Generating properties of biparabolic invertible polynomial maps in three variables

Yu. Bodnarchuk

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2004), pp. 34-39.

Résumé

Invertible polynomial map of the standard 1-parabolic form $x_i\to f_i(x_1,\dots ,x_{n-1})$, $i$, $x_n\to \alpha x_n+h_n(x_1,\dots ,x_{n-1})$ is a natural generalization of a triangular map. To generalize the previous results about triangular and bitriangular maps, it is shown that the group of tame polynomial transformations $TG{A}_3$ is generated by an affine group $AG{L}_3$ and any nonlinear biparabolic map of the form $U_0\cdot q_1\cdot U_1\cdot q_2\cdot U_2,$ where $U_i$ are linear maps and both $q_i$ have the standard 1-parabolic form.

Bibliographie

[1] Shafarevich I., “On some Infinite Dimensional Groups, II”, Izv. AN USSR, Ser. math., 45 (1981), 214–226 | MR | Zbl

[2] Kac V., “Simple irreducible graded Lie algebrais of finite growth”, Izv. AN USSR, Ser. math., 32 (1969), 1323–1367 | MR

[3] Bodnarchuk Yu., “Some extreme properties of the affine group as an automorphism group of the affine space”, Contribution to General Algebra, 13 (2001), 15–29 | MR

[4] Shestakov I., Umirbaev U., The tame and the wild automorphisms of polynomial rings in three variables, Preprint São Paulo University, 1, 2002, 1–35 | MR

[5] Mortimer B., “Permutation Groups containing Affine Groups of the same degree”, J. of London Math. Soc., 15 (1977), 445–455 | DOI | MR | Zbl

[6] Bodnarchuk Yu., “Generating properties of triangular and bitriangular birational automorphisms of an affine space”, Dopovidi NAN Ukraine, 11 (2002), 7–22 | MR

[7] Bodnarchuk Yu., “On affine-split tame invertible polynomial maps in three variables”, Buletinul A.Ş. a R. M., Matematika, 2002, no. 2(39), 37–43 | MR | Zbl