inverse unary functions then we can express the solution by an inverse multivariate function if this equivalent multivariate function is invertible. For an equivalent irreversible multivariate function we introduced relation and consider the equivalent multivariate function as a special multivariate relation then we can express the solution by an inverse multivariate relation which always exists. No one will belief that every one is familiar with multivariate function composition after their reading this paper and knowing the importance of multivariate function composition to express the solution of a general equation. Further more we discuss the possibility of expressing the solution, a multivariate function or multivariate relation, for a general equation by superposition of unary ones. This will involve Hilbert's 13th problem. The topic shown in this paper can be new directions for many mathematicians because the topic is so basic.