A Theorem on Homotopy Paths

Abstract
We consider the set of points x ∈ Rn+1 satisfying H(x) = 0, where H: Rn+1 → Rn is a C1 function and 0 is a regular value. This set, H−1(0), is a C1 one-dimensional manifold, and each component can be described by a curve x(θ). In this note a theorem is proved which is directly related to and motivated by a result due to Eaves and Scarf on piecewise linear functions. This theorem relates the signs of the derivatives xt(θ) to the signs of the determinants of submatrices of the Jacobian matrix H′. Applications to solving nonlinear equations are given.

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