Using a new compactification (toroidal compactification) and desingularization, we obtain a complete characterization of monodromy at infinity for polynomial Newton system of arbitrary degree, in which we establish an equivalence between the monodromy and the non-existence of 1/2-fractional formal invariant curves. Combining the complete characterization with either Darboux integrability or algebraic reducibility of local centers, we obtain conditions for all cases of global center. Furthermore, investigating the asymptotic behavior of the period function of orbits near infinity, we prove the non-isochronicity for the global center, which consequently solves an open problem proposed by Conti.

We characterize isochronous centers of real planar polynomial quadratic-like Hamiltonian systems, i.e., planar differential systems x˙=−Hy(x,y), y˙=Hx(x,y) where H(x,y)=A(x)+B(x)y+C(x)y2 with A(x),B(x),C(x)∈R[x]. Given a polynomial quadratic-like Hamiltonian system with a center at the origin we can determine whether it is isochronous or not. Conversely, we can construct most of the polynomial quadratic-like Hamiltonian systems with an isochronous center, up to the determination of a particular Belyi function. Our results show the strong algebraic properties that are forced by the isochronicity of the center.

As shown in a previous paper, whenever a rational vector field on ℂ^n, n>2, is Liouvillian integrable, then it admits a first integral obtained by two successive integrations from a one-form with coefficients in a finite algebraic extension L of the rational function field K. In the present work we discuss and characterize exceptional vector fields in this class, for which – by definition – the choice L=K is not possible. In particular we show that exceptional vector field exist, giving explicit constructions in dimension three.

We consider complex rational vector fields in dimension n>2 (equivalently, differential forms of degree n-1 in n variables) which admit a Liouvillian first integral. Extending a classical result by Singer for n=2, our main result states that there exists a first integral which is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. The proof uses Puiseux series in a novel way to simplify computations. We also apply this method to give elementary proofs of Singer's theorem for rational one-forms, and of the Prelle-Singer theorem on elementary integrability of rational vector fields.

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

In this section we give another idea related to monodromy. This is the holonomy of the foliation P dy−Qdx = 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 Poincar´e return map for foliations.

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 n in terms of n.

In this chapter we consider the second mechanism which gives rise to centers in polynomial systems: the existence of an algebraic symmetry. After some brief preliminary comments, we shall show that such symmetries can be used to obtain a complete classification of centers in polynomial Li´enard systems.

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 [58], which we follow closely.

Any cubic generic Hamiltonian with at least one period annulus contained in its level curves, as we have explained in Subsection 1.2.1, can be transformed into the normal form

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 separatrix cycle. Monodromytype arguments play an implicit role in both of these approaches. The last section describes attempts to circumvent some of the computational complexity of the center focus problem by working over finite fields. It ends with 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 [59].

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.

Usually, the maximum of the number of limit cycles is denoted by H(n), and is called the Hilbert number. We recall that a limit cycle of system (1.1) is an isolated closed orbit.

In the study of the weak Hilbert’s 16th problem by using Abelian integrals, it is crucial to estimate the number of zeros of them. In this chapter, we introduce several methods to study the number of zeros of the Abelian integrals. In particular for the one defined in (1.11), which is related to the codimension 2 Bogdanov– Takens bifurcation problem, as we explained in Subsection 1.2.2.

Note that the theorem doesn’t say anything about the perturbation terms, it is purely topological. However, in generic cases of (i) we can conclude that the perturbation terms would have to be relatively exact. In Case (ii) we show later that the 1-form given by the perturbation terms will be the sum of a relatively exact term and the pull back of a 1-form on the factorized space.

In this chapter we will explain the relation between the number of zeros of the Abelian integrals and the number of limit cycles of the corresponding planar polynomial differential systems.

In this chapter, we consider one of the two main mechanisms which seem to underlie the existence of centers in polynomial vector fields. The background and history to this topic is covered in detail by Schlomiuk [117].

In this chapter we begin the second part of these notes, looking at some ideas based around the concept of monodromy. 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. If this path is a closed loop, the object often undergoes a non-trivial transformation when the parameter returns to its original value. Understanding these transformations can give great insight into the original problem.

We consider complex rational vector fields that admit a first integral whose logarithmic derivative lies in a finite extension of the rational function field K. In view of the Prelle-Singer theorem, these are the rational vector fields that admit an elementary first integral. Elementary integrable vector fields which are not Darboux integrable – thus the extension field is necessarily a proper extension of K – may be called exceptional by an observation in an earlier paper by Christopher et al. For dimension two we characterize all possible algebraic extension fields underlying the exceptional cases, provide a construction of all exceptional vector fields, and obtain some criteria that restrict the degree of L.

We consider a three dimensional complex polynomial, or rational, vector field (equivalently, a two-form in three variables) which admits a Liouvillian first integral. We prove that there exists a first integral whose differential is the product of a rational 1-form with a Darboux function, or there exists a Darboux Jacobi multiplier. Moreover, we prove that Liouvillian integrability always implies the existence of a first integral that is obtained by two successive integrations from a one-forms with coefficients in a finite algebraic extension of the rational function field.