# Polyregularity of the dot product of slice regular functions

## TL;DR

This paper investigates the S-polyregularity of the dot product of slice regular functions in quaternionic analysis, establishing conditions under which the product maintains S-polyregularity and deriving related linearization theorems.

## Contribution

It provides the first comprehensive analysis of the S-polyregularity of products of slice regular functions, including necessary and sufficient conditions and order determination.

## Key findings

- Product of a slice regular function and a quaternionic polynomial is S-polyregular with known order.
- Necessary and sufficient conditions for the product of two slice regular functions to be S-polyregular.
- Extension of results to products of S-polyregular functions and special dot products.

## Abstract

In this paper, we are concerned with the S-polyregularity the regular dot product of slice regular functions. We prove that the product of a slice regular function and right quaternionic polynomial function is a S-polyregular function and we determinate its exact order. The general case of the product of any two slice regular functions is also discussed. In fact, we provide sufficient and necessary conditions to the product of slice regular functions be a S-polyregular function. The obtained results are then extended to the product of S-polyregular functions and remain valid for a special dot product. As consequences we obtain linearization theorems for such S-polyregular products with respect to the slice regular functions.