# Characterizing injectivity of classes of maps via classes of matrices

## TL;DR

This paper introduces a framework linking the injectivity of various classes of maps to the injectivity of associated matrix classes, enabling unified analysis of properties like monotonicity and composition.

## Contribution

It provides a formalism to characterize injectivity of diverse map classes through matrix properties, extending classical criteria and applying to chemical reaction networks.

## Key findings

- Characterizes injectivity of generalized monomial maps.
- Extends classical injectivity criteria to broader map classes.
- Applies framework to analyze chemical reaction networks.

## Abstract

We present a framework for characterizing injectivity of classes of maps (on cosets of a linear subspace) by injectivity of classes of matrices. Using our formalism, we characterize injectivity of several classes of maps, including generalized monomial and monotonic (not necessarily continuous) maps. In fact, monotonic maps are special cases of {\em component-wise affine} maps. Further, we study compositions of maps with a matrix and other composed maps, in particular, rational functions. Our framework covers classical injectivity criteria based on mean value theorems for vector-valued maps and recent results obtained in the study of chemical reaction networks.