Geodesic
Certain formulas for solutions to trinomial and tetranomial algebraic equations
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 5 (2012) no. 2, pp. 230-238.

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Algebraic equations with one and two parameters are considered. We prove that solutions to such equations can be represented as linear combination of generalized hypergeometric series. This result allows to express (nonlinearly) solutions to cubic and quartic equations by Gauss hypergeometric series.
Mots-clés : algebraic equation
Keywords: Gauss hypergeometric series, generalized hypergeometric series. 
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Evgeny N. Mikhalkin. Certain formulas for solutions to trinomial and tetranomial algebraic equations. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 5 (2012) no. 2, pp. 230-238. http://geodesic.mathdoc.fr/item/JSFU_2012_5_2_a8/

[1] E. N. Mikhalkin, “O reshenii obschikh algebraicheskikh uravnenii s pomoschyu integralov ot elementarnykh funktsii”, Sib. matem. zhurn., 47:2 (2006), 365–371 | MR | Zbl

[2] R. Birkeland, “Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen”, Math. Ztschr., 26 (1927), 566–578 | DOI | MR | Zbl

[3] A. M. Perelomov, “Gipergeometricheskie resheniya nekotorykh algebraicheskikh uravnenii”, Teoreticheskaya i matematicheskaya fizika, 140:1 (2004), 3–13 | MR | Zbl

[4] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, Nauka, M., 1973

[5] H. J. Mellin, “Résolution de l‘équation algébrique générale à l’aide de la fonction gamma”, C. R. Acad. Sci. Paris Sér. I Math., 172 (1921), 658–661 | Zbl

[6] A. Yu. Semusheva, A. K. Tsikh, “Prodolzhenie issledovanii Mellina o reshenii algebraicheskikh uravnenii”, Kompleksnyi analiz i differentsialnye operatory (k 150-letiyu S. V. Kovalevskoi), KrasGU, 2000, 134–146

[7] T. Clausen, “Uber die Fälle, wenn die Reihe von der Form $y=1+\frac\alpha1\cdot\frac\beta\gamma x+\frac{\alpha\cdot\alpha+1}{1\cdot2}\cdot\frac{\beta\cdot\beta+1}{\gamma\cdot\gamma+1}x^2+$ etc. ein quadrat von der Form $z=1+\frac{\alpha'}1\cdot\frac{\beta'}{\gamma'}\cdot\frac{\delta'}{\varepsilon'}x+\frac{\alpha'\cdot\alpha'+1}{1\cdot2}\cdot\frac{\beta'\cdot\beta'+1}{\gamma'\cdot\gamma'+1}\cdot\frac{\delta'\cdot\delta'+1}{\varepsilon'\cdot\varepsilon'+1}x^2+$ etc. hat”, Journ. Reine Ang. Math., 3 (1828), 89–91 | DOI | Zbl

[8] M. Passare, A. Tsikh, “Algebraic equations and hypergeometric series”, The legacy of Niels Henrik Abel, Springer, Berlin–Heidelberg–New York, 2004, 653–672 | DOI | MR | Zbl

[9] V. Bârsan, G. A. Nemnes, “Physical relevance of the Passare–Tsikh solution of the principal quintic equation”, Journ. of Adv. Research in Phys., 2:1 (2011), 1–6