Deformations and Inversion Formulas For Formal Automorphisms in Noncommutative Variables
| dc.creator | Zhao, Wenhua | |
| dc.date | 2005-09-06 | |
| dc.date | 2007-03-15 | |
| dc.date.accessioned | 2026-07-07T12:36:37Z | |
| dc.date.available | 2026-07-07T12:36:37Z | |
| dc.description | Let $z=(z_1, z_2, ..., z_n)$ be noncommutative free variables and $t$ a formal parameter which commutes with $z$. Let $k$ be any unital integral domain of any characteristic and $F_t(z)=z-H_t(z)$ with $H_t(z)\in {k[[t]]< < z >>}^{\times n}$ and the order $o(H_t(z))\geq 2$. Note that $F_t(z)$ can be viewed as a deformation of the formal map $F(z):=z-H_{t=1}(z)$ when it makes sense (for example, when $H_t(z)\in {k[t]< < z >>}^{\times n}$). The inverse map $G_t(z)$ of $F_t(z)$ can always be written as $G_t(z)=z+M_t(z)$ with $M_t(z)\in {k[[t]]< < z >>}^{\times n}$ and $o(M_t(z))\geq 2$. In this paper, we first derive the PDE's satisfied by $M_t(z)$ and $u(F_t), u(G_t)\in {k[[t]]< < z >>}$ with $u(z)\in {k< < z >>}$ in the general case as well as in the special case when $H_t(z)=tH(z)$ for some $H(z)\in {k< < z >>}^{\times n}$. We also show that the formal power series above are actually characterized by certain Cauchy problems of these PDE's. Secondly, we apply the derived PDE's to prove a recurrent inversion formula for formal maps in noncommutative variables. Finally, for the case char. $k=0$, we derive an expansion inversion formula by the planar binary rooted trees. | |
| dc.description | Latex, 30 pages. Following the referee's suggestion, an example has been added and fully discussed. Some references have been updated | |
| dc.identifier | https://arxiv.org/abs/math/0509130 | |
| dc.identifier | http://arxiv.org/abs/math/0509130 | |
| dc.identifier | Internat. J. Algebra Comput., 17 (2007), no. 2, 261--288. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/218156 | |
| dc.subject | General Mathematics | |
| dc.subject | Complex Variables | |
| dc.subject | 14R10, 32H02 | |
| dc.title | Deformations and Inversion Formulas For Formal Automorphisms in Noncommutative Variables | |
| dc.type | text |