# Planar quadratic vector fields with invariant lines of total multiplicity at least five

###### Abstract

In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms of algebraic invariants and comitants and also geometrically, using divisors of the complex projective plane, the class of quadratic differential systems with at least five invariant lines. These conditions are such that no matter how a system may be presented, one can verify by using them whether the system has or does not have at least five invariant lines and to check to which orbit (or family of orbits) it belongs.

Keywords: quadratic differential system, Poincaré compactification, algebraic invariant curve, algebraic affine invariant, configuration of invariant lines.

## 1 Introduction

We consider here real planar differential systems of the form

(1.1) |

where , i.e. are polynomials in over , and their associated vector fields

(1.2) |

Each such system generates a complex differential vector field when the variables range over . To the complex systems we can apply the work of Darboux on integrability via invariant algebraic curves (cf.[7]). For a brief introduction to the work of Darboux we refer to the survey article [23]. Some applications of the work of Darboux in connection with the problem of the center are given in [24].

For the system (1.1) we can use the following definition.

###### Definition 1.1.

An affine algebraic invariant curve of a polynomial system (1.1) (or an algebraic particular integral) is a curve where , , such that there exists satisfying in . We call the cofactor of with respect to the system.

Poincaré was the first to appreciate the work of Darboux [7], which it called ”admirable” (see [17]) and inspired by Darboux’s work, Poincaré wrote two articles [18],[19] where he also stated a problem still open today.

With this brilliant work Darboux open up a whole new area of investigations where one studies how the presence of particular algebraic integrals impacts on global properties of the systems, for example on global integrability. In recent years there has been a surge in activity in this area of research and this article is part of a growing literature in the subject. In particular we mention here [4], [6] and the work of C. Christopher, J.V. Perreira and J. Llibre on the notion of multiplicity of an invariant algebraic curve of a differential system [5].

In this article, which is based on [26], we study systematically the simplest kind of such a structure, i.e. quadratic systems (1.1) possessing invariant lines. Some references on this topic are: [29, 8, 2, 13, 14, 22, 20, 30, 33, 12].

To a line we associate its projective completion under the embedding , . The line is called the line at infinity of the system (1.1). It follows from the work of Darboux that each system of differential equations of the form (1.1) yields a differential equation on the complex projective plane which is the compactification of the complex system (1.1) on (cf. Section 2). The line is an invariant manifold of this complex differential equation.

###### Notation 1.1.

Let us denote by

For the multiplicity of the line at infinity the reader is refereed to Section 2.

We shall call degenerate quadratic differential system a system (1.1) with and .

###### Notation 1.2.

To a system (1.1) in we can associate a point in , the ordered tuple of the coefficients of , and this correspondence is an injection

(1.3) | ||||

The topology of yields an induced topology on QS.

###### Definition 1.2.

We say that an invariant straight line , , for a quadratic vector field has multiplicity if there exists a sequence of real quadratic vector fields converging to , such that each has distinct (complex) invariant straight lines , converging to as (with the topology of their coefficients), and this does not occur for .

###### Proposition 1.1.

[2] The maximum number of invariant lines (including the line at infinity and including multiplicities) which a quadratic system could have is six.

###### Definition 1.3.

We call configuration of invariant lines of a system in the set of all its invariant lines (real or/and complex), each endowed with its own multiplicity and together with all the real singular points of located on these lines, each one endowed with its own multiplicity.

We associate to each system in QSL its configuration of invariant lines. In analogous manner to how we view the phase portraits of the systems on the Poincaré disc (see for example, [11]), we can also view the configurations of real lines on the disc. To help imagining the full configurations, we complete the picture by drawing dashed lines whenever these are complex.

On the class of quadratic systems acts the group of real affine transformations and time rescaling. Since quadratic systems depend on 12 parameters and since this group depends on 7 parameters, the class of quadratic systems modulo this group action, actually depends on five parameters.

It is clear that the configuration of invariant lines of a system is an affine invariant. The notion of multiplicity defined by Definition 1.2 is invariant under the group action, i.e. if a quadratic system has an invariant line of multiplicity , then each system in the orbit of under the group action has an invariant line of the same multiplicity .

In this article we shall consider the case when the system (1.1) has at least five invariant lines considered with their multiplicities.

The problems which we solve in this article are the following:

I) Construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at infinity and including multiplicities. For each orbit exhibit its configuration.

II) Characterize in terms of algebraic invariants and comitants and also geometrically, using divisors or zero-cycles of the complex projective plane, the class of quadratic differential systems with at least five invariant lines. These conditions should be such that no matter how a system may be presented to us, we should be able to verify by using them whether the system has or does not have at least five invariant lines and to check to which orbit or perhaps family of orbits it belongs.

Our main results are formulated in Theorems 5.1 and 6.1. Theorem 5.1 gives a total of 11 distinct orbits of systems with a configuration with exactly six invariant lines including the line at infinity and including multiplicities. Theorem 6.1 gives a system of representatives for 17 distinct orbits of systems with exactly five invariant lines including the line at infinity and including multiplicities. Furthermore we give a complete list of representatives of the remaining orbits which are classified in 13 one-parameter families. We characterize each one of these 13 families in terms of algebraic invariants and comitants and geometrically. As the calculation of invariants and comitants can be implemented on a computer, this verification can be done by a computer.

All quadratic systems with at least five invariant lines including the line at infinity and including multiplicities are algebraically integrable, i.e. they all have the rational first integrals and the phase portraits of these systems can easily be drawn. We leave the discussion of issues related to integrability, as well as the drawing of the phase portraits of the systems we consider here, in a follow up paper of this work.

## 2 Differential equations in of first degree and first order and their invariant projective curves

In [7] Darboux considered differential equations of first degree and first order of the complex projective plane. These are equations of the form

() |

where , , are homogeneous polynomials of the same degree
. These are called equations in Clebsch form . ^{1}^{1}1
Darboux
used the notion of Clebsch connex to define them.

We remark that we can have an infinity of such equations yielding the same integral curves. Indeed, for any ordered triple of homogeneous polynomials of the same degree and for any homogeneous polynomial of degree , the -equation corresponding to

(2.1) |

has the same integral curves as the equation . Two equations determined by polynomials and satisfying (2.1) are said to be equivalent.

###### Theorem 2.1 (Darboux, [7]).

Let be homogeneous polynomials of the same degree over . Then there exists a unique , more precisely

such that if satisfy (2.1) for this then

###### Theorem 2.2 (Darboux, [7]).

Every equation (CF) with is equivalent to an equation

(2.2) |

where are homogeneous polynomials of degree subject to the identity

(2.3) |

###### Definition 2.1 (Darboux, [7]).

An algebraic invariant curve for an equation is a projective curve where is a homogeneous polynomial over such that where is the differential operator

i.e. such that . is called the cofactor of with respect to the equation .

We now show that this definition is in agreement with Definition 1.1, i.e. it includes as a particular Definition 1.1.

To a system (1.1) we can associate an equation (2.2) subject to the identity (2.3). We first associate to the systems (1.1) the differential form

and its associated differential equation .

We consider the map , given by and suppose that . Since and we have:

the pull–back form has poles at and its associated equation can be written as

Then the –form in has homogeneous polynomial coefficients of degree , and for the equations and have the same solutions. Therefore the differential equation can be written as (2.2) where

(2.4) | |||||

and . Clearly , and are homogeneous polynomials of degree satisfying (2.3).

The equation (2.2) becomes in this case

or equivalently

(2.5) |

We observe that is an algebraic invariant curve of this equation according to Definition 2.1, with cofactor . We shall also say that is an invariant line for the systems (1.1).

To an affine algebraic curve , , we can associate its projective completion where . From the indicated above the correspondence between a system (1.1) and equation (2.5) follows the next proposition.

###### Proposition 2.1.

Conversely, starting now with an equation in Clebsch form we can consider its restriction on the affine chart and associate a differential system:

(2.6) |

where , , . The following proposition follows easily by using Euler’s formula for a homogeneous polynomial of degree n.

###### Proposition 2.2.

## 3 Divisors associated to invariant lines configurations

Consider real differential systems of the form:

(3.1) |

with

Let be the 12-tuple of the coefficients of system (3.1) and denote .

###### Notation 3.1.

Let us denote by a point in . Each particular system (3.1) yields an ordered 12-tuple of its coefficients.

###### Notation 3.2.

Let

We denote .

###### Definition 3.1.

We consider formal expressions of the form where or is an irreducible curve of and is an integer and only a finite number of are not zero. Such an expression will be called a zero-cycle of (respectively a divisor of or a divisor of ) if (respectively, belongs to the line , or is an irreducible curve of ). We call degree of the expression the integer . We call support of the set of points such that .

In this section, for systems (3.1) we shall assume the conditions and .

###### Definition 3.2.

Let .

where is the intersection number (see, [9]) of the curves defined by homogeneous polynomials and .

###### Notation 3.3.

(3.2) | ||||

A complex projective line is invariant for the system if either it coincides with or it is the projective completion of an invariant affine line .

###### Notation 3.4.

Let . Let us denote

###### Remark 3.5.

We note that the line is included in for any .

Let , , be all the distinct invariant affine lines (real or complex) of a system . Let be the complex projective completion of .

###### Notation 3.6.

We denote

###### Definition 3.3.

###### Notation 3.7.

(3.3) | ||||

## 4 The main -comitants associated to configurations of invariant lines

On the set of all differential systems of the form (3.1) acts the group of affine transformation on the plane. Indeed for every , we have:

where is a nonsingular matrix, is a matrix over . For every we can form its transformed system :

where

The map

verifies the axioms for a left group action. For every subgroup we have an induced action of on . We can identify the set of systems (3.1) with via the map which associates to each system (3.1) the 12-tuple of its coefficients.

The action of on yields an action of this group on . For every let , where is the 12-tuple of coefficients of . We know (cf. [27]) that is linear and that the map thus obtained is a group homomorphism. For every subgroup of , induces a representation of onto a subgroup of .

###### Definition 4.1.

A polynomial is called a comitant of systems (3.1) with respect to a subgroup of , if there exists such that for every and for every the following relation holds:

where . If the polynomial does not explicitly depend on and then it is called invariant. The number is called the weight of the comitant . If (or ) then the comitant of systems (3.1) is called -comitant (respectively, affine comitant).

###### Definition 4.2.

A subset will be called -invariant, if for every we have .

Let us consider the polynomials

As it was shown in [27] the polynomials

(4.1) |

of degree one in the coefficients of systems (3.1) are -comitants of these systems.

###### Theorem 4.1.

Let be the subgroup of formed by translations. Consider the linear representation of into its corresponding subgroup , i.e. for every , we consider as above .

###### Definition 4.3.

Consider polynomials , and assume that the polynomials are -comitants of systems (3.1) where denotes the degree of the binary form in and with coefficients in . We denote by

the set of the coefficients in of the -comitants , and by its zero set:

###### Definition 4.4.

Let be -comitants of systems (3.1) and homogeneous polynomials in the coefficients of these systems. A -comitant of systems (3.1) is called a conditional -comitant (or -comitant) modulo (i.e. modulo the ideal generated by in the ring ) if the following two conditions are satisfied:

(i) the algebraic subset is -invariant (see Definition 4.2);

(ii) for every ( we have

###### Definition 4.5.

A polynomial , homogeneous of even degree in , has well determined sign on with respect to if for every , the binary form yields a function of constant sign on .

###### Observation 4.2.

We draw the attention to the fact, that if a -comitant of systems (3.1) of even weight is a binary form of even degree in and and of even degree in and also has well determined sign on some -invariant algebraic subset , then this sign is conserved after an affine transformation of the plane and time rescaling.

We now construct polynomials and which will be shown in Lemma 6.3 to be -comitants.

###### Notation 4.3.

Consider the polynomial where , and . Then

###### Proposition 4.1.

Consider distinct directions in the affine plane, where by direction we mean a point . For the existence of an invariant straight line of a system of coefficients corresponding to each one of these directions it is necessary that there exist distinct common factors of the polynomials and over .

Proof: Suppose that is an invariant line for a quadratic system corresponding to . Then we must have such that

(4.3) |

Hence

So is a reducible conic which occurs if and only if the respective determinant . But . The point at infinity of is and so . Hence, the two homogeneous polynomials of degree 3 in , must have the common factor .

###### Remark 4.4.

Consider two parallel invariant affine lines , , of a quadratic system of coefficients . Then , i.e. the T-comitant can be used for determining the directions of parallel invariant lines of systems (3.1).

Indeed, according to (4.3) from the hypothesis we must have

Therefore for the quadratic form in and : we obtain and hence . Then calculations yield: and hence