Solutions of some quasianalytic equations
Volume 129 / 2022
Abstract
In this article we are interested in solving systems of equations $F_j(t,x) = 0$, $1\leq j\leq p$, for $C^\infty $ functions $F_j$ in a neighborhood of $(0,0)\in \mathbb R\times \mathbb R^p$, which are in a given differentiable quasianalytic system $\mathcal C$ (for example, a quasianalytic Denjoy–Carleman class or the class of infinitely differentiable functions definable in a polynomially bounded o-minimal structure). We suppose that the implicit function theorem is true in this system, that is, the equations $F_i(t,x) = 0$ have a unique solution $ x_j(t)$, $j=1,\ldots , p$, in a neighborhood at $ t=0 $ if the Jacobian of the functions $F_i$ with respect to the variables $ x_j$ is nonzero at $ x_j = t = 0$, $j=1,\ldots ,p$. This condition is not necessary for the equations $F_i =0$ to have a solution in the system. An example is the analytic system. A theorem of Artin can be used to give sufficient conditions under which a system of analytic equations has analytic solutions. In the case of a general quasianalytic system the theorem of Artin is not available. In this article weaker conditions will be given. We show that if $ F(t,x) = (F_1(t,x),\ldots ,F_p(t,x)) =0$ has a formal power series solution $ u(t) = ( u_1(t),\ldots , u_p(t))\in (\mathbb R[[t]])^p$, and $\textrm {det}\, D_xF(t, u(t))\neq 0$, then for each $j=1,\ldots ,p$, $ u_j(t)$ is the Taylor expansion at $0\in \mathbb R$ of a function in the system $\mathcal {C}$. We also treat the same problem in the case of more independent variables.
