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Linear-in-degree monomial Rota-Baxter of weight zero and averaging operators on $F[x, y]$ and $F_0[x, y]$

A. Khodzitskii

TL;DR

This work classifies weight-zero Rota-Baxter operators on $F[x,y]$ and $F_0[x,y]$ that arise from linear-in-degree monomial averaging operators. By reducing the problem to a finite set of canonical averaging forms (i)–(iv) and solving resulting non-linear recurrences for the RB coefficients, the authors provide explicit lattice-based descriptions of RB-operators across multiple subcases, including degenerate and nondegenerate shifts. The main contributions are complete patterns for the coefficient arrays $\alpha_{n,m}$ or $\gamma_{n,m}$ under various $p_x,p_y,r,c$ configurations, together with closed-form formulas and divisibility conditions that characterize when nontrivial RB-operators exist. The results advance the structural understanding of RB-operators tied to averaging operators and yield a structured landscape of weight-zero RB-algebras on multivariate polynomial rings.

Abstract

Rota-Baxter operators on the polynomial algebra have been actively studied since the work of S.H. Zheng, L. Guo, and M. Rosenkranz (2015). Monomial operators of an arbitrary weight (2016), as well as injective operators of weight zero on $F[x]$ (2021), have been described. The author described monomial Rota-Baxter operators of nonzero weight on $F[x, y]$ coming from averaging operators (2023) and studied the connection between monomial Rota-Baxter operators and averaging operators (2024). The main result of the current work is the classification of monomial Rota-Baxter operators of weight zero on $F[x,y]$ coming from monomial linear-in-degree averaging operators.

Linear-in-degree monomial Rota-Baxter of weight zero and averaging operators on $F[x, y]$ and $F_0[x, y]$

TL;DR

This work classifies weight-zero Rota-Baxter operators on and that arise from linear-in-degree monomial averaging operators. By reducing the problem to a finite set of canonical averaging forms (i)–(iv) and solving resulting non-linear recurrences for the RB coefficients, the authors provide explicit lattice-based descriptions of RB-operators across multiple subcases, including degenerate and nondegenerate shifts. The main contributions are complete patterns for the coefficient arrays or under various configurations, together with closed-form formulas and divisibility conditions that characterize when nontrivial RB-operators exist. The results advance the structural understanding of RB-operators tied to averaging operators and yield a structured landscape of weight-zero RB-algebras on multivariate polynomial rings.

Abstract

Rota-Baxter operators on the polynomial algebra have been actively studied since the work of S.H. Zheng, L. Guo, and M. Rosenkranz (2015). Monomial operators of an arbitrary weight (2016), as well as injective operators of weight zero on (2021), have been described. The author described monomial Rota-Baxter operators of nonzero weight on coming from averaging operators (2023) and studied the connection between monomial Rota-Baxter operators and averaging operators (2024). The main result of the current work is the classification of monomial Rota-Baxter operators of weight zero on coming from monomial linear-in-degree averaging operators.
Paper Structure (11 sections, 19 theorems, 185 equations)

This paper contains 11 sections, 19 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear-in-degree averaging operators
  4. Cases $(\mathrm{i})$ and $(\mathrm{ii})$
  5. Case $(\mathrm{ii})$
  6. Case $(\mathrm{i})$
  7. Case $(\mathrm{iii})$
  8. Case $p = q = 0$
  9. Case $p + q > 0$ for averaging operators
  10. Case $p + q > 0$ for Rota---Baxter operators
  11. Discussion

Key Result

Lemma 2.1

Let $A$ be an algebra and let $R$ be an RB-operator of weight $\lambda$ on $A$. a) The operator $\alpha^{-1} R$ for any $\alpha \in F^*$ is an RB-operator of weight $\alpha\lambda$ on $A$. b) The operator $\psi^{-1} R \psi$ for any $\psi \in \mathrm{Aut}(A)$ is an RB-operator of weight $\lambda$ on

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Definition 2.2: Monom2
  • Lemma 2.1: GuoMonographBGP
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.3
  • Example 2.1
  • Lemma 2.4
  • ...and 34 more