# On the Factorization of Nonlinear Recurrences in Modules

## TL;DR

This paper develops conditions for factorizing nonlinear higher order recurrences in modules over rings, generalizing linear cases and applying to systems of nonlinear difference equations.

## Contribution

It introduces a framework for semiconjugate factorization of nonlinear recurrences in modules, extending linear recurrence factorization to nonlinear cases over rings.

## Key findings

- Factorization is possible when the ring contains certain units.
- Shared roots of characteristic polynomials enable semiconjugate factorization.
- Results apply to systems of nonlinear difference equations in modules.

## Abstract

For rings R with identity, we define a class of nonlinear higher order recurrences on unitary left R-modules that include linear recurrences as special cases. We obtain conditions under which a recurrence of order k+1 in this class is equivalent to a pair, known as a semiconjugate factorization, that consists of a recurrence of order k and a recurrence of order 1. We show that such a factorization is possible whenever R contains certain sequences of units. Further, if the coefficients of the original recurrence in R are independent of the index then we show that the semiconjugate factorization exists if two characteristic polynomials share a common root that is a unit in R. We use this fact to show that an overlapping factorization of these polynomials in an integral domain R yields a semiconjugate factorization of the corresponding recurrence in the module. These results are applicable to systems of higher order, nonlinear difference equations in direct products of rings. Such systems may be represented as higher order equations in a module over the ring.