Zur numerischen Bestimmung des Abbildungsgrades im \(R^ n\). I (Q1077133)

This paper presents and proves two formulas for the topological degree of a map \(\Phi\) from an n-dimensional polyhedron \(P^ n\) into \(R^ n\). The first formula is based on projecting the image of \(\Phi\) onto the unit n- cube, and depends upon a technical property of a simplicial triangulation of \(P^ n\), and upon computing the algebraic signs of determinants of matrices whose entries are \(\pm 1\). It is: \[ \deg (\Phi,\quad int(P^ n))=(1/k)\sum sgn\{\det [sgn(\Phi (a^ 1)),...,sgn(\Phi (a^ n))]\} \] where the sum runs over the k (n-1)-simplices \([a^ 1,...,a^ n]\) in the appropriate triangulation of the boundary of \(P^ n\). The second formula depends upon projecting the image of \(\Phi\) into a construction called the ''unit octahedron'', for which there is a natural simplicial triangulation. It is: \[ \deg (\Phi,\quad int(P^ n))=2^{-n}\sum \det [g(a^ 1),...,g(a^ n)] \] where the sum is again over the (n-1)- simplices in an appropriate (different) triangulation of \(P^ n\), and the components of \(g(a^ i)\) are either 1, -1, or 0 depending upon the algebraic sign of the components of \(\Phi\) in the region surrounding \(a^ i\). (This is made precise in the paper, as is the meaning of ''appropriate''.) These formulas are new and interesting, as is the ''unit n-octahedron'' construction. However, the paper does not give numerical examples or an algorithm for automatic computation. Also, examples where it is desirable to explicitly compute the Brouwer degree are lacking. The proofs are somewhat involved and technical.