Computing the topological degree of a mapping inRn

SummaryLetP be a connectedn-dimensional polyhedron, and let(1) $$b(P) = \sum\limits_{j = 1}^m {t_j [Y_1^{(j)} ...Y_n^{(j)} ]}$$ be the oriented boundary ofP in terms of orientedn−1 simplexestj[Y1(j) ...Yn(j)], whereYi(j) is a vertex of a simplex andtj=±1. LetF=(f1, ...,fn) be a vector of real, continuous functions defined onP, and letF≠θ≡(0, ..., 0) onb (P). Assume that for 1<μ≦n, andΦμ=(ϕ1, ..., ϕπ) where ϕi=fji,jk≠jl ifk≠l, the setsS(Aμ)={X∈b(P:Φμ (X/|Φμ(X)|=Aμ}∩Hμ (andb (P)−S(Aμ) consist of a finite number of connected subsets ofb(P), for all vectorsAμ=(±1, 0, ..., 0), (0, ±1, 0, ..., 0), ..., (0, ..., 0, ±1) and for all μ−1 dimensional simplexesHμ 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(2) $$d(F,P,\theta ) = \frac{1}{{2^n n!}}\sum\limits_{j = 1}^m {t_j \Delta ({\text{sgn }}F(Y_1^{(j)} ,{\text{ }}...{\text{, sgn }}F(Y_n^{(j)} ))}$$ where thetj andYi(j) are the same as those in (1), where sgnF=(sgnf1, ..., sgnfn), where for a real, sgna=1,0 or −1 ifa>0, =0 or <0 respectively, and where Δ(B1, ...,Bn) denotes the determinant of then×n matrix withi'th rowBi. An algorithm is given for computingd(F, P, θ) using (2), and the use of (2) is illustrated in examples.