Geodesic
The divided differences in theory of differential-difference equatians and in theory of groups
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 3, pp. 25-48 
A. A. Korobov. The divided differences in theory of differential-difference equatians and in theory of groups. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 3, pp. 25-48. http://geodesic.mathdoc.fr/item/VNGU_2006_6_3_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Complete description of nonlinear groups of all automorphisms of finitely generated relatively free groups with a nontrivial law is gave. It is shown that finitely generated soluble-by-finite subgroups of such groups is cause of nonlinearity of these groups. A wide class of such subgroups for any finite rank of the relatively free group is gave. Method of decomposition of the polynomial with integral coefficients into simple (in $\mathbb{Q}[x]$) factors, to prove the last result, is proposed. It is based on property of divided differences. The proof of last result of the article is based on another property of divided differences. It is proved that for a complex matrix A (of degree $n$) with real spectrum the matrix exponential $\exp(A)$ does not belong to linear span of matrices $A^0,A^1,\dots,A^{n-2}$.

[1] A. A. Korobov, “O nelineinykh podgruppakh grupp avtomorfizmov otnositelno svobodnykh grupp”, 2-oi Sib. kongress po prikl. i industr. mat. (Novosibirsk, 1996), 191

[2] O. M. Mateiko, O. I. Tavgen, “Lineinost grupp avtomorfizmov otnositelno svobodnykh grupp”, Mat. zametki, 58:3 (1995), 465–467 | MR | Zbl

[3] A. A. Korobov, “O starshem koeffitsiente matrichnoi eksponenty”, Sovremennye metody teorii funktsii i smezhnye problemy, Materialy konferentsii, VGU, Voronezh, 2005, 125

[4] A. A. Korobov, “Algebraicheskii metod dlya resheniya odnoi zadachi optimalnogo upravleniya”, Algebra i teoriya chisel, 5-aya Mezhd. konf., TGPU, Tula, 2003, 139–140

[5] A. A. Korobov, “Ob algebraicheskom metode dlya resheniya novoi zadachi optimalnogo upravleniya”, Vychisl. tekhnologii, 8:4 (2003), 283–289 | Zbl

[6] A. A. Korobov, “Nekotorye usloviya tochechnoi vyrozhdennosti dlya lineinykh sistem s zapazdyvaniem”, Kompyuternaya logika, algebra i intellektualnoe upravlenie, Trudy Vseros. nauchn. shk., Ir. VTs, Irkutsk, 1995, 230–244

[7] Yu. I. Merzlyakov, Ratsionalnye gruppy, Nauka, M., 1987

[8] Kh. Neiman, Mnogoobraziya grupp, Mir, M., 1969

[9] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, Nauka, M., 1982

[10] G. S. M. Kokseter, U. O. Dzh. Mozer, Porozhdayuschie elementy i opredelyayuschie sootnosheniya diskretnykh grupp, Nauka, M., 1980

[11] R. Lidl, G. Niderraiter, Konechnye polya, Mir, M., 1988

[12] A. G. Kurosh, Kurs vysshei algebry, Gostekhizdat, M., 1946

[13] M. V. Yakovkin, Chislennaya teoriya privodimosti mnogochlenov, AN SSSR, M., 1959

[14] D. K. Faddeev, I. S. Sominskii, Sbornik zadach po vysshei algebre, Nauka, M., 1968

[15] D. K. Fadeev, Lektsii po algebre, Lan, Sankt-Peterburg, 2002

[16] N. S. Bakhvalov, N. P. Zhidkov, G. M. Kobelkov, Chislennye metody, Nauka, M., 2004 | MR

[17] I. V. Proskuryakov, Sbornik zadach po lineinoi algebre, Nauka, M., 1974

[18] J. R. J. Groves, “Varieties of soluble groups and a dichotomy of P. Hall”, Bull. Austral. Math. Soc., 5:3 (1971), 391–410 | DOI | MR | Zbl

[19] J. A. Chang, H. J. Godwin, “A table of irreducible polynomials and theis exponents”, Proc. Cambridge Philos. Soc., 65 (1969), 513–522 | DOI | MR | Zbl

[20] H. G. Zimmer, Computational Problems, Methods and Results in Algebraic Number Theory, Springer-Verlag, Berlin–Heidelberg–New York, 1972 | MR | Zbl

[21] Runge, “Über die Zerlegung ganzzahliger Functionen in Irreductibile Factoren”, J. reine angew. Math., 99 (1885), 89–97

[22] M. Mandl, “Über die Zerlegung ganzer, ganzzahliger Functionen in Irreductibile Factoren”, J. reine angew. Math., 113 (1894), 252–261

[23] O. Perron, “Neue Kriterien für die Irreduzibilität algebraischer Gleichungen”, J. reine angew. Math., 132 (1907), 288–307 | Zbl

[24] E. S. Selmer, “On the irreducibility of certain trinomials”, Math. Scand., 4 (1956), 287–302 | MR | Zbl

[25] N. Obrechkoff, Zeros of polynomials, Marin Drinov Academic Publishing House, Sofia, 2003

[26] W. Ljunggren, “On the irreducibility of certain trinamials and quadrinomials”, Math. Scand., 8 (1960), 81–96 | MR