MULTIPLICITY OF INVARIANT ALGEBRAIC CURVES AND DARBOUX INTEGRABILITY

JAUME LLIBRE^{1} AND JORGE VITÓRIO PEREIRA^{2}

1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 – Bellaterra, Barcelona, Spain. email: jllibre@mat.uab.es

2 Instituto de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina, 110 Jardim Botânico, 22460-320 - Rio de Janeiro, RJ, Brasil. email : jvp@impa.br

Abstract. We define four different kinds of multiplicity of an invariant algebraic curve for a given polynomial vector field and investigate their relationships. After taking a closer look at the singularities and at the line of infinity, we improve the Darboux theory of integrability using these new notions of multiplicity.

1. Introduction

In this paper we want to contribute toward a better understanding of the fascinating relationships between the integrability of a polynomial vector field on the complex plane, and the existence of invariant algebraic solutions for such vector fields. The seminal work on this subject is Darboux’s paper [6] of 1878. He showed how to force integrability from the abundance of invariant algebraic curves. More precisely, he proved that a polynomial vector field of degree d with at least d(d + 1)/2 invariant algebraic curves has a first integral. This theory of integrability received also contributions from the work of Poincaré [11], who mainly was interested in the rational first integrals.

The subject remained almost forgotten until 1979 when Jouanolou [7] published his Lecture Notes. There he shows that if the number of invariant algebraic curves of a polynomial system of degree d is at least [d(d+1)/2]+2, then the vector field has a rational first integral, and in particular all its solutions are algebraic curves.

Since then, the subject has been revitalized and studied under different viewpoints. Thus, for instance, the 1983 work of Prelle and Singer [12], using methods of differential algebra, shows that if a polynomial vector field has an elementary first integral, then it can be computed using only the invariant algebraic curves; i.e., this kind of first integrals can be found using Darboux’s approach. Nine years later Singer [17] studied the existence of a wider class of first integrals, namely Liouvillian first integrals. Roughly speaking, his main result is that the Liouvillian first integrals have integrating factors given by Darbouxian functions.

¿From a more geometrical point of view the work of Scárdua [13] relates the properties of the holonomy groups of invariant algebraic curves with the existence of first integrals. Later on Camacho and Scárdua [1] proposed to find a wider class of mathematical objects generalizing the Liouvillian functions.

In this paper we adopt a more classical point of view, closer to Darboux’s original one, and improve his theory in various aspects. One of the main ideas coming from the work of Darboux, is the control of the space of cofactors associated to the invariant algebraic curves. If the polynomial vector field has degree d, then the cofactors are polynomials of degree at most d − 1, hence are contained in the finite dimensional vector space ℂ_{d−1}[x, y].

In [2] the authors used singular points to reduce the dimension of the space of possible cofactors and consequently, the number of conditions to force Darboux integrability. Here we strength this approach considering not just the singular points, but also their degree of degeneracy.

In [10] the extactic curves for polynomials vector fields are introduced. Essentially, they are the curves of inflection and higher order inflection points for the orbits of the vector field. In the context of integrability they provide new bounds for the number of invariant algebraic curves. These bounds take into account the degree of the invariant algebraic curves, and for a fixed degree they provide, in general, much better bounds than Darboux’s classical approach.

In this work we use the extactic curves to define two notions of algebraic multiplicity. The first (see Subsection 3.2) is more useful to force integrability, while the second (see Section 6) turn out to be a more precise tool to bound the geometric multiplicity, another notion defined here. Finally we show how the geometric multiplicity gives rise to exponential factors, introducing the notion of derivative of an invariant algebraic curve. The germ of the idea of the interaction between exponential factors and geometric multiplicity can be found in Christopher’s paper [3] and in [4, 5]. Motivated by this interaction we define the notion of integrable multiplicity. This is the key notion which allow us to improve the Darboux theory of integrability.

The interplay between the different notions of multiplicity and the Darboux integrability can be sumarized in the following diagram.

Geometric Multiplicity

⟶_{Corollary 3}^{Corollary 2}

Integrable Multiplicity

Proposition 6↓  ↓ ^{Theorem 4(a)}

Algebraic Multiplicity

⟶_{Theorem 4(b)}

Darboux Integrability

The paper is organized as follows. In Section 2 we define the basic notions of algebraic invariant curves, first integrals, integrating factors, exponential factors and extactic curves. Section 3 is responsible for introducing the notions of geometric, algebraic and integrable multiplicities, besides the concept of derivative of an algebraic invariant curve. There we also investigate some relations between them. Section 4 studies the singular points and the line at infinity preparing the background to Section 5, where we improve the Darboux theory of integrability. Finally in Section 6 we define the strong algebraic multiplicity, and comment how it can be used to provide sharper upper bounds for the geometric multiplicity. We conclude with some remarks given in Section 7.

Contents

1. Introduction 1 2. Preliminaries 2 2.1. Vector fields 2 2.2. Invariant algebraic curves and exponential factors 3 2.3. First Integrals and Integrating Factors 3 2.4. Extactic curves 4 3. Notions of multiplicity for invariant algebraic curves 5 3.1. Geometric Multiplicity 5 3.2. Algebraic Multiplicity 6 3.3. Integrable Multiplicity 7 3.4. Relations between the different notions of multiplicity 11 4. The role played by singular points and the line at infinity 11 4.1. Finite singular points 11 4.2. The line at infinity 12 5. Darboux Integrability 13 6. Strong Algebraic Multiplicity 14 7. Final Remarks 15 8. Acknowledgments 15 References 15

2. Preliminaries

2.1. Vector fields.

Definition 1. We say that X is a (polynomial) vector field of (affine) degree d on ℂ^{2} if it can be written in the form

X = a(x, y)\frac{∂}{∂x} + b(x, y)\frac{∂}{∂y} , (1)

where a(x, y) and b(x, y) are polynomials in ℂ[x, y] such that the maximum degree of a and b is d.

Another definition of the degree of a vector field X is the following one: X has degree d if the number of tangencies of X with a generic straight line is d. In general the two definitions do not coincide, but they differ by at most 1. For more details about this see [8].

The set of all vector fields of degree d can be seen as a Zariski open set in the affine space given by the coefficients of the vector fields.

2.2. Invariant algebraic curves and exponential factors.

Definition 2. Let f ∈ ℂ[x, y]. The algebraic curve {f = 0}, or simply f, is said to be invariant by the vector field X when X(f)/f is a polynomial. In that case L_{f} = X(f)/f is called the cofactor of the invariant algebraic curve f.

Observe that the degree of L_{f} is at most the degree of X minus one, and that an invariant algebraic curve of X is formed by solutions of the vector field X. In other words a solution of X has either empty intersection with the zero set of f, or it is entirely contained in {f = 0}.

The next two propositions can be found in [5].

Proposition 1. Let f ∈ ℂ[x, y] and f = f_{1}^{n_{1}} ⋯ f_{r}^{n_{r}} be its factorization in irreducible factors. Then, for a vector field X, f is an invariant algebraic curve with cofactor L_{f} if, and only if f_{i} is an invariant algebraic curve for each i = 1, …, r with cofactor L_{f_{i}}. Moreover L_{f} = n_{1}L_{f_{1}} +⋯+n_{r}L_{f_{r}}.

Definition 3. Let f, g ∈ ℂ[x, y], we say that e = exp(g/f) is an exponential factor of the vector field X of degree d, if X(e)/e is a polynomial of degree at most d − 1. This polynomial is called the cofactor of the exponential factor e, which we denote by L_{e}. The quotient g/f is an exponential coefficient of X.

Note that despite the fact that an exponential factor satisfies the same kind of equation than an invariant algebraic curve, a priori, it has no relation with the solutions of the vector field because its zero set is empty. Although we have the following result.

Proposition 2. If e = exp (g/f) is an exponential factor for the vector field X, then f is an invariant algebraic curve and g satisfies the equation

X(g) = gL_{f} + fL_{e} .

2.3. First Integrals and Integrating Factors. This work is mainly interested in two classes of first integrals, the rational ones, and the Darbouxian; that is a product of complex powers of polynomials and complex powers of exponential of rational functions. We recall that by first integral we mean:

Definition 4. Let U be an open subset of ℂ^{2}. We say that a non constant (multi–valued) function H : U → ℂ is a first integral of a vector field X on U if, and only if, X|_{U}(H) = 0. When H is the restriction of a rational, respectively Darbouxian, function to U then we say that H is a rational, respectively Darbouxian, first integral.

Besides the first integrals we are also interested in the integrating factors. Which can be formally defined as follows.

Definition 5. We say that a non constant (multi–valued) function R : U → ℂ is an integrating factor of a vector field X on U if, and only if, X|_{U}(R) = −divX|_{U} · R. We recall that div denotes the divergence of the vector field.

If we know an integrating factor we can compute a first integral. Reciprocally if H is a first integral of the vector field (1), then there is a unique integrating factor R satisfying

Ra = \frac{∂H}{∂y} and Rb = −\frac{∂H}{∂x} . (2)

Such R is called the integrating factor associated to H.

2.4. Extactic curves. The definitions and results of this subsection comes from the work of the second author. Here we state and prove simplified versions adapted to the complex plane.

Intuitively the objects defined below measure the order of contact of the solutions of a vector field with algebraic curves of given degree. For more information see [10].

Definition 6. If X is a vector field on ℂ^{2} the n–th extactic curve of X, E_{n}(X), is given by the equation

det (\begin{matrix}v_{1} & v_{2} & ⋯ & v_{l} \\ X(v_{1}) & X(v_{2}) & ⋯ & X(v_{l}) \\ ⋮ & ⋮ & ⋯ & ⋮ \\ X^{l−1}(v_{1}) & X^{l−1}(v_{2}) & ⋯ & X^{l−1}(v_{l}) \end{matrix}) , (3)

where v_{1}, v_{2}, …, v_{l} is a basis of the ℂ–vector space of polynomials in ℂ[x, y] of degree at most n (hence l = (n + 1)(n + 2)/2), X^{0}(v_{i}) = v_{i} and X^{j}(v_{i}) = X^{j−1}(X(v_{i})).

Observe that the definition of extactic curve is independent of the chosen basis of the ℂ–vector space of polynomials of degree at most n.

Theorem 1. Let X be a vector field on ℂ^{2}. Then E_{n}(X) = 0 and E_{n−1}(X) ≠ 0 if, and only if, X admits a rational first integral of exact degree n.

Proof: Let p ∈ ℂ^{2} be a non–singular point of X. We may assume that p is the origin of ℂ^{2}. Suppose that the solution passing through it is parametrized, locally, by (x, y(x)). Since E_{n}(X) vanishes identically, the composition of our local solution with the n–Veronese map,

(x, y) → (x^{n}, x^{n−1}y, x^{n−2}y^{2}, …, y^{n}, x^{n−1}, …, y^{n−1}, …, x^{2}, xy, y^{2}, x, y, 1)

is contained in a hyperplane, therefore (x, y(x)) must be contained in an algebraic curve of degree less than or equal to n. The fact that E_{n−1}(X) ≠ 0 implies that the generic solution is of degree at least n. This completes the “only if” part of the statement.

If X admits a first integral of degree n then every invariant curve is of degree at most n and hence every point is a n–inflection point, i.e., E_{n}(X) = 0. Since not every invariant curve has degree n − 1, E_{n−1}(X) ≠ 0.

Proposition 3. Every algebraic curve of degree n invariant by the vector field X is a factor of E_{n}(X).

Proof: Let f be an invariant algebraic curve of degree n. As it was observed the choice of the basis of the ℂ–vector space plays no role in the definitions of extactic curve, therefore we can take v_{1} = f. Since

X(f) = L_{f}f ,
X^{2}(f) = X(L_{f}f)  = (L_{f}^{2} + X(L_{f})) f ,
X^{k}(f) = X(X^{k−1}(f)) = ( polynomial ) f ,

f is a factor of E_{n}(X).

Example 1. If we consider the 2–parameter of vector fields

X_{(t,b)} = (2t^{2} − 2x^{2}) \frac{∂}{∂x} + (b − 4xy − 2t^{3}y^{2}) \frac{∂}{∂y} ,

we have that the first extactic curve of X_{(t,b)} is described by

y (x − t) (x + t) (x + t^{3}y − t\sqrt{\frac{2 + bt}{2}}) (x + t^{3}y + t\sqrt{\frac{2 + bt}{2}}) = yl_{t}^{(1)}l_{t}^{(2)}l_{t}^{(3)}l_{t}^{(4)}.

It follows from Proposition 3 that every invariant straight line, must be contained in the first extactic curve. It can be easily verified that y is not invariant by any vector field of the form X_{(t,b)} with b ≠ 0, and that for every i ∈ {1, 2, 3, 4}, l_{t}^{(i)} is an invariant straight line for X_{(t,b)}.

The above family of vector field appears in the paper [16].

Example 2. The 3–parameter family of vector fields given by

X_{(t,b,d)} = (−y − bx^{2} − dy^{2}) \frac{∂}{∂x} + (x + (t − b)xy) \frac{∂}{∂y} ,

corresponds to a quadratic polynomial vector field having a center at origin. It has been studied in [14].

The first and the second extactic curves, E_{1}(X_{(t,b,d)}) and E_{2}(X_{(t,b,d)}), for X_{(t,b,d)} are

(1 + (t − b)y) · (tbx^{4} − d^{2}y^{4} − 2dy^{3} + (−td + bd)x^{2}y^{2} − 2dx^{2}y − x^{2} − y^{2}) , (4) (1 + (t − b)y)^{4} · x · (d − t + b(2t − 2d)y + tb(b + t)x^{2} + bd(t + b)y^{2}) · p_{16} , (5)

respectively. Here p_{16} denotes a polynomial of degree 16. One can see that the linear factor of (4) and the quadratic factor of (5) are the unique invariant algebraic curves of the respective degree for the vector field X_{(t,b,d)}.

3. Notions of multiplicity for invariant algebraic curves

3.1. Geometric Multiplicity. Let X be a vector field on ℂ^{2} of degree d.

Definition 7. An invariant algebraic curve f of degree n for the vector field X, has l–geometric multiplicity m if, there exists a sequence of vector fields X_{k}, of degree d, converging to X, such that each X_{k} has m distinct invariant algebraic curves of degree at most l, f_{k}^{(1)}, …, f_{k}^{(m)}, converging to f as k goes to infinity. Here, by distinct we mean that pairwise they have no common factors. We denote m by μ_{g,l}(X, f), or if no confusion can arise μ_{g,l}(f).

Finally we define the geometric multiplicity, μ_{g}(X, f), of the invariant algebraic curve f with respect to X as

lim_{l→∞} μ_{g,l}(X, f) .

Remark 1. By convergence we mean the convergence of the coefficients of the polynomials involved, either in the case of vector fields, or in the case of algebraic invariant curves.

Remark 2. In Definition 7 if l < n, then clearly μ_{g,l}(X, f) = 0.

Note that in the definition of geometric multiplicity it is allowed that it becomes infinity. But if this is the case we will show in Proposition 5 that the vector field has a rational first integral.

Example 3. To illustrate the geometric multiplicity of an invariant algebraic curve we shall compute the multiplicity of the invariant straight line l = {x = 0} for the vector field X_{(0,b)} defined in Example 1. ¿From that example we know that for any t ≠ 0, X_{(t,b)} has exactly four distinct invariant straight lines, and hence one can conclude that μ_{g,1}(X_{(0,b)}, L) ≥ 4. Later on we will see that μ_{g,1}(X_{(0,b)}, L) = 4.

Example 4. For the vector fields X_{(0,b,d)} given in Example 2, the invariant straight line 1 − by raised to the second power has 2–geometric multiplicity at least 2. This is due to the fact that if we fix b and d and make t tend to zero, it is possible to check that we have the invariant conic C = d − t + b(2t − 2d)y + tb(b + t)x^{2} + bd(t + b)y^{2} tends to (1 − by)^{2}. For the moment we cannot give upper bounds to the l–geometric multiplicity, that, we will do in the next subsection.

Proposition 4. Let f be an algebraic curve of degree n invariant by the vector field X. Whenever μ_{g,l}(X, f) = m, there exists an analytic 1–parameter family of polynomial vector fields with degree d, X_{t} where t ∈ (−ϵ, ϵ), analytic in t such that for each t ∈ (−ϵ, ϵ) \ {0}, X_{t} has m distinct invariant algebraic curves f_{t}^{(1)}, …, f_{t}^{(m)} of degree at most l converging to f as t tends to zero, and X_{0} = X.

Proof: It follows easily from the fact that the set of vector fields with invariant algebraic curves of degree at most n is a Zariski closed set in the space of vector fields of degree d.

Proposition 5. Let f be an invariant algebraic curve for the vector field X. If μ_{g}(X, f) = ∞, then X admits a rational first integral.

Proof: Suppose X has degree d. Since μ_{g}(X, f) = ∞, there exists an l such that

μ_{g,l}(X, f) > η = \frac{d(d + 1)}{2} + 1 .

¿From Proposition 4 there exists an analytic 1–parameter family X_{t}, t ∈ (−ϵ, ϵ), such that for t ≠ 0, X_{t} has at least η distinct invariant algebraic curves. It is known from [7] (see a very short proof in [5]) that a polynomial vector field of degree d having more than η distinct invariant algebraic curves has a rational first integral. Hence if t ≠ 0, X_{t} admits a rational first integral.

Let S_{i} be the Zariski closed set in the space of all vector fields of degree d with rational first integral of degree at most i. The intersection of S_{i} with the analytic family X_{t} is either all the family or is a discrete set. In the first case the proposition is proved. So suppose that for all i, the intersection of S_{i} with the family X_{t} is a discrete set, and taking their union we obtain an enumerable set. This is in contradiction with the fact this set must be equal to X_{t}, with t ∈ (−ϵ, ϵ) \ {0}.

We point out that Schlomiuk in [15] defines a notion of geometric multiplicity of an invariant algebraic curve with respect to a family of polynomial vector fields. There, besides the use of the topology of the space of coeffiecients, as we do, she uses a kind of Hausdorff topology, see [15]. We also like to remark that her definition is with respect to a fixed family, while ours do not depend on the choice of any particular family. These turn out to be the essential differences between the two notions. They are both relevant, and we choose the present approach because together with the notion of algebraic multiplicity we can provide computable upper bounds of it without considering any particular family.

To make more clear the difference observe that in Example 4, for Schlomiuk the invariant straight line has multiplicity 2, with respect to the considered family . For us the invariant straight line has 2–geometric multiplicity 1 (see Example 6), and its square has 2–geometric multiplicity exactly 2, see Example 9.

3.2. Algebraic Multiplicity.

Definition 8. An invariant algebraic curve f of degree n for the vector field X has l–algebraic multiplicity m when m is the greatest positive integer such that the m–th power of f divides E_{l}(X). We denote m by μ_{a,l}(X, f), or if the context is clear μ_{a,l}(f).

As in the definition of l–geometric multiplicity, if l < n then μ_{a,l}(X, f) = 0.

Proposition 6. The following inequality holds :

μ_{g,l}(X, f) ≤ μ_{a,l}(X, f) .

Proof: Suppose that the degree of f is n, with μ_{g,l}(X, f) = m. Let X_{k} be a sequence of vector fields of degree d converging to X_{0} = X, with m distinct invariant algebraic curves f_{k}^{(1)}, …, f_{k}^{(m)} converging to f. By Proposition 3 the product, f_{k}^{(1)} ⋯ f_{k}^{(m)} divides E_{l}(X_{k}). Since f_{k}^{(1)} ⋯ f_{k}^{(m)} are distinct,

lim_{k→∞} f_{k}^{(1)} ⋯ f_{k}^{(m)} = f^{m} ,

implies that f^{m} divides E_{l}(X).

Example 5. In Example 1 we have calculated the extactic curve of X_{(t,b)}. From those calculations it follows

μ_{a,1}(X_{(t,b)}, l_{t}^{(i)}) = 1, for t ≠ 0 and i = 1, 2, 3, 4 ;
μ_{a,1}(X_{(0,b)}, l) = 4 .

Therefore, by Proposition 6, we obtain the equality

μ_{g,1}(X_{(0,b)}, l) = μ_{a,1}(X_{(0,b)}, l) = 4

mentioned in Example 3. For n ≥ 2 we have μ_{a,n}(X_{(0,b)}, l^{n}) ≥ 4 by Proposition 6 and Example 3, although it can be much more. Using a computer algebra system one can verify that

E_{2}(X_{(t,b)}) = by(x − t)^{4}(x + t)^{4}(−2t^{2} + 2x^{2} − bt^{3} + 4t^{3}xy + 2t^{6}y^{2})^{4} ·
(90t^{3}xy^{3} − 18bt^{3}y^{2} + 18t^{2}y^{2} + 90x^{2}y^{2} − 45bxy + 5b^{2}) ,

consequently μ_{a,2}(l^{2}) = 8.

Example 6. Now we study the algebraic multiplicity of the invariant straight line for the vector field X_{(0,b,d)} given in Example 2 and analyzed in Example 4 under the prism of geometric multiplicity. ¿From the expression of the first extactic curve E_{1}(X_{(t,b,d)}) we get immediately that μ_{a,1}(X_{(0,b,d)}, 1− by) = 1, and consequently by Proposition 6, μ_{g,1}(X_{(0,b,d)}, 1 − by) = 1.

Similarly using the expression of the second extactic curve E_{2}(X_{(t,b,d)}) it is easy to conclude that μ_{a,2}(X_{(0,b,d)}, (1 − by)^{2}) = 3. ¿From Example 4 we obtain

2 ≤ μ_{g,2}(X_{(0,b,d)}, (1 − by)^{2}) ≤ 3 .

Observe that in [15], Schlomiuk also define a notion of algebraic multiplicity. In this case this notion is of complete different nature than ours. Although, there she shows that his notions of geometric and algebraic multiplicity coincides in the particular case of quadratic Hamiltonian vector fields.

3.3. Integrable Multiplicity.

3.3.1. Derivative of an invariant algebraic curve.

Definition 9. Let X_{t}, t ∈ (−ϵ, ϵ), be an analytic 1–parameter family of vector fields of degree d such that, for each t ≠ 0, X_{t} has two distinct invariant algebraic curves f_{t}^{(1)} and f_{t}^{(2)}. Let f be an invariant algebraic curve of X = X_{0}. Assume that f_{t}^{(1)}, f_{t}^{(2)} → f as t → 0, and that

(6)  lim_{t→0} \frac{f_{t}^{(2)} − f_{t}^{(1)}}{t^{j}} = 0 for j = 1, …, r − 1 ,
(7)  lim_{t→0} \frac{f_{t}^{(2)} − f_{t}^{(1)}}{t^{r}} ≠ 0 .

In that case we call the limit (7) a derivative of the invariant algebraic curve f. From now on we denote the ℂ–vector space generated by all derivatives of f by Df.

Although the next series of results are new, the germs of its demonstrations are already presented in Section 5 of [5].

Lemma 1. Using the notation of Definition 9, if (6) and (7) hold we have that

L_{t}^{(2)} = L_{t}^{(1)} + t^{r}L + O(t^{n+1}) ,

where L_{t}^{(i)}, i = 1, 2, are the cofactors of f_{t}^{(i)} with respect to the vector field X_{t}, and L is a polynomial of degree at most d − 1.

Proof: From the following computation

X (\frac{f_{t}^{(2)} − f_{t}^{(1)}}{t^{j}} ) = \frac{L_{t}^{(2)}f_{t}^{(2)} − L_{t}^{(1)}f_{t}^{(1)}}{t^{j}} = \frac{(L_{t}^{(2)} − L_{t}^{(1)}) f_{t}^{(1)}}{t^{j}} + O(t^{r−j}) ,

and (6) we get

lim_{t→0} \frac{L_{t}^{(2)} − L_{t}^{(1)}}{t^{j}} f = 0 ,

whenever j < r. Therefore, we can write

L_{t}^{(2)} = L_{t}^{(1)} + t^{r}L + O(t^{r+1}) ,

and since L_{t}^{(i)}, i = 1, 2, are polynomials of degree at most d − 1 it follows that L has degree at most d − 1.

Theorem 2. Let X be a vector field of degree d. If g is a derivative of an invariant algebraic curve f of X, then g/f is an exponential coefficient of X.

Proof: Since g is a derivative of f there exists an analytic 1–parameter family of vector fields X_{t} of degree d, such that X_{0} = X and for a convenient ϵ there exists for every t ∈ (−ϵ, ϵ) two distinct algebraic curves f_{t}^{(1)}, f_{t}^{(2)} invariant by X_{t}, both converging to f, and satisfying

lim_{t→0} \frac{f_{t}^{(1)} − f_{t}^{(2)}}{t} = g .

We will need the next formulas later on:

X_{t} (\frac{f_{t}^{(2)}}{f_{t}^{(1)}} ) = \frac{X_{t} (f_{t}^{(2)}) f_{t}^{(1)} − X_{t} (f_{t}^{(1)}) f_{t}^{(2)}}{(f_{t}^{(1)})^{2}}

= (L_{t}^{(2)} − L_{t}^{(1)}) \frac{f_{t}^{(2)}}{f_{t}^{(1)}}

(8)  = t^{r}L\frac{f_{t}^{(2)}}{f_{t}^{(1)}} + O(t^{r+1}) .

We define g_{t} as

\frac{f_{t}^{(2)} − f_{t}^{(1)}}{t^{r}} .

¿From (8) it follows that

X_{t} ((\frac{f_{t}^{(2)}}{f_{t}^{(1)}} )^{1/t^{r}} ) = \frac{1}{t^{r}} (\frac{f_{t}^{(2)}}{f_{t}^{(1)}} )^{1/t^{r}} (\frac{f_{t}^{(2)}}{f_{t}^{(1)}} )^{−1} X_{t} (\frac{f_{t}^{(2)}}{f_{t}^{(1)}} )

= (\frac{f_{t}^{(1)} + t^{r}g_{t}}{f_{t}^{(1)}} )^{1/t^{r}} (\frac{f_{t}^{(1)} + t^{r}g_{t}}{f_{t}^{(1)}} )^{−1} (L\frac{f_{t}^{(2)}}{f_{t}^{(1)}} + O (t))

= (1 + t^{r} \frac{g_{t}}{f_{t}^{(1)}} )^{1/t^{r}} (1 + O (t^{r}) ) (L\frac{f_{t}^{(2)}}{f_{t}^{(1)}} + O (t)) .

These equalities imply that

X_{t} ((1 + t^{r} \frac{g_{t}}{f_{t}^{(1)}} )^{1/t^{r}} ) = (L\frac{f_{t}^{(2)}}{f_{t}^{(1)}} + O (t)) (1 + t^{r} \frac{g_{t}}{f_{t}^{(1)}} )^{1/t^{r}} .

Finally taking the limit in this last equality when t tends to zero we obtain

X (exp(g/f)) = L exp(g/f) .

Consequently g/f is an exponential coefficient of X, and this completes the proof of the theorem.

Corollary 1. Given an invariant algebraic curve f of the vector field X with μ_{g}(X, f) > 1, then exists at least one exponential coefficient having non–trivial denominator f_{1}, such that f_{1} divides f.

To clarify the notions of derivative of an invariant algebraic curve we now study some examples.

Example 7. Consider the 2–parameter family X_{(t,b)} given in Example 1. In Example 3 we showed that μ_{g,1}(X_{(0,b)}, x) = 4. The vector field X_{(t,b)} has exactly four invariant straight lines, which we called l_{t}^{(i)} for i = 1, …, 4. To obtain derivatives of the line x we have to compute the limits

lim_{t→0} \frac{l_{t}^{(i)} − l_{t}^{(j)}}{t} , for i, j ∈ \{1, 2, 3, 4\} and i ≠ j .

Doing such computations we have

(9)  lim_{t→0} \frac{l_{t}^{(2)} − l_{t}^{(1)}}{t} = lim_{t→0} \frac{(x + t) − (x − t)}{t} = 2 ,

(10)  lim_{t→0} \frac{l_{t}^{(3)} − l_{t}^{(1)}}{t} = lim_{t→0} \frac{(x + t^{3}y − t\sqrt{1 + bt/2}) − (x − t)}{t} = 0 ,

(11)  lim_{t→0} \frac{l_{t}^{(4)} − l_{t}^{(1)}}{t} = lim_{t→0} \frac{(x + t^{3}y + t\sqrt{1 + bt/2}) − (x − t)}{t} = 2 .

¿From equation (9) or (11) we obtain the same exponential coefficient, namely 2/x. Since the limit (10) is zero, we compute

lim_{t→0} \frac{l_{t}^{(3)} − l_{t}^{(1)}}{t^{2}} = lim_{t→0} \frac{(x + t^{3}y − t\sqrt{1 + bt/2}) − (x − t)}{t^{2}} = −\frac{b}{4} .

And from that we obtain the exponential coefficient −b/(4x). We note that this last exponential coefficient only differs by multiplication by a constant from the previous one, and hence they provide essentially a unique exponential factor.

To find others exponential coefficients we consider products of two invariant straight lines and do similar computations. For example, if we take the products of l_{t}^{(1)}l_{t}^{(3)} and l_{t}^{(2)}l_{t}^{(4)}, and calculate

lim_{t→0} \frac{l_{t}^{(1)}l_{t}^{(3)} − l_{t}^{(2)}l_{t}^{(4)}}{t} = −4x ,

we obtain as exponential coefficient −4x/x^{2} = −4/x, which again gives the same exponential factor as before. On the other hand if we compute

lim_{t→0} \frac{l_{t}^{(1)}l_{t}^{(2)} − l_{t}^{(3)}l_{t}^{(4)}}{t^{3}} = −2xy + \frac{b}{2} ,

and we get as exponential coefficient (−2xy + b/2)/x^{2}, which gives rise to a new exponential factor.

Example 8. Now, we want to study the derivative of the square of the invariant straight line for the vector field X_{(0,b,d)}, already studied in Examples 2, 4 and 6. The careful reader must have noted that in fact the conic do not tend to (1 − by)^{2}, but in fact tend to d · (1 − by)^{2} when t tends to zero. So from the definition of derivative of an multiple invariant curve, we obtain that one of the derivatives (this can be the only one) of (1 − by)^{2} is given by

g = lim_{t→0} \frac{d(1 − by)^{2} − C_{(t,b,d)}}{t} = 1 + 2(d − b)y − b^{2}x^{2} − 3bdy^{2} .

Therefore g/(1 − by)^{2} is an exponential coefficient of X_{(0,b,d)}.

3.3.2. Notion of integrable multiplicity.

Definition 10. Let f be an invariant algebraic curve of the vector field X. We denote by C_{exp}(f), the ℂ–vector subspace of the ℂ–vector space of rational functions on ℂ^{2}, formed by all exponential coefficients of X of the form g/f^{k} where g ∈ ℂ[x, y] and k runs over all the non–negative integers.

Proposition 7. The following inclusion holds

C_{exp}(f) ⊇ ⟨\frac{Df}{f} , \frac{Df^{2}}{f^{2}} , …, \frac{Df^{n}}{f^{n}} , … ⟩ .

Here the right hand side of the previous expression denotes the minimal ℂ–vector subspace of the space of rational functions that contains all the vector spaces Df^{n}/f^{n} for any positive integer.

Proof: It is an easy consequence of Theorem 2.

Definition 11. An invariant algebraic curve f for the vector field X, has integrable multiplicity m if

dim_{ℂ} C_{exp}(f) = m − 1 .

We denote m by μ_{i}(X, f), or if no confusion is possible μ_{i}(f).

The minus one in the above equality is due to the fact that we also take into account the invariant algebraic curve f in the notion of integrable multiplicity.

With this new notions, we are able to reformulate Corollary 1, as follows.

Corollary 2. Let f be an invariant algebraic curve of the vector field X. If μ_{g}(f) > 1, then μ_{i}(f) > 1.

3.4. Relations between the different notions of multiplicity. The next result can be find as Theorem 6.3 in [4]. We include it here, because it shows how to force geometric multiplicity from integrable multiplicity.

Proposition 8. Suppose a vector field X has an exponential factor exp(g/f), where g = 0 and f = 0 are non-singular and have normal crossings with themselves and the line at infinity. Then the vector field is of the form

(a_{o}f^{2} − a_{1}ff_{y} − a_{2}(g_{y}f − gf_{y})) \frac{∂}{∂x} + (a_{3}f^{2} + a_{1}ff_{x} + a_{2}(g_{x}f − gf_{x})) \frac{∂}{∂y} , (12)

where deg(a_{i}) satisfy the expected bounds. In particular, the exponential can be seen to be the limit of the invariant curves f and f +ϵg as ϵ tends to zero in the family of vector fields X_{ϵ} = p_{ϵ} \frac{∂}{∂x} +q_{ϵ} \frac{∂}{∂y}, where

p_{ϵ} = a_{0}f(f + ϵg) − (a_{1} + ϵ^{−1}a_{2})f(f + ϵg)_{y} + ϵ^{−1}a_{2}f_{y}(f + ϵg) ,
q_{ϵ} = a_{3}f(f + ϵg) + (a_{1} + ϵ^{−1}a_{2})f(f + ϵg)_{x} − ϵ^{−1}a_{2}f_{x}(f + ϵg) ,

which tends to (12) as ϵ tends to zero.

¿From Proposition 8 we get easily the next corollary.

Corollary 3. If f is an invariant algebraic curve of a vector field X and g is a derivative of f such that f and g satisfies the hypothesis of Proposition 8, then μ_{g}(X, f) ≥ 2.

We do not know if it is true or not that μ_{i}(X, f) = sup_{n} μ_{g}(X, f^{n}).

4. The role played by singular points and the line at infinity

4.1. Finite singular points. Let I ⊂ ℂ[x, y] be an ideal and p = (x_{0}, y_{0}) a point of ℂ^{2}. Denote by T I_{p} the ideal generated by all monomials (x − x_{0})^{i}(y − y_{0})^{j} which appears with nonzero coefficient in the Taylor series expansion around p of some polynomial f ∈ I.

Now, let X be a vector field of degree d, written in the form (1), and I be ideal generated by a and b. Consider the following exact sequence of ℂ–vector spaces

0 ⟶ ker π_{p} ⟶ ℂ_{d−1}[x, y] ⟶^{π_{p}} \frac{ℂ_{d−1}[x, y]}{T I_{p} ∩ ℂ_{d−1}[x, y]} ⟶ 0 ,

where π_{p} is the natural projection and ℂ_{d−1}[x, y] is the ℂ–vector space of polynomials of degree at most d − 1. Note that ker π_{p} = ℂ_{d−1}[x, y] if, and only if, p is a non-singular point of X.

Proposition 9. If f is an invariant algebraic curve of X not passing through p, then L_{f} ∈ ker π_{p}.

Proof: Without loss of generality we can assume that p = (0, 0). Since f do not pass through p, after multiplying f by a suitable complex number, it can be written as f = 1 + \overline{f}, where \overline{f}(0, 0) ≠ 0.

Suppose that π_{p}(L_{f}) ≠ 0. In this case denote by M = {m_{1}, …, m_{k}} the set of all monomials in L_{f} with nonzero coefficients such that π_{p}(m_{i}) ≠ 0.

We say that m_{i} < m_{j} if there exist (r, s) ∈ ℤ_{+}^{2} \{(0, 0)} satisfying x^{r}y^{s}m_{i} = m_{j}. Here ℤ_{+} denotes the set of all non–negative integers. Observe that < defines a partial order in the finite set M.

Take a minimal element of M with respect to this partial order and denoted it by m_{∗}. Then from the equation

a\frac{∂f}{∂x} + b\frac{∂f}{∂y} = L_{f} + \overline{f}L_{f} ,

one can see that m_{∗} appears with nonzero coefficient on the right hand side, and with zero coefficient on the left hand side (because m_{∗} ∉ T I_{p}). Therefore we obtain a contradiction and the proof is completed.

Corollary 4. If g/f is an exponential coefficient of X with f(p) ≠ 0, then L_{e} ∈ ker π_{p}, where e = exp(g/f).

Proof: The proof can be made using similar arguments to the previous proposition. The main observation is that from the equality

X (exp (\frac{g}{f} )) = L_{e} exp (\frac{g}{f} ) , (13)

we obtain

a (f \frac{∂g}{∂x} − g \frac{∂f}{∂x} ) + b (f \frac{∂g}{∂y} − g \frac{∂f}{∂y} ) = L_{e}f^{2} = L_{e} (f^{2}(p) + h.o.t.) .

Definition 12. We call ker π_{p} the space of admissible cofactors of invariant algebraic curves of X not passing through p.

4.2. The line at infinity. We say that the line at infinity is invariant by X if xb_{d} − ya_{d} ≠ 0, where a_{d} and b_{d} are the homogeneous part of degree d of the components of the vector field (1). This is due to the fact that when we extend the vector field to a foliation of the complex projective plane ℂP (2), the line at infinity turns to be invariant.

When xb_{d} − ya_{d} ≡ 0 the vector field X given by (1) can be written in the form

X = \overline{a}\frac{∂}{∂x} + \overline{b}\frac{∂}{∂y} + h (x\frac{∂}{∂x} + y \frac{∂}{∂y} ) , (14)

where \overline{a} and \overline{b} are polynomials of degree at most d − 1 and h is a homogeneous polynomial of degree d − 1. In fact h = a_{d}/x = b_{d}/y. The homogeneous polynomial h defines a set of points in ℂP (1), this set is formed by the tangencies of the induced foliation with the line at infinity. The relevance of such tangencies for our study can be seen in the next proposition.

Proposition 10. Suppose the line at infinity is not invariant. Let L be a cofactor and L_{d−1} be its homogeneous part of highest degree. Then the following statements hold.

(a) If L is a cofactor of an invariant algebraic curve f of X, then L_{d−1} = deg(f)h.

(b) If L is a cofactor of an exponential factor e = exp(g/f) of X, then L_{d−1} = 0.

Proof: Let k be the degree of f, and we denote by f_{k} the homogeneous part of higher degree of f. Since X(f) = L_{f}f and using Euler’s formula for homogeneous polynomials, we get

\overline{a}\frac{∂f}{∂x} + \overline{b}\frac{∂f}{∂y} + h (x\frac{∂}{∂x} + y \frac{∂}{∂y} ) (f − f_{k}) + khf_{k} = L_{f}f .

Comparing the homogeneous parts of highest degree in this equality it is easy to see that khf_{k} = L_{d−1}f_{k}. And consequently statement (a) is proved.

Denote by l the degree of g, and as usual g_{l} is the homogeneous part of highest degree of g. From (13) it follows that

(\overline{a} + hx) (f \frac{∂g}{∂x} − g \frac{∂f}{∂x} ) + (\overline{b} + hy) (f \frac{∂g}{∂y} − g \frac{∂f}{∂y} ) = Lf^{2} .

Equalizing the higher homogeneous parts as before, we are able to write

h (x (f_{k} \frac{∂g_{l}}{∂x} − g_{l} \frac{∂f_{k}}{∂x} ) + y (f_{k} \frac{∂g_{l}}{∂y} − g_{l} \frac{∂f_{k}}{∂y} )) = L_{d−1}f_{k}^{2} .

Applying Euler’s formula again we arrive at the following equation

(l − k)hf_{k}g_{l} = L_{d−1}f_{k}^{2} .

Comparing the degrees of both sides we get that the degree of f must be equal to the degree of g, i.e., l = k. Hence we can conclude the statement (b).

5. Darboux Integrability

The next theorem, without taking into account the exponential factors, was proved by Darboux in [6]. This later improvement was made in [4, 5]. Its proof follows from straightfoward computations, for more details see [5].

Theorem 3. Let X be a vector field. If X admits p distinct invariant algebraic curves f_{i}, for i = 1, …, p, and q independent exponential factors e_{j}, for j = 1, …, q. Then the following statements hold.

(a) If there are λ_{i}, ρ_{j} ∈ ℂ not all zero such that ∑_{i=1}^{p} λ_{i}L_{f_{i}} + ∑_{j=1}^{q} ρ_{j}L_{e_{j}} = 0, then the (multi–valued)

function f_{1}^{λ_{1}} ⋯ f_{p}^{λ_{p}}e_{1}^{ρ_{1}} ⋯ e_{q}^{ρ_{q}} is a first integral of the vector field X.

(b) If there are λ_{i}, ρ_{j} ∈ ℂ not all zero such that ∑_{i=1}^{p} λ_{i}L_{f_{i}} + ∑_{j=1}^{q} ρ_{j}L_{e_{j}} = −divX, then the function

f_{1}^{λ_{1}} ⋯ f_{p}^{λ_{p}}e_{1}^{ρ_{1}} ⋯ e_{q}^{ρ_{q}} is an integrating factor of X.

Definition 13. Let X be a vector field of degree d, and S ⊂ ℂ^{2} a finite set of points (eventually empty). The restricted cofactor space with respect to S, Σ_{S}, is defined by

Σ_{S} = ( ⋂_{p∈S} ker π_{p}) ⋂ Γ ,

where

Γ = \begin{cases}ℂ_{d−1}[x, y] & if the line at infinity is invariant, \\ ℂ_{d−2}[x, y] ⊕ h · ℂ & otherwise. \end{cases}

We recall that h is the homogeneous polynomial of degree d − 1 describing the tangencies of the vector field X with the line at infinity, which appears in formula (14).

In [2] the notion of independent singular points was introduced in order improve the Darboux theory of integrability. Recalling that each point p = (x_{0}, y_{o}) in ℂ^{2} defines a maximal ideal, m_{p} = ⟨x − x_{0}, y − y_{0}⟩ ⊂ ℂ[x, y]. We can say that p_{1}, …, p_{r} are independent singular points if

dim (∩_{i=1}^{r}m_{p_{i}}) ∩ ℂ_{d−1}[x, y] = dimℂ_{d−1}[x, y] − r .

With this notation, it is easy to prove the following result.

Proposition 11. Assume that S = {p_{1}, …, p_{r}} are independent singular points of a vector field of degree d. Then the following inequality holds

dim Σ_{S} ≤ dim ℂ_{d−1}[x, y] − r .

Theorem 4. (Integrability Theorem) Let X be a vector field of degree d. Assume that X has f_{1}, …, f_{q} distinct invariant algebraic curves, and that p_{1}, …, p_{r} are singular points of X. Suppose that f_{j}(p_{k}) ≠ 0 for j = 1, …, q and k = 1, …, r. Then following statements hold.

(a) Let μ = ∑_{j=1}^{p} μ_{i}(X, f_{j}) and σ = dim Σ_{S}, where S = {p_{1}, …, p_{r}}.

(a.1) If μ ≥ σ + 2, then X has a rational first integral. (a.2) If μ ≥ σ + 1, then X has a Darbouxian first integral. (a.3) If μ ≥ σ, then X has either a Darbouxian first integral or a Darbouxian integrating factor.

(b) If there exists a positive integer l such that

∑_{j=1}^{p} deg(f_{j})μ_{a,l}(X, f_{j}) ≥ n_{l}(X) ,

then X admits a rational first integral. If we denote the line at infinity as l_{∞}, then

n_{l}(X) = \begin{cases}\frac{d(l^{4} + 6l^{3} + 11l^{2} + 6l) − l^{4} − 2l^{3} + l^{2} − 6l}{8} , & if l_{∞} is invariant, \\ \frac{d(l^{4} + 6l^{3} + 11l^{2} + 6l) − 2l^{4} − 8l^{3} − 10l^{2} − 4l}{8} , & otherwise. \end{cases}

Proof: The hardest part of the proof has already been made in the previous sections, and here this is reflected in the way that the assumptions in the statements of the theorem are presented. Now to prove parts (a.2) and (a.3) is just a matter of counting dimensions and using Theorem 3. More precisely, one has just to observe that all possible cofactors are contained in Σ_{S} (see Propositons 9, 10 and Corollary 4). Hence if μ is at least σ we can assure that the hypothesis of statement either (a) or (b) of Theorem 3 are fullfilled, and hence (a.3) follows.

In the case that μ is at least σ + 1 we can garantee that the assumption of statement (a) of Theorem 3 holds, therefore (a.2) holds.

When μ is at least σ + 2, from (a.2) we can obtain two independent Darbouxian first integrals, say H_{1} and H_{2}. We can see easily that the integrating factor R_{i} associated to log H_{i} is a rational function. Since the quotient of two integrating factors is a first integral, the statement (a.1) follows from the independence of H_{1} and H_{2}.

Statement (b) can be proved observing that the degree of E_{l}(X), together with Theorem 1 and Proposition 3 give bounds for the number of invariant algebraic curves of degree at most l. For more details see [10].

Corollary 5. If there exists a positive integer l such that

∑_{j=1}^{p} deg(f_{j})μ_{g,l}(X, f_{j}) ≥ n_{l}(X) ,

then X admits a rational first integral.

6. Strong Algebraic Multiplicity

Definition 14. If X is a vector field on ℂ^{2} the n–th extactic ideal of X, EI_{n}(X), is generated by σ_{(k_{1},... ,k_{l})}, where

det (\begin{matrix}X^{k_{1}}(v_{1}) & X^{k_{1}}(v_{2}) & ⋯ & X^{k_{1}}(v_{l}) \\ X^{k_{2}}(v_{1}) & X^{k_{2}}(v_{2}) & ⋯ & X^{k_{2}}(v_{l}) \\ ⋮ & ⋮ & ⋯ & ⋮ \\ X^{k_{l}}(v_{1}) & X^{k_{l}}(v_{2}) & ⋯ & X^{k_{l}}(v_{l}) \end{matrix}) , (15)

where 0 ≤ k_{1} < k_{2} < ⋯ < k_{l} and the k_{i}’s are integers, v_{1}, v_{2}, …, v_{l} is a basis of ℂ_{n}[x, y] (so l = (n + 1)(n + 2)/2).

Definition 15. An invariant algebraic curve f of degree n for the vector field X has l–strong algebraic multiplicity m when m is the smallest positive integer such that the m–th power of f belongs to the extactic ideal, EI_{l}(X). We denote m by μ_{sa,l}(X, f), or if the context is clear μ_{sa,l}(f).

Proposition 6 can be easily generalized to give the pair of inequalities,

μ_{g,l}(X, f) ≤ μ_{sa,l}(X, f) ≤ μ_{a,l}(X, f) . (16)

Example 9. We have showed, in Example 6, that the square of the invariant straight line 1 − by of the vector field X_{(0,b,d)} has 2–algebraic multiplicity 3, and hence we know that its 2–geometric multiplicty is either 2 or 3. With just two generators of the second extactic ideal EI_{n}(X_{(0,b,d)}), namely σ_{(0,1,... ,5,6)} and σ_{(0,1,... ,5,7)}, one can see that the 2–strong algebraic multiplicity is exactly 2. Here we use the inequality (16), and also from it follows that the 2–geometric multiplicity is exactly 2.

7. Final Remarks

Although we studied vector fields in the complex plane, all results here extend easily to real polynomial vector fields defined on the real plane. To do that one has just to think the real vector fields as been complex, and after obtaining a complex first integral or complex integrating we can return to the real world using the results presented in Section 3 of [9].

We also remark that it is possible to generalize our results to codimension 1 foliations on ℂ^{n} given by polynomial 1–forms. To see a hint how to do that we suggest that reader consult the work of Jouanolou [7].

To conclude we would like to emphasize that at first sight the concept of geometric multiplicity is not computable, neverless the algebraic and strong algebraic multiplicity gives a computational approach to bound such multiplicity. We believe that in some sense the strong algebraic multiplicity contains all information necessary to obtain the geometric multiplicity.

8. Acknowledgments

We want to thank the organizers of the Workshop on asymptotic series, differential algebra and finiteness problems in non–linear dynamical systems held at the Centre de Recherces Mathématiques – Université de Montréal. Their support during our stay allowed us to have a great atmosphere to conceive this paper. Specially we want to express our gratitude to Dana Schlomiuk and Nicolas Vulpe for providing examples that permitted us to test the different notions of multiplicity.

The first author is partially supported by DGES grant number PB96–1153 and by CICYT grant number 1999SGR 00399.

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