# Constructing Faithful Homomorphisms over Fields of Finite Characteristic

## TL;DR

This paper develops explicit algebraic homomorphisms that preserve transcendence degree over finite fields, extending prior characteristic zero results to positive characteristic fields under certain conditions.

## Contribution

It introduces the first construction of faithful homomorphisms over finite fields for classes of polynomials with bounded inseparable degree, generalizing previous characteristic zero results.

## Key findings

- Constructed explicit faithful maps for some polynomial classes in finite fields.
- Extended algebraic independence tests to positive characteristic fields.
- First generalization of Jacobian-based methods to finite characteristic settings.

## Abstract

We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken, Mittmann and Saxena (2013), and exploited by them, and Agrawal, Saha, Saptharishi and Saxena (2016) to design algebraic independence based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields was unknown due to the failure of the Jacobian criterion over finite characteristic fields. Building on a recent criterion of Pandey, Sinhababu and Saxena (2018), we construct explicit faithful maps for some natural classes of polynomials in the positive characteristic field setting, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken et al. and Agrawal et al. in the positive characteristic setting.