On the Polya Permanent Problem over Finite Commutative Rings

Abrantes Malaquias Belo Caiúve

doi:doi:10.11648/j.ml.20241002.12

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On the Polya Permanent Problem over Finite Commutative Rings

Abrantes Malaquias Belo Caiúve^*

Published in Mathematics Letters (Volume 10, Issue 2)

Received: 24 September 2024 Accepted: 16 October 2024 Published: 18 November 2024

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Abstract

In this paper we address the Polya permanent problem that was first raised in the second decade of the last century. Despite this, it continues to be treated in several surveys, of which we highlight the studies that point out Polya’s permanent problem over finite fields. Unlike previous papers, we focus on finite commutative rings, and to this end, we start by considering a commutative ring with identity R and its decomposition into a direct sum of finite local rings. Next we suppose that the characteristic of each residue field Fiis different from two, and we proof that if n is greater than or equal to 3, then no bijective map Φ from M_n(R) to M_n(R) transforms the permanent into a determinant. We developed this technique to estimate the order of the general linear group of degree n over a finite commutative ring with identity. The paper begins with the introduction where we present the title, the preliminaries that help the understanding of the following subject, then we talk about the unit permanent and unit determinant in M_n(R), we demonstrate the main result and conclusions. Regarding the methodology, we use the previous results on finite fields and the structure of finite commutative rings and also radical theory of rings.

Published in	Mathematics Letters (Volume 10, Issue 2)
DOI	10.11648/j.ml.20241002.12
Page(s)	19-23
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Determinant, Permanent, Nilradical, Finite Ring

References

[1]	Pólya, G. Aufgabe 424. Archiv der Mathematik und Physik. 20(3) 1913, 271.
[2]	Gibson, P. M. Conversion of the permanent into the determinant. Proceedings of the American Mathematical Society. Volume 27. 1971, 471-476. https://doi.org/10.1090/S0002-9939-1971-0279110-X
[3]	Gathen, J. von zur. Permanent and Determinant. Linear Algebra and its applications. New York, Elsevier Science Publishing Co., Inc., 96: 87-100, 1987.
[4]	McCuaig, W. Pólya’s Permanent Problem. The electronic journal of combinatorics. Volume 11, Issue 1, 2004. https://doi.org/10.37236/1832
[5]	Tarakanov, V.; Zatorskii, R. A relationship between determinants and permanents. Academic Journal. Mathematical Notes. Volume 85, Issue 1, 2. 2009, p. 267. https://doi.org/10.1134/S0001434609010301
[6]	Dolinar, G; Guterman, A. E.; Kuzma, B; Orel, M. On the Polya permanent problem over finite fields. European Journal of Combinatorics, Volume 32, Issue 1, 2011, pp. 116-132. https://doi.org/10.1016/j.ejc.2010.07.001
[7]	Dolinar, G.; Guterman, A. E; Kuzma, B. On the gibson barrier for the pólya problem. Journal of Mathematical Sciences. Volume 185. 2012. pp. 224-232 https://doi.org/10.1007/s10958-012-0912-2
[8]	Budrevich, M. V.; Guterman, A. E. Permanent has less zeros than determinant over finite fields. Theory and applications of finite fields. American Mathematical Society, Contemporary Mathematics 579, 33-42. Providence, RI, 2012. https://dx.doi.org/10.1090/conm/579/11516
[9]	Guterman, A. On the Pólya Conversion Problem for Permanents and Determinants. Journal of Mathematical Sciences. Volume 197. 2014, pp. 782-786. https://doi.org/10.1007/s10958-014-1762-x
[10]	Li, W; Zhang, H. On the Permanental Polynomials of Matrices. Bulletin of the Malaysian Mathematical Sciences Society. Volume 38. 2015, pp 1361-1374. https://doi.org/10.1007/s40840-014-0093-x
[11]	Budrevich, M; Dolinar, G; Guterman, A.; Kuzma, B. Lower bounds for Pólya’s problem on permanent. International Journal of Algebra and Computation. Volume 26. No. 06. 2016, pp. 1237-1255. https://doi.org/10.1142/S0218196716500521
[12]	Guterman, A. E.; Spiridonov, I. A. Permanent Polya problem for additive surjective maps. Linear Algebra and its Applications Volume 599. 2020, pp. 140-155. https://doi.org/10.1016/j.laa.2020.04.001
[13]	Sanguanwong, R; Rodtes, K. A remark on an equivalence of two versions of Polya’s permanent problem. Mathematics - Rings and Algebras. arXiv:2012.01278v1 [math.RA], 2020. https://doi.org/10.48550/arXiv.2012.01278
[14]	Sanguanwong, R; Rodtes, K. Determinants and permanents of power matrices. Bulletin of the Australian Mathematical Society, Volume 105, Issue 1. 2022, pp. 37-45. https://doi.org/10.1017/S0004972721000356
[15]	Atiyah, M. F.; MacDonald, I. G. Introduction to commutative algebra. London: Addison-Wesley Publishing Company, 1969, pp. 2-3.
[16]	McDonald, B. R. Finite rings with identity. Pure and Applied Mathematics. Volume 28. New York: Marcel Dekker, Inc. 1974, pp. 94-97.

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Caiúve, A. M. B. (2024). On the Polya Permanent Problem over Finite Commutative Rings. Mathematics Letters, 10(2), 19-23. https://doi.org/10.11648/j.ml.20241002.12

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ACS Style

Caiúve, A. M. B. On the Polya Permanent Problem over Finite Commutative Rings. Math. Lett. 2024, 10(2), 19-23. doi: 10.11648/j.ml.20241002.12

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AMA Style

Caiúve AMB. On the Polya Permanent Problem over Finite Commutative Rings. Math Lett. 2024;10(2):19-23. doi: 10.11648/j.ml.20241002.12

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@article{10.11648/j.ml.20241002.12,
  author = {Abrantes Malaquias Belo Caiúve},
  title = {On the Polya Permanent Problem over Finite Commutative Rings},
  journal = {Mathematics Letters},
  volume = {10},
  number = {2},
  pages = {19-23},
  doi = {10.11648/j.ml.20241002.12},
  url = {https://doi.org/10.11648/j.ml.20241002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20241002.12},
  abstract = {In this paper we address the Polya permanent problem that was first raised in the second decade of the last century. Despite this, it continues to be treated in several surveys, of which we highlight the studies that point out Polya’s permanent problem over finite fields. Unlike previous papers, we focus on finite commutative rings, and to this end, we start by considering a commutative ring with identity R and its decomposition into a direct sum of finite local rings. Next we suppose that the characteristic of each residue field Fiis different from two, and we proof that if n is greater than or equal to 3, then no bijective map Φ from Mn(R) to Mn(R) transforms the permanent into a determinant. We developed this technique to estimate the order of the general linear group of degree n over a finite commutative ring with identity. The paper begins with the introduction where we present the title, the preliminaries that help the understanding of the following subject, then we talk about the unit permanent and unit determinant in Mn(R), we demonstrate the main result and conclusions. Regarding the methodology, we use the previous results on finite fields and the structure of finite commutative rings and also radical theory of rings.},
 year = {2024}
}

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TY  - JOUR
T1  - On the Polya Permanent Problem over Finite Commutative Rings
AU  - Abrantes Malaquias Belo Caiúve
Y1  - 2024/11/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ml.20241002.12
DO  - 10.11648/j.ml.20241002.12
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 19
EP  - 23
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20241002.12
AB  - In this paper we address the Polya permanent problem that was first raised in the second decade of the last century. Despite this, it continues to be treated in several surveys, of which we highlight the studies that point out Polya’s permanent problem over finite fields. Unlike previous papers, we focus on finite commutative rings, and to this end, we start by considering a commutative ring with identity R and its decomposition into a direct sum of finite local rings. Next we suppose that the characteristic of each residue field Fiis different from two, and we proof that if n is greater than or equal to 3, then no bijective map Φ from Mn(R) to Mn(R) transforms the permanent into a determinant. We developed this technique to estimate the order of the general linear group of degree n over a finite commutative ring with identity. The paper begins with the introduction where we present the title, the preliminaries that help the understanding of the following subject, then we talk about the unit permanent and unit determinant in Mn(R), we demonstrate the main result and conclusions. Regarding the methodology, we use the previous results on finite fields and the structure of finite commutative rings and also radical theory of rings.
VL  - 10
IS  - 2
ER  -

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Author Information

Abrantes Malaquias Belo Caiúve

Department of Exact Mathematical Sciences, Instituto Superior de Ciências de Educação do Cuanza-Sul (ISCED CS), Sumbe, Angola

Contact Email

http://orcid.org/0000-0001-8640-6295

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Plain Text BibTeX RIS

APA Style

Caiúve, A. M. B. (2024). On the Polya Permanent Problem over Finite Commutative Rings. Mathematics Letters, 10(2), 19-23. https://doi.org/10.11648/j.ml.20241002.12

Copy | Download

ACS Style

Caiúve, A. M. B. On the Polya Permanent Problem over Finite Commutative Rings. Math. Lett. 2024, 10(2), 19-23. doi: 10.11648/j.ml.20241002.12

Copy | Download

AMA Style

Caiúve AMB. On the Polya Permanent Problem over Finite Commutative Rings. Math Lett. 2024;10(2):19-23. doi: 10.11648/j.ml.20241002.12

Copy | Download

@article{10.11648/j.ml.20241002.12,
  author = {Abrantes Malaquias Belo Caiúve},
  title = {On the Polya Permanent Problem over Finite Commutative Rings},
  journal = {Mathematics Letters},
  volume = {10},
  number = {2},
  pages = {19-23},
  doi = {10.11648/j.ml.20241002.12},
  url = {https://doi.org/10.11648/j.ml.20241002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20241002.12},
  abstract = {In this paper we address the Polya permanent problem that was first raised in the second decade of the last century. Despite this, it continues to be treated in several surveys, of which we highlight the studies that point out Polya’s permanent problem over finite fields. Unlike previous papers, we focus on finite commutative rings, and to this end, we start by considering a commutative ring with identity R and its decomposition into a direct sum of finite local rings. Next we suppose that the characteristic of each residue field Fiis different from two, and we proof that if n is greater than or equal to 3, then no bijective map Φ from Mn(R) to Mn(R) transforms the permanent into a determinant. We developed this technique to estimate the order of the general linear group of degree n over a finite commutative ring with identity. The paper begins with the introduction where we present the title, the preliminaries that help the understanding of the following subject, then we talk about the unit permanent and unit determinant in Mn(R), we demonstrate the main result and conclusions. Regarding the methodology, we use the previous results on finite fields and the structure of finite commutative rings and also radical theory of rings.},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - On the Polya Permanent Problem over Finite Commutative Rings
AU  - Abrantes Malaquias Belo Caiúve
Y1  - 2024/11/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ml.20241002.12
DO  - 10.11648/j.ml.20241002.12
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 19
EP  - 23
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20241002.12
AB  - In this paper we address the Polya permanent problem that was first raised in the second decade of the last century. Despite this, it continues to be treated in several surveys, of which we highlight the studies that point out Polya’s permanent problem over finite fields. Unlike previous papers, we focus on finite commutative rings, and to this end, we start by considering a commutative ring with identity R and its decomposition into a direct sum of finite local rings. Next we suppose that the characteristic of each residue field Fiis different from two, and we proof that if n is greater than or equal to 3, then no bijective map Φ from Mn(R) to Mn(R) transforms the permanent into a determinant. We developed this technique to estimate the order of the general linear group of degree n over a finite commutative ring with identity. The paper begins with the introduction where we present the title, the preliminaries that help the understanding of the following subject, then we talk about the unit permanent and unit determinant in Mn(R), we demonstrate the main result and conclusions. Regarding the methodology, we use the previous results on finite fields and the structure of finite commutative rings and also radical theory of rings.
VL  - 10
IS  - 2
ER  -

Copy | Download