Abstract
In this paper, we study a local property of the zero set of a differentiable map F : R^<n+d> → R^n. We prove that, under a regular value condition, for each x∈ F^<-1>(o), there exist a neighborhood U of x and a sign c ∈ {-1, 1} such that sign det[x(p)^T_<σ_d> = c ・ sgnσ・ sign det D_xF(x(p))_<σ^n> for all permutation σ of degree (n + d), where p is a d-dimensional parametrization parameter vector of the zero set F^<-1>(o) in an open subset V of R^d and [x(p) ^T__<σ_d>] := (∂x_j/∂p_l)T(j ∈ σ^<-1>(n + 1,…, n+ d), l ∈ {1,…, d}), D_xF(x(p))_<σ^n> = [∂F_i(x(p))/∂x_k](i ∈ {1,…,n},k ∈ σ^<-1>(1,…,n)). This results naturally leads to an index theory. We show a local property of the change of the Morse index and the orientation of critical point set w.r.t. the multiparametric function f :R^<n+d> → R. Finally, we discuss the change of the stationary index of the equality constrained multiparametric nonlinear programs.
