Part I: Background and Existing Work 8
4.2 Brouwer Degree
The Brouwer degree is, broadly speaking, a particular instance of the topological degree where the topological space is specified as Rn.
Conceptually, the degreedeg(F, D, y)of a mappingF :Rn→Rn, whereDis a bounded open subset of Rn and y ∈ Rn, is the excess of the number of points of F−1(y)∩D at which the Jacobian determinant of F is positive over those at which it is negative. An integer result, it can be regarded (and utilised) as an estimate to the number of solutions toF(x) = y inD. In particular, a non-zero result guarantees the presence of at least one such solution. Given a system of equationsF = 0and a boxX,deg(F,X,0)thus estimates the number solutions (i.e. roots of F) within X.
Definition 4.1 (Brouwer Degree). For some n, let D be a bounded open subset of Rn, F :cl(D) → Rn be continuous, andy ∈Rn\F(∂D). Under these conditions, the Brouwer (topological) degree is a function
deg:{(F, D, y)} →Z (4.1) satisfying the following:
• Identity function property: deg(id, D, y) = 1 if y∈D.
• Additive property: If D =D1 ∪D2, where D1 and D2 are disjoint, or if D1 and D2
are disjoint sets such thaty6∈ F(D\(D1∪D2)), then
deg(F, D, y) = deg(F, D1, y) +deg(F, D2, y). (4.2)
• Homotopic invariance property: If F,G :cl(D)→Rn are continuous, and homotopic with homotopy Ht(x) =H(t, x), then if Ht−1(y) does not intersect the boundary ofD for all values oft∈[0,1], deg(F, D, y) =deg(G, D, y).
See Subsection 5.3.5 for a definition and further discussion of homotopies. It can be shown (and should be clear) that these conditions define the Brouwer degree in a meaningful way for all possible choices of argument satisfying the stated restrictions. Where the restrictions are violated, for exampley∈ F(∂D), it is undefined.
4 Topological Degree
Complex Functions
The above definition for Brouwer degree only caters for real-valued functions of a real variable in a finite number of dimensions. However, it can easily be adapted to cater for complex functions F∗ :Cn→ Cn. Such a function F∗ may be regarded as a function from R2n to R2n and then the degree is defined as before. Indeed, complex functions have an additional beneficial property when treated in this fashion: an analytic function will be orientation preserving(i.e. its Jacobian determinant is always non-negative). Therefore the degree will simply count the number of solutions inD, with multiplicity.
4.2.1 Properties of the Brouwer Degree
The following are fundamental properties of the Brouwer degree. They can be of use in guiding the construction of algorithms to compute the degree.
Additive Property
This property (4.2) holds axiomatically and provides that the topological degree is defined as a summation of values for all roots present in a given region.
Root-Counting Property
IfF(x) =y has no solutions in D, thendeg(F, D, y) = 0. Otherwise, if all the solutions of F(x) =y inD are non-singular andF has continuous derivatives, then deg(F, D, y) is the sum of the signs of the Jacobian determinants at all the solutions:
deg(F, D, y) = X
x∈F−1(y)
sgn(det(JF(x))). (4.3)
This property is meaningful for cases with continuously differentiable functions and non-singular solutions. In the case of non-singular solutions, the Brouwer degree is less well-defined
— here it merely indicates the existence of solutions.
Boundary Determinism
The Brouwer degreedeg(F, D, y)is determined by y and the values of F on the boundary of D only. This property is crucial since it means that the degree can (in principle) be computed merely by analysing the behaviour of the functionF on the boundary of a region, irrespective of the complexity of the behaviour ofF within the region in question.
Figure 4.1 illustrates this property for a simple systemf1 = 0, f2= 0 inR2; the ordering of the zero sets around the boundary of the region determines whether either an even or odd number of solutions are present, not the behaviour of the functions within the box. In the left-hand case, there must be an even number of solutions (since the degree sums to zero in this particular example, half have a positive Jacobian determinant and the other
4 Topological Degree
are zero solutions; in an alternative case (where the zero set of f1 is a dashed line) there might be two solutions. Conversely, in the right-hand case, there must be an odd number of solutions.
D D
f (x)=01 f (x)=01
f (x)=02 f (x)=02
Figure 4.1: Boundary determinism inR2: even and odd numbers of solutions.
Continuity
This property also holds axiomatically (see Definition 4.1, homotopic invariance property).
This provides that the Brouwer degreedeg(F, D, y)is an integer which is (locally) continuous with respect toF and y. In other words, it is constant for small peturbations in F andy.
Suppose that Y is a box which contains points y1 and y2. Then if F−1(Y) does not intersect the boundary ofD,deg(F, D, y1) =deg(F, D, y2).
This illustrates that a smooth change in either y or F, in such a way as to not move solutions toF(x) =y across the boundary ofD, leaves the degreedeg(F, D, y) unchanged.
In practice this means that sufficiently good numerical approximations toF and y will be enough to compute the degreeexactly, given sufficient precision. This property is completely ruined in the case where a solution lies on the boundary ofD, which is why these cases are explicitly excluded.
4.2.2 Example
This is a simple system of equations taken from Aberth [Abe94]. F :R3 →R3 is given by f1 = x21+x22−x3,
f2 = x22+x23−x1, f3 = x23+x21−x2.
F has roots at(0,0,0)and (12,12,12), the former with negative Jacobian determinant and the latter with positive Jacobian determinant. Hence:
• deg(F,([−14,14],[−14,14],[−14,14]),0) = −1. This (box) region encloses only the first root, so the degree is the sign of its Jacobian determinant, −1.
4 Topological Degree
• deg(F,([14,34],[14,34],[14,34]),0) = 1. This region encloses only the second root, so the degree is the sign of its Jacobian determinant,1.
• deg(F,([1,2],[1,2],[1,2]),0) = 0. This region encloses neither of the roots, so the degree is0.
• deg(F,([−1,1],[−1,1],[−1,1]),0) = 0. This region encloses both roots, so the degree is the sum of the signs of the Jacobian determinants,0. This example neatly illustrates a potential cause for confusion — it can be difficult to discern whether there are no roots, or roots with Jacobian determinants of opposite sign which cancel out.