Kreĭn, M. G.; Naĭmark, M. A. The method of symmetric and Hermitian forms in the theory of separation of the roots of algebraic equations. (English) Zbl 0584.12018 Linear Multilinear Algebra 10, 265-308 (1981); errata 12, 77 (1982). Summary: The present publication (translated from the Russian by O. Boshko and J. L. Howland) is in the nature of a survey; therefore the presentation is similar to articles in mathematical encyclopedias. We do not intend to give a systematic presentation of all relevant problems on the basis of some single method, but rather attempt to include different methods of establishing the various propositions, indicating their characteristic differences. All basic methods and results are presented in detail; moreover, the indications given in the paper will allow a more or less experienced mathematician to construct proofs by himself of nearly all the results given. Section headings: 1. The Hermite-Jacobi method; 2. The fundamental properties and applications of Bezoutiants; 3. The Hermite method for the separation of complex roots and its development; 4. Association with some problems of function theory. Cited in 2 ReviewsCited in 103 Documents MSC: 12D05 Polynomials in real and complex fields: factorization 15A63 Quadratic and bilinear forms, inner products 12E12 Equations in general fields 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) Keywords:symmetric forms; Hermitian forms; algebraic equations; survey; Hermite-Jacobi method; Bezoutiants; Hermite method; separation of complex roots × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Achiezer N., Jzv. Akad. Nauk S.S.S.R. pp 1169– (1931) [2] Baur I., Math, Ann. 50 (6) pp 241– (1898) · JFM 29.0071.02 · doi:10.1007/BF01448065 [3] Biehler, J. Reine Angew.Math. 87 (27) pp 350– (1879) [4] Borchardt C., J. Math. Pures Appl. 12 (5) pp 50– (1847) [5] Borchardt C., J. Reine Angew. Math. 53 (5) pp 281– (1857) · ERAM 053.1400cj · doi:10.1515/crll.1857.53.281 [6] Borchardt C., J. Reine Angew. Math. 53 (5) pp 367– (1857) · ERAM 053.1410cj · doi:10.1515/crll.1857.53.367 [7] Brioschi F., Nouv. Ann. de Math. 15 (10) pp 264– (1856) [8] Carathéodory C., Rend. Circ. Mat. Palermo 32 (36) pp 193– (1911) · JFM 42.0429.01 · doi:10.1007/BF03014795 [9] Cauchy A., J. Ecole Polytechn. 15 (17) pp 176– (1737) [10] Cayley A., J. Math. Pures Appl. 13 (24) (1737) [11] Cayley A., J. Reine Angew. Math. 53 (13) pp 366– (1857) · ERAM 053.1409cj · doi:10.1515/crll.1857.53.366 [12] Cohn A., Math. Z. 14 (30) pp 110– (1922) · JFM 48.0083.01 · doi:10.1007/BF01215894 [13] Darboux G., Bull. Sci. Math. 8 (17) pp 92– (1875) [14] Eneström G., Kass. Öfversigt af Kongl. Vetenskaps-Akademiens Förhandingar 50 (34) pp 405– (1893) [15] Eneström G., Tôhoku Math. J. 18 (34) pp 34– (1920) [16] Frobenius G., S-B. preuss. Akad.Wiss. 18 (34) pp 241– (1894) [17] Fujiwara M., Töhoku Math. J 8 (5) pp 78– (1915) [18] Fujiwara M., Math. Z. 24 (25) pp 161– (1926) · JFM 51.0098.01 · doi:10.1007/BF01216772 [19] Fujiwara M., Japan. J. Math. 3 pp 9– (1925) [20] Fujiwara M., Tóhoku Math. J. 25 (29) (1925) [21] Grave D., Hesse, Univ. Izv. Kiev 3 (6) pp 1– (1903) [22] Grave, D. 1914.Elements of Higher Algebra, Vol. 691, 23,27–29. Kiev. (in Russian) [23] Gundelfinger S., J. Reine Angew. Math. 91 (6) pp 221– (1881) [24] Gursa, E.Course of mathematical analysis30 part I, (in Russian) [25] Hattendorff K., Nachr. Ges. Wiss. 91 (10) pp 779– (1873) [26] Hattendorff K., 1 Aufl., Leipzig 663 (10) pp 17– (1929) [27] Herglotz G., Ber. Verh.saechs. Akad., Leipzi 8 (36) pp 501– (1911) [28] Herglotz G., Math. Z. 19 (31) pp 26– (1923) [29] Hermite C., C.R. Acad. Sci. Paris 35 (6) pp 52– (1852) [30] Hermite C., C. R. Acad. Sci., Paris 36 (6) pp 294– (1853) [31] Hermite C., J. Reine Angew. Math. 52 (6) pp 39– (1856) · ERAM 052.1365cj · doi:10.1515/crll.1856.52.39 [32] Hermite C., Bull. Soc. Math. France 7 (27) pp 128– (1879) [33] Hurwitz A., Math. Ann. 46 (27) pp 273– (1895) · JFM 26.0119.03 · doi:10.1007/BF01446812 [34] Jacobi G., J. Reine Angew. Math. 14 (12) pp 281– (1835) · ERAM 014.0522cj · doi:10.1515/crll.1835.14.281 [35] Jacobi G., J. Reine Angew Math. 15 (24) pp 285– (1836) · ERAM 015.0513cj · doi:10.1515/crll.1836.15.285 [36] Jacobi G., J. Reine Angew. Math. 53 (6) pp 265– (1857) · ERAM 053.1397cj · doi:10.1515/crll.1857.53.265 [37] Jacobi G., J. Reine Angew. Math. 53 (7) pp 275– (1857) · ERAM 053.1399cj · doi:10.1515/crll.1857.53.275 [38] Jamamoto J., Tohoku Math. J. 33 (20) (1930) [39] Jentzsch R., Arch. d. Math. u. Phys. 17 (27) pp 105– (1910) [40] Joachimstal F., J. Reine Angew. Math. 48 (8) pp 386– (1854) · ERAM 048.1300cj · doi:10.1515/crll.1854.48.386 [41] Kakeya S., Tbhoku Math. J. 2 (34) pp 140– (1912) [42] Krein M., Rec. Math. 40 (29) pp 271– (1933) [43] Krein M., Comm. de la Soc. Math, de Kharkov 10 (23) pp 33– (1934) [44] Kronecker L., C. R. Acad. ScL Paris 68 (10) pp 1078– (1869) [45] Kronecker, Monatsber. preuss. Akad. Wiss pp 117– (1873) [46] Lagrange, Qeuvres 24 pp 3– (1867) [47] Laguerre E., J, Reine Angew. Math. 89 (27) pp 339– (1880) [48] Landau, E. 1916.Darstellung und Begruendung einiger neuerer Ergebnisse der Funktionentheorie, 110Berlin: 1 Aufl. [49] Landau, E. 1929.Darstellung und Begruendung einiger neuerer Ergebnisse der Funktionentheorie, Vol. 34, 122Berlin: 2 Aufl. [50] Liénard A., Revue de math. spéc. 21 (29) pp 153–155– (1911) [51] Liénard et A., J. Math. Pures Appl. 10 (25) pp 291– (1914) [52] Markov A., Trans. Acad. Sci. 74 (21) pp 1– (1894) [53] Markov A., Trans. St. Petersburg Math. Soc. 26 (33) (1899) [54] 1899.Some Generalizations of the Same Theorem, Vol. 33, 36–40. Ibid. (in Russian) [55] Nevanlinna R., Ann. Acaa Sc. Fenn. 18 (36) pp 1– (1922) [56] Nevanlinna, R. 1929.Ueber beschraenkte analytische Funktionen, A Vol. 32, 1–75. Ibid. 7 [57] Orlando L., Atti. Accad. Naz. Lincei 19 (29) pp 801– (1910) [58] Orlando, L. 1910. ”Nuovo osservazioni sul problema di Hurwitz”. Vol. 19, Ibid. · JFM 41.0125.03 [59] Orlando, L. 1910.Sull’equazione alle semisomme e sul teorema di Hurwitz, Vol. 19:2, 317–321. Ibid. · JFM 41.0125.04 [60] Orlando, L. 1910.Sopra alcune question relative al problema di Hurwitz, Vol. 19:2, 430–434. Ibid. · JFM 41.0125.05 [61] Orlando, L. 1911.Sulla dimonstrazione elementare del teorema di Hurwitz, Vol. 20:1, 742–745. Ibid. · JFM 42.0111.03 [62] Orlando L., Math. Ann. 71 pp 233– (1911) · JFM 42.0111.04 · doi:10.1007/BF01456650 [63] Petr K., Bull intern. 11 pp 14– (1906) [64] Pick G., Math. Ann. 77 (35) pp 7– (1916) [65] Rouché E., J. Ecole Polytechn 22 (30) pp 217– (1862) [66] Routh E., A Treatise on the Stability of a Given State of Motion · JFM 17.0315.02 [67] Routh, E. 1892.The Advanced Part of a Treatise on the Dynamics of a System of Rigid BodiesVol. 431, 29–29. London Part II [68] Schur I., J. Reine Angew. Math. 148 (30) pp 122– (1918) [69] Schur I., Z. Angew. Math. Mech. 1 (29) pp 307– (1921) · JFM 48.0082.03 · doi:10.1002/zamm.19210010405 [70] Stodola A., Schweizer Bau-Zeitung 23 (29) (1921) [71] Sturm C., Bull de Ferussac 11 (1829) [72] Sturm C., Mémoires présentés par divers savants á l’ Acad. sci. France 6 pp 273– (1835) [73] Sturm C., a German translation with notes by A. Loewy is available in Ostwalds ”Klassiker der exakten Wiss.” 143 [74] Sturm C., Leipzig 66 pp 5– (1904) [75] Sturm et C, J. Math. Pures Appl. 1 (29) pp 278– (1836) [76] Sturm C., J. Math. Pures Appl. 1 (27) pp 290– (1836) [77] Sturm C., J. Math. Pures Appl. 7 (10) pp 356– (1842) [78] Sylvester J., Philos. Mag. 15 (7) pp 429– (1839) [79] Sylvester J., Philos. Mag. 4 (6) pp 138– (1852) [80] Sylvester J., Philos. Trans. Roy. Soc. London 143 (7) pp 407– (1853) · doi:10.1098/rstl.1853.0018 [81] Sylvester J., Philos. Mag. 446 (23) (1853) [82] Takagi T., Japan. J. Math. 1 (37) pp 83– (1924) [83] Takagi T., Japan. J. Math. 2 (37) pp 13– (1925) [84] Fujiwara M., Tôhoku Math. J. 25 (37) pp 27– (1925) [85] Grommer J., J. Reine Angew. Math., Bd. 144 (37) (1914) [86] Krawchuk M., Transact, of the Physics and Mathematics Department (1) (1914) [87] Kritikos N., Math. Ann., Bd. 81 (1) (1914) [88] Tschebotareff N., Math. Ann. Bd. 99 (1) pp 660– (1928) · JFM 54.0351.02 · doi:10.1007/BF01459119 [89] Bauer, Bieberbaeh. 1928.Vortesungen ueber Algebra334–334. Leipzig [90] Fricke, R. 1924.Lehrbuch der Algebra468–468. Braunschweig · JFM 24.0395.01 [91] Grave, D. 1914.Elements of Higher Algebra691–691. Kiev (in Russian) [92] Haupt, O. 1929.Einfuehrung in die Algebra663–663. Leipzig B. B. I, II [93] Netto, E. 1896.Vorlesungen ueber Algebra388–388. Leipzig B. I. [94] Pascal, E. 1910. ”Repertorium der hoeheren Analysis”. 527–527. Leipzig und Berlin B. 1, 2, Aufl. [95] Serret, J. 1866. ”Cours d’Falgébre supérieure”. 613–613. Paris t.I. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.