Piecewise linear approximation of smooth compact fibers

• Authors: Eugene L. Allgower;Andrew J. Sommese
• Venue: Journal of Complexity
• Year: 2002

Let H: RN → Rm be a twice differentiable map. Let N := n+m. Assume that the fiber F := H-1(y0) of H over a point y0 ∈ Rm is a compact n-dimensional differentiable manifold and that ∂H is of maximal rank in a neighborhood of F. Works of Allgower and Gnutzmann, and of Allgower and Schmidt give an algorithm to find a global piecewise linear approximation of F based on a triangulation of RN. Several authors have given algorithms for obtaining a piecewise linear approximation of F in a way that does not depend on the triangulation in RN. This effects a reduction of the combinatorial complexity from O(N!) to O(n!), but the approximations are not global. In this paper, a probability one algorithm is given, which, given H, F, N, m as above, uses homotopy continuation to construct a differentiable map H˜: R2n+1 → Rn+1 with a compact differentiable manifold F:= H˜-1(0), which is diffeomorphic to F by an explicit diffeomorphism, and with ∂H˜ of maximal rank in a neighborhood of F˜. This reduces the problem when N 2n + 1 to the known piecewise linear algorithm in R2n+1 at the simpler expense of carrying out a well-behaved homotopy continuation starting at vertices of the simplices of the triangulation of a neighborhood of π(F) ⊂ R2n+1, where π: RN → R2n+1 is a generic linear projection. Consequently, a global algorithm is obtained with a combinatorial complexity of O((2n + 1)!).