Magnification Spaces: A nonstandard approach to inverse mapping theorems

TL;DR

This paper introduces a novel nonstandard infinitesimal approach called magnification spaces to provide transparent proofs of inverse mapping theorems, including cases with degenerate mappings, expanding the theoretical toolkit in differential analysis.

Contribution

It develops a new infinitesimal order of magnitude and overflow technique for nonnumerical proofs of inverse mapping theorems, applicable to degenerate cases and minimal regularity.

Findings

Provided a transparent proof of Behrens and Nijenhuis inverse mapping theorem

Extended inverse mapping results to mappings with vanishing linear parts

Suggested further applications of the magnification spaces approach

Abstract

This paper develops an infinitesimal order of magnitude coupled with overflow technique that allows nonnumerical proofs of nondegenerate and degenerate inverse mapping theorems for mappings minimally regular at a point. This approach is used first to give a transparent proof of the inverse mapping theorem of Behrens and Nijenhuis and then is deployed to prove an inverse mapping result for mappings whose linear part vanishes at the differentiable point. We finish by indicating further possible capacities of this approach.