Problem:

Given F : B ⊂ IR^{N} → IR^{n}.

Determine Z(F) := Set of zeros of F.

Assumptions

• B is n-dimensional smooth manifold in IR^{N}.

• F is smooth.

• Z(F) finite, all zeros nondegenerate, i.e. F is transverse to {0}.

• No zeros on the boundary ∂B of B.

Solution by trajectory method

• Define implicitely a 1-dimensional set K(F) such that Z(F) ⊂ K.

• Trace K numerically, stringing all the zeros encountered.

How to define K: Take V ⊂ IR^{n}, 1-dimensional, such that 0 ∈ V, and consider F^{−1}(V) ⊂ B. Under slight regularity conditions this will be a set of “curves” in B, which contains Z(F).

How should V be choosen? There are obviously good choices, namely such that Z(F) is contained in a single component of F^{−1}(V) !

Definition Let V be a k-dimensional linear subspace in IR^{n}. Then we call

F_{V} (F) := F^{−1}(V)

a k-Newton leaf for F. ◇

Let V, W be two linear subspaces in IR^{n}. Then:

• If V ⊂ W, then F_{V} (F) ⊂ F_{W}(F).

• F_{{0}}(F) = Z(F) ⊂ F_{V} (F) of IR^{n}.

• F_{V} (F) ∩ F_{W}(F) = F_{V ∩W}(F)

• V ∩ W = {0} ⟹ F_{V} (F) ∩ F_{W}(F) = Z(F). We call a 1-Newton leaf a Newton trajectory.

Problem in application: Newton leaves may be disconnected!

Why the name Newton leaf? Newtons method: x_{n+1} := x_{n}−t_{n}F′(x_{n})^{−1}F (x_{n}). Continuous version: ẋ = −F′(x)^{−1}F (x). Desingularized version: ẋ = −F̃′(x)F (x). If a trajectory starts on a particular Newton leaf, it stays on it.

Practical aspect: Since any Newton leaf contains all zeros of F, we can reduce the n-di-mensional problem of calculating Z(F) to a kdimensional (ideally a 1-dimensional) one. So, for practical application we need V such that • V has low dimension, and

• Z(F) contained in single component of F_{V} (F). ⟹ Study the geometrical and topological properties of Newton leaves.

• Through any point x ∈ B \ Z(F) passes exactly one Newton trajectory.

• F_{V} (F) is a manifold, if F is transverse to V.

• Bifurcations are possible in points, where F is not transverse to V. Bifurcation points for Newton trajectories are called extraneous singularities. (The structure of the set ExS of these points has been studied by Jongen, Jonker and Twilt.)

• Newton leaves constitute a “foliation”.

Some pictures

4

Theorem 1 Let 1 ≤ k ≤ n − 1. Then any compact component of a k-Newton leaf for F which does not intersect ∂B contains an even number of zeros of F. Theorem 2 Let F_{V} (F) be a k-leaf disjoint from ∂B, 1 ≤ k ≤ n. Let U ⊂ F_{V} (F) be bounded, connected and open in F_{V} (F). Let the boundary ∂_{F_{V} (F)}U be contained in the union of finitely many Newton leaf components for F and suppose this boundary contains no zeros of F. Then

1) U contains an even number of zeros of F.

2) If ∂U ≠ ∅ and U contains zeros of F, then U contains an extraneous singularity of F.

3) If U contains no zeros and no extraneous singularities, then U and \overline{U} admit fixed point free maps.

Examples:

In application we do not want Newton leaves without zeros. How do these look like? If ExS = ∅ they cannot be ≃ S^{n−1} because of 3) above. But how about the torus S^{1} × S^{1}? Generally: If ExS(F) = ∅, can there be compact (n−1)-leaves containing no zeros? Theorem 3 Let F_{V} (F) ⊂ IR^{n} be a compact, regular component of a k-Newton leaf for F : IR^{n} → IR^{n} containing no zeros. Let τ : IR^{n} \ Z(F) → S^{n−1} be the trajectory map x ↦ F (x)/∥F (x)∥.

Then τ|_{F_{V} (F)} : F_{V} (F) → V ∩ S^{n−1} ≃ S^{k−1} is onto for k > 1 and constant for k = 1.

From this, connections to 1-sphere-bundles over S^{k−1} and to the homotopy groups π_{k−2}(O_{2}) give:

Such F_{V} (F) are ≃ S^{1} for k = 1 and ≃ S^{1} × S^{k−1} for 2 ≤ k ≠ 3. For k = 3 the situation is more complicated because of the existence of the Hopf-fibering S^{3} → S^{2}. But we have more structure: Consider the exterior of F_{V} (F) and vary V…

Further connections with algebraic topology etc. Results on connective structure between k-leaves and zeros of F.

Summary:

• k-Newton leaves are inverse images of k-di-mensional linear subspaces. Under slight regularity conditions on F, nearly all k-Newton leaves are k-dimensional submanifolds. Bifurcations may occur in extraneous singularities.

• Every Newton leaf contains all zeros of F. Thus Newton leaves can lead to a substantial reduction of the problem dimension.

• Newton leafs constitute a foliation: The settheoretic relations among the linear subspaces in IR^{n} carry over to relations among Newton leaves.

• Newton leaves have connections to global optimization, differential and algebraic topology (especially sphere- and k-plane bundles over spheres and projective spaces and generalized fiberings by compact and noncompact manifolds), global analysis (like global implicit function theorems, etc).

• The study of Newton leaves has just begun. There are many open questions and interesting generalizations; but also a number of nice and interesting results available.