# Algorithm for studying polynomial maps and reductions modulo prime number

## TL;DR

This paper explores an algorithm for inverting polynomial automorphisms using reductions modulo prime numbers, employing Segre homotopy, and introduces a method to recover inverses over integers from their prime reductions.

## Contribution

It presents a novel method to compute polynomial automorphism inverses over integers via their prime reductions, expanding on previous algorithms and introducing Segre homotopy techniques.

## Key findings

- The algorithm effectively retrieves inverses from prime reductions.
- Segre homotopy aids in understanding polynomial automorphisms.
- Examples demonstrate the practical effectiveness of the approach.

## Abstract

In our previous paper an effective algorithm for inverting polynomial automorphisms was proposed. Also the class of Pascal finite polynomial automorphisms was introduced. Pascal finite polynomial maps constitute a generalization of exponential automorphisms to positive characteristic. In this note we explore properties of the algorithm while using Segre homotopy and reductions modulo prime number. We give a method of retrieving an inverse of a given polynomial automorphism $F$ with integer coefficients form a finite set of the inverses of its reductions modulo prime numbers. Some examples illustrate effective aspects of our approach.