Certain solutions to a system of scalar equations with a vector parameter (Q1335907)

Let \(f: \mathbb{C}^{n+ m}\mapsto \mathbb{C}^ n\) where \(f= (f_ 1,\dots, f_ n)\) is a mapping, analytic at the origin and let the Jacobian of \(f\) vanish at zero, \(f(0)= 0\). The paper studies the problem of finding ``small solutions'', \(z= z(u)\), where \(\| z(u)\|\to 0\) as \(\| u\|\to 0\), to the system (1) \(f_ i(z_ 1,\dots, z_ n, u_ 1,\dots, u_ m)= 0\) \((1\leq i\leq n)\) and how these solutions are concurrently derived from the ``small solutions'' of the system (2) \(f_ i(z_ 1,\dots, z_ n, 0,\dots, u_ k, 0,\dots, 0)= 0\) (\(1\leq i\leq n\) and \(1\leq k\leq m\)). It is well known that components of each ``small solution'' to system (2) are representable as absolutely convergent fractional powers of the parameter \(u_ k\). Solutions to system (1) are constructed from simple solutions of system (2). Simple solutions requires \(u_ k\) have a non-zero Jacobian in a deleted neighborhood of zero. Assuming simple solutions to (2), the author constructs them. The process leads to the primary theorem of the paper whereby one assumes there is a \(k\in \mathbb{N}\) such that system (2) has a simple solution and an additional number of inequalities are satisfied. Then the substitution of simple solutions \(u_ k\), to be \(u_ k= v^ r_ k\), \(u_ i= v_ i v^{r_ i}_ k\), (\(1\leq i\leq m,\;i\neq k,\;r\in \mathbb{N},\;r_ i\in \mathbb{N}\)) yields a solution to system (1) where each component is representable as a power series. Several wonderful examples illustrating the results are given such as \[ \ddot X+ 2u_ 1 X+ (X)(1- X^ 2)= u_ 2\sin wt. \] {}.