K[x_1,...,x_n]-algebras of multivariate polynomial heart-based rhotrix sets

Joel Nathaniel B. Pondoc and Aljean Faye P. Lobitaña
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 395–429
DOI: 10.7546/nntdm.2026.32.2.395-429
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Authors and affiliations

Joel Nathaniel B. Pondoc
College of Mathematics and Natural Sciences, University of Northwestern Mindanao
Labuyo, Tangub City, 7214 Misamis Occidental, Philippines

Aljean Faye P. Lobitaña
College of Mathematics and Natural Sciences, University of Northwestern Mindanao
Labuyo, Tangub City, 7214 Misamis Occidental, Philippines

Abstract

This study constructs and investigates the m-dimensional multivariate polynomial heart-based rhotrix (“mph-rhotrix”) algebras over an arbitrary commutative and unitary polynomial ring in n indeterminates, K[x_1,…,x_n], with entries in the same ring, where m=2k+1, k\in \N:=\{0,1,2,\ldots\}, n\in\mathbb{Z}^+, denoted by \textbf{CRAPol}(K,m,n,x), which is commutative, and \textbf{NRAPol}(K,m,n,x), which is noncommutative, and their generalized forms, \textbf{gCRAPol}(K,m,n_1,n_2,x) and \textbf{gNRAPol}(K,m,n_1,n_2,x), respectively, which consider rhotrix entries in K[x_1,…,x_{n_1}] and are over an arbitrary commutative unitary polynomial ring K[x_1,…,x_{n_2}]. This study is motivated by existing developments in rhotrix theory and the lack of formal investigation into rhotrix algebras and their tensor products. This paper first establishes the set \mathcal{R}_m(K,n,x) of m-dimensional mph-rhotrices, proving that it forms an associative commutative unitary K[x_1,…,x_{n}]-algebra. Eventually, the tensor products of the algebras were defined and examined. The results of this study enrich the theoretical landscape of rhotrices by expanding it on the realm of algebras. Furthermore, this study is theoretical in scope and does not cover practical applications, although the authors recommend a critical exploration of potential extensions and applications of the results found herein.

Keywords

  • Rhotrix
  • Heart-based
  • Polynomial
  • Algebra
  • Tensor product

2020 Mathematics Subject Classification

  • 13C05
  • 13C60
  • 16D10

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Manuscript history

  • Received: 14 March 2026
  • Revised: 18 June 2026
  • Accepted: 27 August 2026
  • Online First: 28 August 2026

Copyright information

Ⓒ 2026 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Pondoc, J. N. B., & Lobitaña, A. F. P. (2026). K[x_1,...,x_n]-algebras of multivariate polynomial heart-based rhotrix sets. Notes on Number Theory and Discrete Mathematics, 32(2), 395-429, DOI: 10.7546/nntdm.2026.32.2.395-429.

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