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Journal of Interdisciplinary Mathematics

(Since 1998)
Open Access ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502
CiteScore 2025: 1.9 (Q2)

CiteScore 2025: 1.9 (Q2)

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Journal of Interdisciplinary Mathematics
Open Access Research Article

Non-archimedean SEL Egyptian fraction expansion and characterizations of rational elements of function fields

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pp. 441–456Vol. 28Issue 2March 2025DOI: 10.47974/JIM-1747XML
Received:
08 Nov 2022
Published Online:
15 Mar 2025
Article type:
Research Article
Language:
EN
Article no.:
JIM-1747
Pages:
441–456

Abstract

For a complete field equipped with a discrete non-archimedean valuation, we propose the SEL Egyptian fraction expansion, a generalization of the non-archimedean Sylvester series, Engel series, and Lüroth series expansions for its elements. Using this expansion, characterizations of rational elements of the two function fields, each completed with a prime-adic valuation and the degree valuation, respectively, are also established. These results generalize the existing rationality characterizations via the above three series expansions for elements of such function fields.

Keywords

Engel series expansionLüroth series expansionSylvester series expansionSEL Egyptian fraction expansionDiscrete-valued non-archimedean fieldFunction fields

Subject Classifications

11A6711S23

References

[1] J. W. S. Cassels, Local fields, Cambridge University Press, Cambridge (1986).
[2] R. Cohen, “Egyptian fraction expansions,” Math. Mag., vol. 46, pp. 76-80 (1973).
[3] P. Erdös, A. Rényi, and P. Szüsz, “On Engel’s and Sylvester’s series,” Ann. Univ. Sci. Budapest. Eötvös. Sect. Math., vol. 1, pp. 7–32 (1958).
[4] M. Janthawee and N. R. Kanasri, “SEL Egyptian fraction expansion and characterizations of rational numbers,” J. Discret. Math. Sci. Cryptogr., vol. 24, pp. 277–298 (2021).
[5] V. Laohakosol, T. Chaichana, J. Rattanamoong, and N. R. Kanasri, “Engel series and Cohen-Egyptian fraction expansions,” Int. J. Math. Math. Sci., Article ID 865705, 15 pages (2009).
[6] V. Laohakosol, N. Kanasri, and A. Harnchoowong, “Characterizing rational elements using Knopfmachers’ expansions in function fields,” Thai J. Math., vol. 4, pp. 223–244 (2006).
[7] V. Laohakosol and N. Rompurk, “A characterization of rational elements by Lüroth-type series expansions in the p-adic number field and in the field of Laurent series over a finite field,” Acta Arith., vol. 122, pp. 195–205 (2006).
[8] M. E. Mays, “A worst case of the Fibonacci-Sylvester expansion,” J. Combin., vol. 1, pp. 141–148 (1987).
[9] P. J. McCarthy, Algebraic extension of fields, Dover Publications, New York (1991).
[10] O. Perron, Irrationalzahlen, De Gruyter, Berlin (1939).

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