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Enumerative Geometry versus Counting Problems of Low Degree Polynomial Vector Fields.

Dedicated to Professor Solomon Marcus with affection and gratitude.

1 Introduction

In this article we consider real autonomous differential systems

[mathematical expression not reproducible] (1.1)

where p, q [member of] R[x,y], i.e. p, q are polynomials in x, y over R and their associated vector fields

[mathematical expression not reproducible] (1.2)

We call degree of a system (1.1) (or of a vector field (1.2)) the integer n = max(deg p, deg q). In particular we call quadratic a differential system (1.1) with n = 2 and we denote by QS the class of all such systems. We call cubic a differential system (1.1) with n = 3 and we denote by CS the class of all such systems. A system (1.1) is non-degenerate if the two polynomials p(x,y),q(x,y) have no non-constant common factor.

Problems on these systems are easy to state and for this reason they are appealing. But they are at the same time extremely hard to solve. In fact one of the most elusive problem on Hilbert's list of 23 problems given at ICM in Paris in 1900 [34], is the second part of Hilbert's 16th problem, which from now on we shall simply denote by H16.

The purpose of this article is to introduce the perspective of Enumerative Geometry in the theory of low degree planar polynomial vector fields, searching to establish connections with this much larger and much older area of research, and to give a short survey of results as well as some open problems belonging to what may be called the initial stages of an Enumerative Geometry of planar polynomial vector fields.

In section 2 we consider the second part of H16 and other related hard open problems. These problems are dauntingly hard even for the case of quadratic differential systems. Thus even famous mathematicians like Petrovski and Landis "proved" a theorem in 1958 about these systems in [50] which failed to be true as it was shown in two independent works, one by Chen and Wang [22] (1979) and the other by Shi Song Ling in [69] (1980). Furthermore statements which fail to be true continue to appear in the literature on these systems. The purpose of this section is to discuss some facts about these problems which show how hard these problems really are.

In section 3 we give a quick overview of Enumerative Geometry and its history and state Hilbert's 15th problem on Schubert calculus.

In section 4 we propose a new interpretation for Hilbert's words "the same method of continuous variation of coefficients" in Hilbert's statement of his 16th problem.

In section 5 we view a number of questions on low degree planar polynomial vector fields from the perspective of Enumerative Geometry. The results obtained form a body of work which initiates an Enumerative Geometry of low degree planar polynomial differential systems. We present results of an algebraic nature as well as some "counting problems" in this area of research whose solution are either algebraic or algebraic up to a point and then need some analytic tools to be completely solved. The algebro-geometric results are connected with Darboux theory of integrability which provides us with Darboux or Liuovillian first integrals of the corresponding systems. We include a long list of references on these questions.

2 On the second part of Hilbert's 16th problem and other related hard open problems

Hilbert's 16th problem has two parts and its first part is on the topology of algebraic curves and surfaces. For a survey on the first part of this problem see Part I of [77]. In the second part, Hilbert says:

"In connection with this purely algebraic problem, I wish to bring forward a question which, it seems to me, may be attacked by the same method of continuous variation of coefficients, and whose answer is of corresponding value for the topology of families of curves defined by differential equations. This is the question as to the maximum number and position of Poincare boundary cycle (cycles limites) for a differential equation of the first order and degree of the form dy/dx = Y/X where X and Y are rational integral functions of the n-th degree in x, y."

To each such equation we can associate a differential system (1.1) where p and q are respectively X and Y and viceversa to such a system we can associate a differential equation. Hence the second part of H16 is also about such differential systems (1.1) or about vector fields (1.2). This problem has not been solved, even for the case of quadratic differential systems.

At the turn of the twenty-first century when this problem became 100 years old, Ilyashenko wrote an article with the impressive title "Centennial history of Hilbert's 16th problem" [38]. In fact there is hardly anything in this article about Hilbert's 16th problem, as originally stated by Hilbert. We do not even find Hilbert's statement of the problem. Can this be explained?

One way to explain this is exactly the sheer daunting difficulty of the problem. The centennial history of this problem would be very short if one did not choose to talk instead about other problems which were inspired by H16 and this is exactly what Ilyashenko chose to do. The following title would have been a great deal closer to the content of the article: "Centennial history of problems arising from the second part of Hilbert's 16th problem".

On behalf of the International Mathematical Union, Arnold asked a number of mathematicians to describe some great problems for the twenty first century. In response to this, Smale gave a list (see [70]) of 18 problems and his problem 13 on this list is H16 which he stated as follows:

Problem 13 on Smale's list [70]: Consider the differential equations in [R.sup.2]

[mathematical expression not reproducible] (2.1)

where p, q [member of] R[x,y], i.e. p, q are polynomials in x, y. Is there a bound K on the number of limit cycles of the form K [less than or equal to] [d.sup.q] where d is the maximum of the degrees of P and Q, and q is a universal constant?

Smale added: "This is a modern version of the second half of Hilbert's 16th problem. Except for the Riemann Hypothesis, it seems to be the most elusive problem of Hilbert's problems."

Thus Smale placed H16 next to the most celebrated and hardest problem in mathematics, the Riemann hypothesis.

Since H16 is an open problem even in its simplest case of quadratic vector fields, people looked for somewhat "simpler" problems and thus a number of offsprings of H16 appeared. Ilyashenko's article is essentially about these offsprings and presents a very good survey on most of them.

"Most of them" because there is at least one problem, the finiteness part of Hilbert's 16th problem for quadratic systems for which his presentation is flawed as we explain further below.

From the way Hilbert stated H16 it is clear that he believed that each polynomial differential system has a finite number of limit cycles, i.e. isolated periodic orbits in the set of periodic orbits. In fact this was not yet proven at that time. Thus the first problem arising from H16 is the individual finiteness problem which asks for a proof that any individual system (2.1) has a finite number of limit cycles.

The first individual finiteness result was obtained by Poincare in 1881 [51] (Theoreme XVII of Chapter VI, p. 60). But this result does not treat the general case. For a discussion of this result we refer the reader to [71] where Sotomayor proves the following version of Poincare's theorem: Polynomial differential systems whose compactifications have only hyperbolic equilibria and no graphics, are generic and have at most finitely many periodic orbits (for the notion of graphic see also [28]).

Dulac gave a "proof in the general case in 1921 ([27]) but Ilyashenko found an error in Dulac's proof (see [35]).

The individual finiteness theorem for the particular case of quadratic differential systems was proved by Chicone, Shafer and Bamon. Firstly Chicone and Shafer proved in [25] the individual finiteness theorem in the special case when the systems have only finite graphics and in [13] Bamon extended this result by including also infinite graphics, i.e. those with non-empty intersection with the line at infinity.

The individual finiteness theorem in the general case was finally proved independently by Ilyashenko in [37] and by Ecalle in [29]. Actually Ecalle and Ilyashenko proved a more general theorem saying that any analytic vector field on a closed surface has a finite number of limit cycles.

From the way he stated the problem it is clear that Hilbert also believed that not only do we have a finite number of limit cycles for each polynomial system but also that if we fix n then there is a finite bound for the number of limit cycles occurring in polynomial differential systems of degree n. Hilbert then asked to determine the maximum number of limit cycles in the family of systems which are of degree n. It is customary to denote this hypothetical maximum of the number of limit cycles occurring in polynomial systems of degree n by H(n) and to call this the Hilbert number.

Thus a second problem arising from Hilbert's 16th problem is the following: Prove the finiteness part of Hilbert's 16th problem, i.e. prove that for any natural number n there exists a natural number [N.sub.n] such that for every polynomial differential system (S) of the type (1.1) of degree n, we have LC(S) [less than or equal to] [N.sub.n] where LC{S) is the number of limit cycles of the system (S). Assuming the finiteness part holds then the Hilbert number H(n) is clearly the smallest one of the numbers [N.sub.n].

In his article "Hilbert's 16th problem" published in Nature [72] in 1987 Ian Stewart remarks: "A proof of finiteness of the number of limit cycles for polynomials of degree higher than two would be a major step towards solving Hilbert's 16th problem. But Hilbert asks for more: namely the exact number H(n)".

In fact the finiteness part of H16 is not even proved for quadratic differential systems. However a program for solving this problem was devised by Roussarie in [58]. This program was further delineated by Dumortier, Roussarie and Rousseau in [28]. Via a compactness argument, they reduced this problem to the problem of proving finite cyclicity of 121 graphics occurring in quadratic differential systems (for the notion cyclicity and of graphic see [28]). Twenty eight years have passed since this program was formulated and we still do not have a proof of the finiteness part of Hilbert's 16th problem for quadratic differential systems, this in spite of many articles written on this topic. At the beginning, from the 121 graphics, groups of graphics fell from the list due to proofs provided by the investigators. But as the singularities of these graphics became more and more degenerate, their finite cyclicities became much harder to prove. This gives us a measure of how hard it is to make progress on H16.

In [38] Ilyashenko gives a very brief summary of this program. He says: "Quadratic vector fields are relatively simple, amidst other polynomial vector fields. For instance, any closed phase curve of such a vector field is convex and contains no more than one singular point inside. It is realistic to try to list all the polycycles that may occur for quadratic vector fields and to prove their finite cyclicity. The first step in this program was done in [28], where a complete list of 121 polycycles that may occur for quadratic vector fields is presented."

Ilyashenko is highly regarded due to his outstanding work [37] and other results, as well as his contribution as a builder of a school. But his work and judgment in [38] is not flawless. In fact his statement mentioned above is flawed on several points and I shall highlight below, step by step, these points.

Firstly, Hilbert's 16th problem is a global problem: it is a statement about the whole class of vector fields of a fixed degree n. So in particular it is a global problem for the whole quadratic class, i.e. for vector fields of degree n = 2. In connection with this special case of Hilbert's 16th problem people often say that "quadratic systems are simple" and Ilyashenko is no exception. While it is true that quadratic systems are simple, this is entirely beside the point here because even in the finiteness part of Hilbert's 16th problem for quadratic systems we are not concerned with just individual quadratic systems but with the whole class of quadratic differential systems. As it turns out this class is a very complicated object of study, so complicated that Hilbert's 16th problem seems to be daunting even for this special case and as we have seen, even the finiteness part of this problem is still an open problem, after a period of 28 years of a lot of intense work, done by several scientists.

Secondly Ilyashenko's statement that "a complete list of 121 polycycles that may occur for quadratic vector fields is presented in [28]" is wrong in two ways. The authors of [28] call "polycycles" only those graphics which possess a first return map. Their list is of 121 graphics and not all of these graphics are polycycles in the above sense. Figure 1 and Figure 2 show two graphics, one of which is a polycycle while the other is not and it indicates the graphic ([F.sup.2.sub.3]) in [28].

Other examples of graphics which are not polycycles in [28] are ([F.sup.1.sub.2]), and ([I.sup.2.sub.29]) as well as many others of the 121 graphics in the list.

Thirdly, this is not a complete list of graphics which may occur in quadratic vector fields. Examples of graphics in quadratic systems not in the given list can easily be found in [64] in Pictures 5.26 and 5.29 as well as many others.

The centennial history of Hilbert's 16th problem in [38] not only mentions very few things about this problem as stated by Hilbert, but also even among the few things mentioned around this problem some are not at all correct. An article which pays a great deal more attention to Hilbert's thinking is [45]. Li Jibin starts by providing the full statement of Hilbert's 16th problem, including its first part on which Li gives an account in section 2. In section 3 Li considers H16. In section 5 we find the distinction between graphics and polycycles in Fig.9. The article is rather rich and includes a section 8 on the rate of growth of the potential Hilbert number H(n) with n.

Referring to the second part of Hilbert's 16th problem, Ilyashenko and Yakovenko wrote the following paragraph in [39]:

"One might ask why Hilbert had chosen polynomial families as the subject for investigation concerning limit cycles. Perhaps the reason was that the universal polynomial family is the only constructive family of line fields on the plane which extends to a family of line fields on the sphere."

We cannot be sure about the reasons which made Hilbert state his 16th problem about polynomial vector fields the way he stated it. We can only advance a hypothesis as the authors of the above paragraph have done and one can say they have a rather good argument. But is their hypothesis truly convincing? Not really, if one is aware of Hilbert's mathematical trajectory prior to 1900.

David Hilbert (1862-1943) did his first mathematical work in classical Invariant Theory, a major area of research in the second part of the 19th century. In classical invariant theory one studies polynomials which are invariant under transformations belonging to the general linear group GL(n, C) or one of its subgroups. A problem on which mathematicians worked in the 19th century was the problem of proving the existence of a finite basis of the invariant polynomials of n-forms under the action of the group of linear transformations over the complex numbers. Hilbert solved this problem in 1890 with an existential proof but this was followed by a more constructive proof given by Hilbert in 1893.

Hilbert's thesis topic was on Invariant Theory (1885) and he has several other papers in invariant theory [32], [33] all concerning invariants of forms, binary forms or n-forms of order m and forms are polynomials. So for over ten years Hilbert worked on polynomial n-forms. Furthermore in his list, Hilbert stated three problems involving polynomials in their statements: his 14th problem, his 15th problem and the first part of his 16th problem. So, a much more convincing hypothesis seems to be that Hilbert stated the second part of his 16th problem on polynomial differential equations simply because polynomials formed his familiar realm, in which he worked for a rather long period of time and on which he stated the three problems preceding H16.

3 On Enumerative Geometry, the principle of continuity and Hilbert's 15th problem

Enumerative geometry is an old subject initiated by Appolonius in 200 BC. Apollonius asked the question: what is the number of circles tangent to three given circles? The answer is: eight. A generalization of Apollonius' problem is the following: find the number of irreducible conies (ellipses, parabolas, hyperbolas) which are tangent to 5 given circles. The correct answer, found by Chasles, is 3264. Another problem, solved by Schubert in his book [68] is: find the number of twisted cubics, tangent to 12 quadric surfaces. Schubert won a gold medal from the Royal Danish Academy in 1875 for the solution of this problem which is the number 5,819,539,783,680. Another problem of enumerative geometry is: count the number of lines meeting four given lines in three-space. Schubert solved this problem as follows: he altered the position of the 4 given lines so that the first and second lines intersect and the third and fourth lines intersect. Then we immediately see that the line joining the two intersection points intersects the four lines as does the line of intersection of the two planes spanned by the lines. Schubert concludes, via the principle of conservation of number, that two or infinity is in general the answer to this question. According to this principle if a certain number of solutions is expected in one case, then there will be the same number in all cases or this number is infinite. In a bit more precise way, the principle of conservation of number asserts that the number of solutions of an enumerative problem remains the same when the parameters of the conditions are specialized, provided that the number remains finite and is weighted by the multiplicities of appearance of solutions (see [42]). This is also called the principle of special position.

Schubert's principle of special position or of conservation of number is a generalization of Poncelet's principle of continuity, phrased by Poncelet in [57] as follows:

"If one figure is derived from another by continuous change and the latter is as general as the former, then any property of the first can be asserted at once for the second figure."

Poncelet did not explain how "as general as the former" is to be interpreted. As indicated in [78], Poncelet's principle of continuity was used extensively in enumerative geometry. But as Zeuthen and Pieri said in [78], Poncelet "failed to give a proper foundation to his principle, therefore failed to establish its limits of validity". It was for this reason that the principle of continuity was attacked by Cauchy and this had some negative consequences. Nevertheless the principle continued to be used and its history can be found in [78]. In part because of this controversy, Schubert renamed it as the principle of conservation of number.

The principle of continuity is a formulation of the law of continuity, an heuristic principle introduced by Leibnitz and based on earlier work done by Kepler. Leibnitz formulated a philosophical principle as the law of continuity in response to attacks on his calculus. He expressed this principle as follows:

In any supposed transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included (for more detail see [44]).

Another application of the principle of conservation of number is Bezout's theorem. First let us consider two curves [C.sub.1],[C.sub.2] defined by two polynomial equations over C, [C.sub.1] : f (x, y) = 0 and [C.sub.2] : g (x, y) = 0. How many common points have these curves? In the case when / = [f.sub.1].[f.sub.2]...[f.sub.d] and g = g1-g2...gd' where [f.sub.i] and [g.sub.j] are all of degree 1, then clearly we have that [C.sub.1] intersects [C.sub.2] in a maximum of d d' points which occurs when all affine lines [f.sub.i] = 0 intersect all affine lines [g.sub.j] at finite points. To have Bezout's theorem we need to consider i) curves over the complex projective plane and ii) that each intersection point must be counted with its multiplicity. The curves are then defined by two homogeneous polynomials F(X, Y, Z) and G(X, Y, Z) over C.

Bezout's theorem: the number of common points in the projective plane over C of the two curves [C.sub.1] : F(X, Y, Z) = 0 and [C.sub.2] : G(X, Y, Z) = 0, counted with multiplicity, is d-d' where d, d' are respectively the degrees of the polynomials F and G.

The theorem involves the concept of intersection multiplicity i(p;[C.sub.i],[C.sub.2]) at a point p [member of] Cof the two curves [C.sub.1], [C.sub.2].

Bezout's theorem was stated for the first time in 1720 by Maclaurin and it was proved in 1764 by Bezout.

Other problems of enumerative geometry are: find the number of conies tangent to five given conies. Schubert found this number to be 3264 which is the correct number (Steiner claimed the answer is 7776); find the number of quadric surfaces tangent to 9 given quadric surfaces in 3-space. This number is 666,841,088. These numbers were obtained by Schubert in [68].

In general, Enumerative Geometry has the goal of answering questions of the type: How many geometric figures of a fixed kind and which satisfy certain given conditions do we have?

Schubert's treatise, entitled Kalkiil des Abzalenden Geometrie (Calculus of Enumerative Geometry) [68], dealing with such problems of enumerative geometry was published in 1879. The reprint edition was published in 1979 by Springer. The Introduction to this reprint edition was written by Steven Kleiman, an authority on enumerative geometry.

In this introduction Kleiman says : "In enumerative geometry, the figures and conditions are assumed to be expressible by algebraic equations. Moreover, imaginary numbers are used, indiscriminately mixed with real ones, and the points at infinity and those at finite distance are accepted on equal footing in projective space.... In fact, this procedure is natural and fruitful. For example, Bezout's theorem, that the number of points of intersection of two algebraic curves is equal to the product of their degrees, would obviously suffer greatly if restricted to the real affine plane."

In connection with these observations of Kleiman we need to stress that although the systems we are interested in, are real vector fields over [R.sup.2], the algebraically closed field C intervenes in an essential way. For example, just like a real polynomial equation f(x) = 0 of degree n has at most n real roots, a real non-degenerate polynomial differential system (1.1) of degree n has at most [n.sup.2] real finite singular points, i.e. common roots of p(x, y) = 0 and q(x, y) = 0. Another example is in the calculation, via the theory of Darboux over the complex field (see the last section and [26]), of real first integrals of real systems (1.1) with the help of the complex first integrals of the same systems but viewed this time as complex systems. (Any such real system generates a system over the complex plane.)

As indicated above, the counting problems in enumerative geometry are expressible in algebraic equations. The (algebraically expressed) figures depend on parameters and the finite number of conditions which the figures must satisfy, yield algebraic equations on the parameters. Then by elimination theory, the number of equations in several parameters may be reduced to just one equation in one variable parameter. This way the problem is finally reduced to applying the fundamental theorem of algebra.

Schubert was the first to have developed a calculus for answering such questions of enumerative geometry ([68]). As Kleiman says in [42], regarding this calculus, Schubert "makes the first (and really the only) attempt at a systematic, progressively deeper theory of enumerative geometry..". Kleiman later adds: "The Calculus works: There is no doubt about that. Why it works is another matter, a serious one, which Hilbert took for his 15th problem."

The fifteenth problem on Hilbert's list is entitled Rigorous Foundation of Schubert's Enumerative Calculus. Hilbert states the problem as follows:

"To establish rigorously and with an exact determination of the limits of validity those geometrical numbers which Schubert especially has determined on the basis of the so-called principle of special position, or conservation of number, by means of the enumerative calculus developed by him."

Hilbert then adds: "Although the algebra of today guarantees, in principle, the possibility of carrying out the processes of elimination, yet for the proof of the theorems of enumerative geometry decidedly more is requisite, namely, the actual carrying out of the process of elimination in the case of equations of special form in such a way that the degree of the final equations and multiplicity of their solutions may be foreseen."

Schubert's calculus and Hilbert's 15th problem motivated much of the work done in intersection theory which is an essential part of algebraic geometry. "The foundations of the calculus were first secured by van der Waerden" (from [41]). The work of van der Waerden is based on (topological) intersection theory [76]. Also from [41]: "Entirely algebraic, rigorous treatments of the foundations of the calculus have become possible with the development of algebraic intersection theories." Intersection theory was founded by Maclaurin in 1720 with the statement of Bezout's theorem and as indicated by Kleiman in [43] it "remained centered around this theorem for a century and a half". Schubert's book and his Calculus produced a revolutionary change (see [43]). He was the first to introduce explicitly intersection rings. Referring to Scubert's calculus Kleiman says "...the abstract essence of the calculus is part of the general intersection theory of contemporary algebraic geometry" ([43]). For a modern presentation of intersection theory see [30].

Enumerative geometry continues to be a very active area of research. An enumerative problem which stirred an interaction between physics and mathematics is the following: Compute the number [N.sub.d] of degree d rational curves (i.e. curves which admit a rational parametrization) on a quintic hyper-surface given by the equation [x.sub.1.sup.5] + [x.sub.2.sup.5] + [x.sub.3.sup.5] + [x.sub.4.sup.5] + [x.sub.5.sup.5] = 0 in projective 4-space. [N.sub.1] = 2875 counts the lines on the quintic hyper-surface and this is a classical result obtained in the 19th century by Schubert. The result [N.sub.2] = 609, 250 was obtained in 1985 by Katz (see [40], [12]). These numbers [N.sub.d] were first predicted by physicists (see [21]) by mirror symmetry. They grow exponentially. For example [N.sub.10] = 704, 288,164, 978, 454, 686,113, 488, 249, 750 (see [12] for all numbers [N.sub.d] for d [less than or equal to] 10). The author knows from Kleiman that the physicists work with physical quantities which are equal to [N.sub.d]'s provided that [N.sub.d]'s are finite (see also [40]). However, so far we only know for sure that the [N.sub.d]'s are finite if d < 10. The physiscists considered these numbers lumped all together in the generating function

[mathematical expression not reproducible]

which has a physical interpretation. The number [N.sub.d] is an example of a Gromov Witten invariant (see [40]). Through the development of the Gromov-Witten theory, the numbers became part of a conjecture proved by Givental in [31].

4 What did Hilbert mean by "the same method of continuous variation of coefficients" in the statement of his 16th problems?

As indicated in section 2, when stating the problem H16, Hilbert referring to the first part of his f 6th problem said:

"In connection with this purely algebraic problem, I wish to bring forward a question which, it seems to me, may be attacked by the same method of continuous variation of coefficients, and whose answer is of corresponding value for the topology of families of curves defined by differential equations."

What did Hilbert have in mind when in the above phrase he said : "...may be attacked by the same method of continuous variation of coefficients?"

An answer to this question was ventured by Christopher in his review MR20858I8 (2005f:3408I) in MathSciNet of the author's paper [60] where Christopher says: "It is interesting that Hilbert himself, when stating the second part of the 16th problem, gives the connection to the first part not in algebraic terms, but with the thought that it might also yield to the method of "continuous variation of parameters"; that is, it would seem, bifurcation theory".

The mathematical term "bifurcation" was introduced by Poincare in 1885 in [52]. But in 1900, when Hilbert proposed his problems, there was no "bifurcation theory" yet. In fact what could be called bifurcation theory began to be developed much later. Some isolated results appeared in the 1930's (see for example [56] and [14], both published in the Soviet Union.) A systematic approach to develop a theory of bifurcation was initiated by A.A. Andronov in Gorki where he moved from Moscow in 1931 and where he built a school in non-linear dynamical systems. The book "Theory of Bifurcations of Dynamic Systems on a plane" was written later by Andronov together with E.A. Leontovich, I.I. Gordon and A.G. Maier ([1]). So rather than referring to a non-existent bifurcation theory in 1900, it is more plausible that Hilbert referred to the principle of continuity. In Hilbert's 15th problem the principle of special position, or conservation of number is explicitly included and as indicated in the preceding section, this principle is no other than the principle of continuity renamed in this way by Schubert. The principle of continuity may also apply to the first part of Hilbert's 16th problem which is a problem about algebraic curves and surfaces. But how about H16?

A polynomial differential system could have algebraic limit cycles but in general limit cycles are non-algebraic, transcendental objects. Analytic techniques necessarily must be part of the game. But it would be a mistake to consider algebraic-geometric methods as outside the game. For example the problem of classifying the global geometric configurations of singularities occurring in the whole quadratic class was solved (see the next section) using only algebraic methods. There are over 1800 such possible configurations. This is the first truly global classification result on this family of systems. But even going beyond such purely algebraic studies, to limit cycles theory, algebra creeps in. For example, in the order k degenerate Hopf bifurcation theorem for [C.sup.[infinity]] systems with a singular point subject to certain conditions (pure imaginary eigenvalues and order k of the singularity), in giving bounds to the number of limit cycles arising from the singularity in [C.sup.[infinity]] perturbations, the application of the Malgrange-Weierstrass Preparation Theorem leads to the factorization of the displacement function (attached to the first return map) through a polynomial which holds the key to the maximum number of such limit cycles: the degree of this polynomial is k and it is the maximum number of such limit cycles, and we may add counted with multiplicity, like in the theorem of Bezout.

5 Enumerative Geometry versus Counting Problems of low degree polynomial vector fields

In the second section of this article we saw how very difficult Hilbert's problem H16 and some finiteness problems arising from it are. In general, limit cycles are elusive objects. Even problems on them which have been solved fail sometimes to be digested by the mathematical community. Ecalle and Ilyashenko wrote two books on their independent proofs of the individual finiteness theorem for limit cycles but as Smale said in [70] "These two papers have yet to be thoroughly digested by the mathematical community."

In H16 as well as some of the finiteness problems arising from H16 we not only have to deal with limit cycles but also with the fact that these problems are essentially of a global nature. For example H16 for quadratic systems involves the whole class QS, an object which is very complicated. On the other hand, as we indicate further below, we can make progress on the algebraic theory of these systems, and also on global counting problems which could involve apart from algebraic methods some analytic techniques. We have seen that enumerative geometry has traditionally been an algebraic-geometric theory. Polynomial vector fields are objects of a mixed nature: i) they are algebraic because they are defined by polynomials; ii) they are also analytic objects; their solutions are analytic functions of the time and in general, their images in the plane do not lie on algebraic curves; iii) their phase portraits are topological objects. The point ii) weighs a great deal more than i) in the theory of these systems. Still, there are questions we can answer about low degree systems which rely mainly on i). In fact we can ask the question: "How far can we go in the study of polynomial vector fields by relying on their algebraic properties?" Even questions which are not of an algebraic nature can sometimes be answered by using algebraic methods extended by some analytic arguments.

In 1878 Darboux published a paper [26] on invariant algebraic curves of polynomial differential systems and their relation to the integrability of these systems. Poincare was enthusiastic about this work which he called "admirable" (see [53]) and inspired by Darboux' work, Poincare wrote two papers [54], [55] and in [54] he stated a problem which is still open today. In his paper [26], Darboux talks about polynomial differential equations over the complex projective plane. However his theory also applies to real polynomial differential systems over the affine plane since any such system generates a differential system over the complex affine plane. Darboux' definition of an invariant algebraic curve is the following:

Let f(x, y) be a polynomial in x, y over C. We say that the curve f(x, y) = 0 is invariant for a polynomial differential system (1.1) if and only if there exists a polynomial K(x, y) in C[x, y] such that the following is an identity:

[mathematical expression not reproducible] (5.1)

Besides being beautiful, the theory of Darboux is also important because this theory is about intergability of the systems. In [26] Darboux gave sufficient condition for integrability of polynomial vector fields in terms of their invariant algebraic curves.

Theorem 5.1. (Darboux [26]) Suppose we have a differential system (1.1) of degree n which possesses k invariant algebraic curves [f.sub.i](x,y) = 0 with [f.sub.i](x,y) [member of] C[x,y], [f.sub.i](x,y) irreducible over C, i [member of] {1,2,...,K} and k [greater than or equal to] n(n + l)/2 + l. Then there exist complex numbers [[lambda].sub.1],..., [[lambda].sub.k]] such that [??] is a first integral of the system.

This theorem, which gives us a criterion for integrability of the system whenever we have sufficiently many invariant algebraic curves, stimulated much research during the past twenty-five years producing many results which we cannot survey here. We only limit ourselves to mentioning two short, compact modern presentations of the work of Darboux, namely [59] and [23].

We are interested here in developments on the theory of polynomial differential systems, which are of a similar nature to those in enumerative geometry. It seems these are the initial stages of an Enumerative Geometry of Polynomial Differential Systems starting with some counting problems on low degree polynomial vector fields. Algebraic-geometric methods are at the basis of this enumerative geometry of polynomial vector fields. But apart from these, in solving counting problems on these systems, analytic tools would also need to intervene.

To state these problems we need the concept of configuration of invariant algebraic curves of a polynomial differential system (1.1).

By a configuration of invariant algebraic curves of a polynomial differential system (1.1) which has a finite number invariant algebraic curves over C and a finite number of singular points, finite or infinite, we mean the set of all these curves, including the line at infinity, each one of them endowed with its own multiplicity and together with all the real singular points of this system located on these invariant curves, each one of these singularities endowed with its own multiplicity.

The following are questions on quadratic vector fields which can be viewed from the perspective of enumerative geometry:

* 1) How many distinct configurations of invariant lines, including the line at infinity, of total multiplicity 6 (respectively 5, 4) of non-degenerate systems in QS with a finite number of singularities at infinity do we have? (The maximum number of invariant straight lines which a non-degenerate quadratic system could have is 6.) * 2) How many distinct configurations of invariant affine straight lines could nondegenerate systems in QS have, in case their line at infinity is filled up with singularities?

* 3) How many possible configurations of invariant hyperbolas and invariant straight lines could systems in QS have? Here we assume that we always have at least an invariant hyperbola. We may thus have systems with only hyperbolas and systems which apart from hyperbolas also possess invariant straight lines.

* 4) How many distinct configurations of invariant straight lines occur in Lotka-Volterra differential systems, i.e. in systems of the form dx/dt = x(ax + by + c), dy/dt = y(Ax + By + C) with a, b, c, A,B,C [member of] R?

* 5) How many global geometric configurations of singularities could systems in QS have?

All problems listed above for QS have been answered and we list below these answers as well as the references.

* 1) It is clear from the start that this is a problem of enumerative geometry. Indeed, saying that a line is invariant of a given differential system in QS can be written as an algebraic equation of the form (5.1). We also have an incidence structure, namely the real singularities located on the invariant line. The condition that an algebraic curve is invariant is in particular a condition of tangency, i.e. the vector field is tangent to the curve at its non-singular points on the curve. These are geometric conditions. We also have the notion of algebraic multiplicity of an invariant curve of a polynomial system (see [24]).

This problem was completely solved in [61] and [63]. Systems in this class which possess invariant lines of total multiplicity 6, including the line at infinity, have a total of II possible configurations of invariant straight lines. Systems in this class which have invariant straight lines of total multiplicity 5 have a total of 30 possible such configurations. These results were obtained in [61]. Systems in this class which possess invariant lines of total multiplicity 4 have a total of 45 possible configurations of invariant straight lines. This result was obtained in [63].

* 2) This problem was completely solved [65] and the answer to this question is 9.

* 3) This problem was completely solved in [48] and in [49]. More precisely it was solved for quadratic systems with 3 distinct singularities at infinity on the real projective plane in [48] yielding 162 configurations, and for the case of quadratic differential systems with either just one or two distinct real singularities at infinity in [49] yielding 43 distinct configurations.

* 4) The answer to this question was shown to be 65 in [67].

* 5) This question is much harder than the preceding four questions and it required several years of work to obtain its solution. There are over 1800 geometric configurations of singularities which were obtained in joint work by Artes, Llibre, Schlomiuk and Vulpe. This work will be published in a book, based on the following papers by these authors [3], [10], [9], [4], [7], [5], [2] as well as on the previously written papers [62], [11]. Part of the results, namely those for systems which have 4 distinct finite singularities, real or complex will only be included in the book. This was the hardest part of the work.

The notion of geometric configuration of singularities, defined in [4] for both finite and infinite singularities is roughly speaking expressed by distinguishing the following two cases:

1) If we have a finite number of infinite singular points and a finite number of finite singularities we call geometric configuration of singularities, finite and infinite, the set of all these singularities each endowed with its own multiplicity together with the local phase portraits around real singularities endowed with additional geometric structure involving the concepts of tangent, order and blow-up equivalences as defined in [2].

2) If the line at infinity Z = 0 is filled up with singularities, in each one of the charts at infinity X [not equal to] 0 and Y [not equal to] 0, the corresponding system in the Poincare compactification is degenerate and we need to do a rescaling of an appropriate degree of the system, so that the degeneracy be removed. The resulting systems have only a finite number of singularities on the line Z = 0. In this case we call geometric configuration of singularities, finite and infinite, the set of all points at infinity (they are all singularities) in which we single out the singularities at infinity of the "reduced" system, taken together with their local phase portraits and we also take the local phase portraits of finite singularities each endowed with additional geometric structure described in [4].

The work done in the references contains much more than just the numbers asked in the above mentioned list of questions. Thus for each one of the first four questions, the specific configurations of invariant curves are drawn on pictures and for the problem 5) the geometric configurations are also explicitly specified and necessary and sufficient conditions for the realization of each one of the configuration are given. In fact the final result for question 5) is a theorem of classification of quadratic differential systems according to their global geometric configurations of singularities, finite and infinite. This is thus the first truly global classification result of this family. Although at this stage we are probably very far from determining the Hilbert number H(2), we have though a beginning of a global theory of quadratic systems.

The presence of invariant algebraic curves produced the Darboux or Liouvillian integrability of all the systems involved in problems 1) and 2) as well as for quite a number of cases for problems 3) and 4) and the first integrals are explicitly given (see for example [66]).

Questions of an algebraic nature resulting from the algebraic nature of polynomial vector fields such as the above listed problems form an area of research which is likely to grow. Connections between these questions to two famous open problems both stated by Poincare, one in the 1885 and another in 1891 are a motivation for pursuing such questions in this research area.

Cubic differential systems are much harder to study than quadratic ones. Thus while quadratic systems with a center are integrable and have no limit cycles, in cubic systems centers could coexist with limit cycles. We have necessary and sufficient conditions for center for the quadratic systems, but the problem of finding necessary and sufficient conditions for a center is still open for cubic differential systems in spite of numerous papers written in this area. Although this is a very hard problem which defied us for over a hundred years, this did not discouraged research on other problems about cubic systems. In particular we mention below some problems of an enumerative geometry of these systems:

1) How many distinct configurations of invariant lines of total multiplicity 9 of systems in CS do we have?

2) The same question as above for systems in CS but for invariant lines of total multiplicity 8.

3) The same question as above for systems in CS but for invariant lines of total multiplicity 7.

The number 9 in problem I) is the maximum number of invariant straight lines which a non-degenerate cubic system could have. Problem 1) was

completely solved in [46] and [15]. The answer to this question is 24. The problem 2) was completely solved in a series of papers of Bujac and Vulpe (see [19], [16], [17], [20], [18]). The total number of configurations of invariant lines of total multiplicity 8 is 51. Results on this problem but involving only the total parallel multiplicity 7 are contained in [73], [74], [75]. Hence these are also results of total multiplicity 8 if we include in our counting the line at infinity. The problem 3) is not completely solved.

Another question, which is not yet solved, is:

Q) How many distinct configurations of invariant hyperbolas could non-degenerate systems in CS have?

We list here also counting problems whose solutions may involve, apart from algebraic geometric methods also some simple additional analytic arguments:

1) How many phase portraits do quadratic systems with invariant straight lines of total multiplicity 6 (respectively 5, 4) do we have?

2) How many topologically distinct phase portraits in the neighborhood of infinity do planar quadratic differential systems with a finite number of infinite singularities have?

3) How many phase portraits do quadratic systems with the line at infinity filled up with singularities have?

4) How many phase portraits do Lotka-Volterra systems have?

Here are the answers to these question and the references where these answers can be found: 1) The answer to this question was found in [64] and [66]. Quadratic systems with invariant lines of total multiplicity 6 have a total of 11 phase portraits. Everyone of the 11 configurations of invariant lines leads to just one phase portrait. In the case the total multiplicity of invariant straight lines is 5 then we have 31 topologically distinct phase portraits. In the case of total multiplicity 4 we have a total of 69 topologically distinct phase portraits.

2) This problem was solved in [47] and [62] where it was established that the total number of phase portraits in the neighborhood of infinity for this class is 40.

3) We have a total of 11 topologically distinct phase portraits for systems in QS with the line at infinity filled up with singularities for a total of 9 distinct configurations of affine invariant lines. The result can be found in [65].

4) There are exactly 112 topologically distinct Lotka-Volterra systems: 60 of them with exactly three invariant lines, all simple; 27 portraits with invariant lines with total multiplicity at least four; 5 with the line at infinity filled up with singularities; 20 phase portraits of degenerate systems.

Due to Darboux' theory we proved the integrability of all the systems in class 1). All systems with invariant lines of total multiplicity 6 possess rational first integral while all systems with invariant lines of total multiplicity 5 possess Darboux first integrals. These results were proved in [64]. All systems in this class which possess invariant straight lines of total multiplicity 4 possess a Liouvillian first integral. This result was proved in [66].

Since systems in the problem 3) have "enough" invariant lines to ensure integrability, in [65] it was also shown that all these systems are integrable via the method of Darboux, having cubic polynomials as inverse integrating factors.

Apart from the theory of integrability of Darboux, to obtain the results it was necessary to define geometric as well as algebraic invariants and to use the invariant theory of polynomial differential systems, along with standard ODE methods.

Algorithms for computer computation to obtain the configurations of invariant curves as well as for obtaining the global geometric configuration of singularities are also given in the above mentioned papers and bifurcation diagrams in the 12-dimensional space of coefficients of systems in QS were also included. These algorithms are based on computing algebraic invariants, modulo the group of affine transformations and time rescaling, of the polynomial vector fields subjected to the specific problems involved.

The following are open problems for cubic differential systems:

1) How many topologically distinct phase portraits of cubic differential systems with the line at infinity filled up with singularities do we have?

2) How many topologically distinct phase portraits of cubic differential systems with invariant lines of total multiplicity 9 (respectively 8,7) do we have?

3) How many topologically distinct phase portraits in the neighborhood of infinity do cubic systems have?

ACKNOWLEGEMENTS The author is very grateful to Steven Kleiman for precious conversations and precious comments in letters exchanged, regarding Enumerative Geometry. She is very grateful to her collaborator Nicolae Vulpe for providing references on his work with Cristina Bujac and also for some technical help with the manuscript. The author thanks Jaume Llibre and Jorge Sotomayor for discussions around Poincare's individual finiteness theorem and for references. She is also grateful to Sergei Yakovenko for the reference [77]. This work was supported by NSERC.

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Dana SCHLOMIUK

Departement de Mathematiques et de Statistiques,

Universite de Montreal,

C.P. 6128, Succursale Centre-ville,

Montreal, QC, H3C 3J7 Canada

E-mail: dasch@dms.umontreal.ca
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