A note on solutions of nonlinear equations with singular Jacobian matrices (Q1089746)

Let \(x^*\) be a solution of the nonlinear equation \(F(x)=0\) where the Jacobian of \(F: R^ n\to R^ n\) has rank n-2. The author considers an unfolded, augmented equation of the form \[ g(x)= \begin{pmatrix} c F(x)- \beta_ 1e_ i- \beta_ 2e_ j \\ F_ x(x)h \\ e^ T_ 1h-1 \\ e^ T_ 2h \end{pmatrix} \tag{1} \] and shows that, under appropriate assumptions and for suitable parameters, the Jacobian of G is nonsingular at \(x^*\) and hence (1) can be used to compute this solution. Such unfoldings and augmentations of singular problems have been studied extensively in connection with numerical bifurcation problems, but the author does not refer to that literature.