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Skew-convex function rings and evaluation of skew rational functions

Masood Aryapoor

TL;DR

This work extends evaluation techniques from skew polynomials to skew rational functions by introducing skew-convex function rings built from the skew product. It develops a general framework for evaluating quotients of skew polynomials through minimal representations and Ore localization, and proves a product formula that governs the evaluation of products at a fixed point. The paper also establishes structural connections between skew-convex rings and endomorphism rings under transitive group actions, and provides explicit criteria for invertibility and evaluation on centrally finite skew fields. These contributions broaden the toolbox for noncommutative polynomial evaluation and offer new avenues for applying skew polynomial theory in algebra and related areas.

Abstract

The product formula for evaluating products of skew polynomials is used to construct a class of rings. As an application, we present a method of evaluating quotients of skew polynomials.

Skew-convex function rings and evaluation of skew rational functions

TL;DR

This work extends evaluation techniques from skew polynomials to skew rational functions by introducing skew-convex function rings built from the skew product. It develops a general framework for evaluating quotients of skew polynomials through minimal representations and Ore localization, and proves a product formula that governs the evaluation of products at a fixed point. The paper also establishes structural connections between skew-convex rings and endomorphism rings under transitive group actions, and provides explicit criteria for invertibility and evaluation on centrally finite skew fields. These contributions broaden the toolbox for noncommutative polynomial evaluation and offer new avenues for applying skew polynomial theory in algebra and related areas.

Abstract

The product formula for evaluating products of skew polynomials is used to construct a class of rings. As an application, we present a method of evaluating quotients of skew polynomials.
Paper Structure (10 sections, 21 theorems, 40 equations)

This paper contains 10 sections, 21 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Skew-convex function rings
  3. Skew-convex function rings
  4. Structure of skew-convex function rings
  5. Skew-invertible functions
  6. Evaluation of skew rational functions
  7. Basic definitions
  8. Evaluation of the skew rational function $(T-b)^{-1}$
  9. Evaluation of skew rational functions over centrally finite skew fields
  10. The product formula for skew rational functions

Key Result

Lemma 2.1

Let $f,h,g\colon X\to K$ be arbitrary functions. Then: (1) The constant function $1$ is a unit for $\diamond$, that is, $f=f\diamond 1=1\diamond f$. (2) $(f+g)\diamond h= f\diamond h+ g\diamond h$, that is, the right distributive law (with respect to pointwise addition) holds for the skew product.

Theorems & Definitions (59)

  • Definition 2.1
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Example 2.1
  • Example 2.2
  • ...and 49 more