Geodesic
On the solution of algebraic equations of small degrees by square radicals
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 3, pp. 5-16 Cet article a éte moissonné depuis la source Math-Net.Ru

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The work is devoted to the search for constructive analytical expressions for the roots of algebraic equations of the third-sixth degree by the coefficients of the equations. Relationships are obtained for the coefficients at which the roots of the equations are represented most simply, for example, rationally. Rational expressions for multiple roots are given. A condition is found under which the polynomial of the sixth degree in the canonical form can be represented by the product of polynomials of the third degree in the canonical form. Particular attention is paid to the symbolic expression of the roots of equations by square radicals from the coefficients. A method for solving equations using defining (generating, related to the original) equations is proposed. All the presented expansions are true of polynomials with arbitrary complex coefficients.
Keywords: solution by radicals, Cardano formulas, roots of polynomials, recurrent equations, defining equations. 
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N. S. Astapov. On the solution of algebraic equations of small degrees by square radicals. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 3, pp. 5-16. http://geodesic.mathdoc.fr/item/VYURM_2022_14_3_a0/

[1] Y. Mochimaru, “Reciprocal solution of a quartic equation”, International Journal of Pure and Applied Mathematics, 14:2 (2004), 207–210 | MR | Zbl

[2] D. Hebison-Evans, Solving Quartics and Cubics for Graphics, Technical Report TR94-487, Basser Dept. Computer Science, Univ. Sydney, 2004

[3] S.I. Schmakov, “A Universal Method of Solving Quartic Equations”, International Journal of Pure and Applied Mathematics, 71:2 (2011), 251–259 | MR

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

[5] L.I. Galieva, I.G. Galyautdinov, “Ob odnom klasse uravnenii, razreshimykh v radikalakh”, Izvestiya vuzov. Matematika, 2011, no. 2, 22–30 | MR | Zbl

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

[7] M.E. Zelenova, “Reshenie polinomialnykh uravnenii v pole algebraicheskikh chisel”, Vestn. Mosk. Un-ta. Seriya 1. Matematika, mekhanika, 2014, no. 1, 25–29 | MR | Zbl

[8] V.I. Shmoilov, M.V. Khisamutdinov, G.A. Kirichenko, “Reshenie algebraicheskikh uravnenii metodom Rutiskhauzera-Nikiportsa”, Vestnik NGU. Seriya: Matematika, mekhanika, informatika, 15:1 (2015), 63–79 | Zbl

[9] I.A. Antipova, E.N. Mikhalkin, A.K. Tsikh, “Ratsionalnye vyrazheniya dlya kratnykh kornei algebraicheskikh uravnenii”, Matematicheskii sbornik, 209:10 (2018), 3–30 | MR | Zbl

[10] Yu.V. Trubnikov, M.M. Chernyavskii, “Lokalizatsiya i nakhozhdenie reshenii trekhchlennykh algebraicheskikh uravnenii”, Matematicheskie struktury i modelirovanie, 2020, no. 2(54), 65–85

[11] S.B. Gashkov, “O slozhnosti integrirovaniya ratsionalnykh drobei”, Trudy MIAN, 218, 1997, 122–133 | Zbl

[12] L.M. Zubov, A.N. Rudev, “Kriterii silnoi elliptichnosti uravnenii ravnovesiya anizotropnogo lineino-uprugogo materiala”, Prikladnaya matematika i mekhanika, 80:6 (2016), 686–721 | Zbl

[13] L.D. Akulenko, S.V. Nesterov, “Izgibnye kolebaniya dvizhuschegosya sterzhnya”, Prikladnaya matematika i mekhanika, 72:5 (2008), 759–774 | MR | Zbl

[14] L.D. Akulenko, D.V. Georgievskii, S.V. Nesterov, “Spektr poperechnykh kolebanii uchastka dvizhuschegosya sterzhnya pri vozdeistvii prodolnoi nagruzki”, Izv. RAN. MTT, 2015, no. 2, 139–144

[15] A.I. Zemlyanukhin, A.V. Bochkarev, “Nelineinoe summirovanie stepennykh ryadov i tochnye resheniya evolyutsionnykh uravnenii”, Izvestiya vuzov. Matematika, 2018, no. 1, 34–41 | MR | Zbl

[16] A.O. Vatulyan, E.L. Kossovich, D.K. Plotnikov, “O nekotorykh osobennostyakh indentirovaniya treschinovatykh sloistykh struktur”, MTT, 2017, no. 4, 94–100

[17] V.V .Vasilev, “K zadache ustoichivosti tsilindricheskoi obolochki pri osevom szhatii”, Izv. RAN. MTT, 2011, no. 2, 5–15

[18] M.A. Ilgamov, “Vliyanie davleniya okruzhayuschei sredy na izgib tonkoi plastiny i plenki”, DAN, 476:4 (2017), 402–405 | MR

[19] N.S. Astapov, I.S. Astapov, “Sravnitelnyi analiz reshenii algebraicheskikh uravnenii tretei i chetvertoi stepeni”, Sibirskii zhurnal chistoi i prikladnoi matematiki, 16:1 (2016), 14–28 | MR | Zbl

[20] F. Klein, Elementarnaya matematika s tochki zreniya vysshei, V 2 t., v. 1, Arifmetika, algebra, analiz, Nauka, M., 1987, 431 pp. | MR

[21] F. Klein, Lektsii ob ikosaedre i reshenii uravnenii pyatoi stepeni, URSS, M., 2004, 337 pp. | MR