Computing the topological degree of a mapping inR ⁿ

Summary

LetP be a connectedn-dimensional polyhedron, and let

$$b(P) = \sum\limits_{j = 1}^m {t_j [Y_1^{(j)} ...Y_n^{(j)} ]}$$

((1))

be the oriented boundary ofP in terms of orientedn−1 simplexest _j[Y ₁ ^(j) ...Y _n ^(j)], whereY _i ^(j) is a vertex of a simplex andt _j=±1. LetF=(f ¹, ...,f ⁿ) be a vector of real, continuous functions defined onP, and letF≠θ≡(0, ..., 0) onb (P). Assume that for 1<μ≦n, andΦ _μ=(ϕ¹, ..., ϕ^π) where ϕⁱ=f ^ji,j _k≠j _l ifk≠l, the setsS(A _μ)={X∈b(P:Φ _μ (X/|Φ _μ(X)|=A _μ}∩H _μ (andb (P)−S(A _μ) consist of a finite number of connected subsets ofb(P), for all vectorsA _μ=(±1, 0, ..., 0), (0, ±1, 0, ..., 0), ..., (0, ..., 0, ±1) and for all μ−1 dimensional simplexesH _μ onb(P). It is shown that ifm is sufficiently large, and

sufficiently small, thend (F, P, θ), the topological degree ofF at θ relative toP, is given by

$$d(F,P,\theta ) = \frac{1}{{2^n n!}}\sum\limits_{j = 1}^m {t_j \Delta ({\text{sgn }}F(Y_1^{(j)} ,{\text{ }}...{\text{, sgn }}F(Y_n^{(j)} ))}$$

((2))

where thet _j andY _i ^(j) are the same as those in (1), where sgnF=(sgnf ¹, ..., sgnf ⁿ), where for a real, sgna=1,0 or −1 ifa>0, =0 or <0 respectively, and where Δ(B ₁, ...,B _n) denotes the determinant of then×n matrix withi'th rowB _i. An algorithm is given for computingd(F, P, θ) using (2), and the use of (2) is illustrated in examples.