Two remarks on continuation procedures for solving some nonlinear equations

Let F be a mapping from R ⁿ⁺¹ into R ⁿ, x(t) be a parametrization for ∣t ? t₀ ∣ < c, of a smooth solution branch of the equation F(x) = 0, A(t) be the jacobian (n + 1) associated to a standard continuation method used for computing the solution branch. We study for a class of singularities at x(t₀) the behaviour, near t = t₀, of the determinant of A(t).