A generalization of the implicit function theorem for mappings from \({\mathcal R}^{n+1}\) into \({\mathcal R}^ n\) and its applications
This paper is devoted to the proof of a generalization of the implicit function theorem to mappings from \(R^{n+1}\) into \(R^ n\). Under some appropriate nondegeneracy condition it is shown that the zero set of such a mapping around the origin (say) is the union of a finite number of curves. An upper bound for this number is derived from Bezout's theorem. In comparison with earlier results in the same direction by various authors, the main improvement is the observation that the curves actually have one more degree of regularity than expected at the origin. in the applications to one-parameter bifurcation problems, this improvement is important because it allows one to assumme one fewer degree of regularity of the data on the one hand and leads to a homogeneous exposition of all the cases on the other hand. Further, the ''almost'' constructive proofs have led to new algorithms of computation of the bifurcated branches which are described and studied elsewhere.
scientific article; zbMATH DE number 3848258
scientific article