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In this chapter I want to give a general background to the center-focus problem, and then to show why the problem is interesting: both in what it tells us about the distinctive algebraic features of polynomial vector fields, and also in the simple concrete estimates it gives of the number of limit cycles which can exist in these vector fields.

In this chapter, we consider one of the two main mechanisms which seem to underlie the existence of centers in polynomial vector fields. We only hint at the historical side, which is covered in detail by Schlomiuk [57].

In this chapter we want to prove that Darboux integrability corresponds to the notion of Liouvillian integrability, or “solution by quadratures”.

In this chapter we consider the second mechanism which gives rise to centers in polynomial systems: the existence of an algebraic symmetry.

In this chapter we give an extended example of a non-trivial classification of centers which involves both Darboux and symmetry mechanisms for producing a center. Further details can be found in [26], which we follow closely.

In this chapter we begin the second part of these notes, looking at some ideas based on the concept of . Very roughly, this is the study of how objects depending on a parameter, and which are locally constant in some sense, change as the parameter moves around a non-trivial path. This idea is particulary appropriate for the center-focus problem, as the essence of this problem is about trying to make global extensions of local information. For example, we might naïvely hope to be able to extend the local first integral at the origin to a global first integral. This is not possible in general, but even if we could do so, the first integral would certainly ramify as a global object. Our desire would then be to read off some important information about the system from this global ramification.

As is well known, the second part of Hilbert’s 16th problem is concerned with bounding the number of limit cycles in a polynomial system (1.2) of degree in terms of . This is a very hard problem, but Arnold has suggested a “Weak Hilbert’s 16th problem” which seems far more tractable: to find a bound on the number of limit cycles which can bifurcate from a first-order perturbation of a Hamiltonian system, where the Hamiltonian, , is a polynomial of degree + 1 and the perturbation terms, and are polynomials of degree .

We want to show that in the case of Hamiltonians of the form where () is a polynomial of degree , the existence of a tangential center implies that either is relatively exact, or the polynomial () is . That is, it can be expressed as a polynomial of a polynomial, () = (()), in a non-trivial way.

In this section we give another idea related to monodromy. This is the holonomy of the foliation = 0 associated to the system (1.2) in the neighborhood of an invariant curve. This object, roughly speaking, is the nonlinear analog to the monodromy of the solutions of a linear differential equation as they turn around a singular point. Alternatively, it can be thought of as a kind of Poincaré return map for foliations.

In this final chapter I want to mention briefly three other approaches to the general center-focus problem. In the first, we try to identify whole components of the center variety by finding their intersections with specific subsets of parameter space and then showing that the type of center is “rigid”. In the second approach, we try to see the consequences of a center on its bounding graphic. Monodromytype arguments play an implicit role in both of these approaches. The last section describes an experimental approach to the center-focus via intensive computations using modular arithmetic and an application of the Weil conjectures. It makes a fitting conclusion to our range of monodromy techniques, since the arithmetic analog of monodromy was an essential ingredient in Deligne’s proof of the Weil conjectures [27].