ON THE NUMBER OF N–DIMENSIONAL INVARIANT SPHERES IN POLYNOMIAL VECTOR FIELDS OF C

We study the polynomial vector fields X = n+1


Introduction and statement of the main results
Let X be the polynomial vector field in C n+1 defined by where every P i is a polynomial of degree m i in the variables x 1 , . . . , x n+1 with coefficients in C. We say that m = (m 1 , . . . , m n+1 ) is the degree of the polynomial field, we assume without loss of generality that m 1 ≥ · · · ≥ m n+1 . We recall that the polynomial differential system in C n+1 of degree m associated with the vector field X is written as dx i dt = P i (x 1 , . . . , x n+1 ), i = 1, . . . , n + 1.
By the Darboux theory of integrability we know that the existence of a sufficiently large number of invariant algebraic hypersurfaces guarantees the existence of a first integral for the polynomial vector field X which can be calculated explicitly, see for instance [4,6]. As usual C[x 1 , . . . , x n+1 ] denotes the ring of all polynomials in the variables x 1 , . . . , x n+1 and coefficients in C. We recall that an invariant algebraic hypersurface of X is a hypersurface f (x 1 , . . . , x n+1 ) = 0 with f ∈ C[x 1 , . . . , x n+1 ] such that there exists a polynomial K ∈ C[x 1 , . . . , x n+1 ] satisfying X f = ∇f · X = Kf . This polynomial K is called the cofactor of the invariant algebraic hypersurface f = 0, and ∇f denotes the gradient of the function f . Note that if the vector field X has degree m, then any cofactor has at most degree m 1 − 1. If the degree of f is 1 then the hypersurface f = 0 is an invariant hyperplane. Here we study the algebraic hypersurfaces f = (x 1 − a 1 ) 2 + . . . + (x n+1 − a n+1 ) 2 − r 2 = 0 that are invariant n-dimensional spheres. Now we introduce one of the best tools in order to look for invariant algebraic hypersurfaces. Let W be a vector subspace of the space of polynomials C[x 1 , . . . , x n+1 ] generated by the polynomials v 1 , . . . , v l . The extactic algebraic hypersurface of X associated with W is . From the properties of the determinants and of the derivation we know that the definition of extactic algebraic hypersurface is independent of the chosen basis of W . The notion of extactic algebraic hypersurface ε W (X ) is important here for two reasons. First it allows to detect when an algebraic hypersurface f = 0 with f ∈ W is invariant by the polynomial vector field X , see the next proposition proved in [3] for polynomial vector fields in C 2 ; the extension to C n+1 is easy and it is presented here. Second ε W (X ) is also important because it allows to define and compute easily the multiplicity of an invariant algebraic hypersurface. Proposition 1.1. Let X be a polynomial vector field in C n+1 and let W be a finitely generated C-vector subspace of C[x 1 , . . . , x n+1 ] with dim(W ) > 1. Then every algebraic invariant hypersurface f = 0 for the vector field X , with f ∈ W , is a factor of the polynomial ε W (X ).
The number of invariant straight lines for polynomial vector fields in C 2 has been studied by several authors, see [1]. We know that for polynomial vector fields of degree (2, 2) the maximal number of invariant straight lines is 5. Xiang Zhang [8] and Sokulski [7] proved that the maximal number of real invariant straight lines for polynomial vector fields in R 2 of degrees (3,3) and (4,4) are 8 and 9, respectively. Later on Llibre and Medrado in [5] generalized these results to invariant hyperplanes of polynomial vector fields in C n+1 .
We want to know the maximum number of n-dimensional invariant spheres that a polynomial vector field in C n+1 can have. We begin this study by analyzing first the case C 2 . That is, given a polynomial vector field X = P ∂ ∂x + Q ∂ ∂y with P and Q polynomials in the variables x and y of degree m = (m 1 , m 2 ), m 1 ≥ m 2 and f = f (x, y) = (x − a) 2 + (y − b) 2 − r 2 = 0 a circle with center (a, b) and radius r > 0, we find an upper bound for the maximum number of invariant circles f = 0 of the polynomial vector field X , and we determine if this bound is reached or not in the class of all polynomial vector fields with degree (m 1 , m 2 ).
The multiplicity of an invariant circle f = (x − a) 2 + (y − b) 2 − r 2 = 0 is the greatest positive integer k such that f k divides the polynomial ε W (X ) with W generated by 1, x 2 , x, y 2 , y. When we study the maximum number of concentric invariant circles of a polynomial vector field X , doing a translation of the center, we can assume without loss of generality, that the center is at the origin. Then the multiplicity of an invariant circle centered at the origin f = x 2 + y 2 − r 2 = 0 is the greatest positive integer k such that f k divides the polynomial ε W (X ) with W generated by 1, x 2 , y 2 .
Below we present the main results of this article in C 2 and in the next section in C n+1 . In both cases we provide upper bounds for the number of invariant ndimensional spheres concentric or not that a polynomial vector field in C 2 or C n+1 can possess. Moreover we show that these bounds are in general not reached. We also determine the exact upper bound for the number of invariant circles concentric or not that polynomial vector fields of degree (2, 2) in C 2 can have. (a) The number of invariant circles of X taking into account its multiplicity is at most 4m 1 + m 2 − 2.
(b) The number of concentric invariant circles of X taking into account its multiplicity is at most Corollary 1.1. Assume that a polynomial vector field X in C 2 of degree m = (m, m) has a finite number of invariant circles. Then the following statements holds.
(a) The number of invariant circles of X taking into account its multiplicity is at most 5m − 2.
(b) The number of concentric invariant circles of X taking into account its multiplicity is at most [(3m + 1)/2].
Proposition 1.2. The bounds given by Theorem 1.1 are not reached.
Assume that a polynomial vector field X in C 2 of degree (2, 2) has a finite number of invariant circles. Then the following statements holds.
(a) The number of invariant circles of X taking into account its multiplicity is at most 2.
(b) The number of concentric invariant circles of X taking into account its multiplicity is at most 1.
Now we generalize the previous results to invariant n-dimensional spheres of polynomial vector fields in C n+1 .
The multiplicity of an invariant n-dimensional sphere The multiplicity of an invariant n-dimensional sphere with center at the origin Assume that a polynomial vector field X in C n+1 with n ≥ 1 of degree m = (m 1 , . . . , m n+1 ) with m 1 ≥ · · · ≥ m n+1 has finitely many invariant n-dimensional spheres. Then the following statements hold.
(a) The number of invariant n-dimensional spheres of X taking into account its multiplicity is at most The number of concentric invariant n-dimensional spheres of X taking into account its multiplicity is at most Assume that a polynomial vector field X in C n+1 with n ≥ 1 of degree m = (m, . . . , m) has finitely many invariant n-dimensional spheres. Then the following statements hold.
(b) The number of invariant n-dimensional spheres of X taking into account its multiplicity is at most All the results in C 2 are proved in section 2, and the results in C n+1 are shown in section 3.

Invariants circles
First we shall show that any invariant algebraic hypersurface is a factor of a convenient extactic algebraic hypersurface.
Proof of Proposition 1.1. Let f = 0 be an invariant algebraic hypersurface X such that f ∈ W . As was observed, the choice of the basis v 1 , . . . , v l of W plays no role in the definition of extactic algebraic hypersurface, therefore we can take v 1 = f in (1.1).
Using induction we shall see that X k (f ) = K k f being K k a polynomial for k = 1, 2, . . . In fact, if k = 1 we have X (f ) = K 1 f where K 1 is the cofactor of the invariant algebraic hypersurface f = 0. Suppose that X k (f ) = K k f being K k a polynomial, and see the case k + 1. We have Thus f appears in all terms in the first column of the determinant ε W (X ), and therefore f is a factor of the polynomial ε W (X ).
Proof of Theorem 1.1. In order to prove the statement (a) we take the subspace W generated by the polynomials {1, x 2 , x, y 2 , y}. Then if f = 0 is a circle we have that f ∈ W .
By Proposition 1.1 if f = 0 is an invariant circle of X , then f is a factor of the polynomial Note that the degree of the polynomial a ij of the previous matrix verifies that deg(a ij ) = deg(a i−1 j ) + m 1 − 1 if i = 2, 3, 4, 5 and j = 2, 3, 4, 5. In fact, every time that we apply the polynomial vector field X to a polynomial, the degree of the resultant polynomial is determined by the first term that multiplies P 1 (polynomial of higher degree) and decreases by 1 for the derivation due to the application of the vector field. For instance, since the degree of X (x) is m 1 the degree of X 2 (x) is 2m 1 − 1.
Note that the previous degree corresponds to the degree of the polynomial X 4 (x 2 ) X 3 (x)X 2 (y 2 )X (y), which corresponds to a permutation of four elements with maximal degree in the expansion of the determinant (2.1).
Since the polynomial ε W (X ) can have at most its degree divided by 2 factors of the form f (x, y) = (x − a) 2 + (y − b) 2 − r 2 , by Proposition 1.1 it follows statement (a) of the theorem. Now consider a set of concentric circles. Doing an appropriate change of coordinates we can consider that the equations of these circles are the form x 2 +y 2 −r 2 = 0. Thus all these circles can be written in the form f = 0 with f ∈ W , where W is the C-vector subspace of C[x, y] generated by 1, x 2 , y 2 . Therefore by Proposition 1.1 if f = 0 is one of these circles, then f is a factor So the degree of the polynomial ε W (X ) is 2m 1 + m 2 + 1 and therefore the statement (b) of the theorem holds.
Proof of Corollary 1.1. The proof is obtained from Theorem 1.1 by a simple calculation. Now we show that the bounds given by Theorem 1.1 are not reached. For this we need the following lemma which is proved in [2]. Recall that an algebraic curve f = f (x, y) = 0 is non-singular if there are points at which f and its first derivatives f x and f y are all zero.
Lemma 2.1. Assume that a polynomial differential system has a non-singular invariant algebraic curve f = 0. If (f x , f y ) = 1, then the polynomial differential system can be written into the forṁ where A, B and C are suitable polynomials.
Proof of Proposition 1.2. We shall prove that the maximum number of invariant circles that a quadratic vector field can have is less than 4m 1 + m 2 − 2. We also will prove that the maximum number of invariant concentric circles of a quadratic vector field is less than m 1 + [(m 2 + 1)/2]. Consequently the two upper bounds of Theorem 1.1 are not reached for quadratic vector fields.
Suppose that a quadratic vector field X has an invariant circle, doing an affine change of coordinates, if necessary, the center of the circle can be translated to the origin and its radius can be taken equals 1, i.e. it can be written as x 2 + y 2 − 1 = 0. Therefore, Lemma 2.1 states that the quadratic differential system associated the field X can be written in the form (2.2) with A, B ∈ R and C = (ax + by + c)/2, this isẋ = A(x 2 + y 2 − 1) + y(ax + by + c), y = B(x 2 + y 2 − 1) − x(ax + by + c), Note that renaming the coefficients of this quadratic system we obtain the quadratic system (2.3) with B = 0. So we can work with the quadratic systeṁ x = A(x 2 + y 2 − 1) + y(ax + by + c),ẏ = −x(ax + by + c).
Consider the circle with center (p, q) and radius r, i.e. f (x, y) = (x − p) 2 + (y − q) 2 − r 2 = 0. If f = 0 is an invariant circle of X , then it verifies X f = Kf , where K = K(x, y) = Dx + Ey + F . Therefore, we obtain (2.4) We will analyze the two cases that interest us, solutions where we get concentric invariant circles and, in the general case solutions which give rise to any invariant circle that the quadratic vector field X can have.
If A = 0 then also D = 0, and the resulting system associated with the vector field X isẋ = y(c + ax + by),ẏ = −x(c + ax + by), which has all the circles f (x, y) = x 2 + y 2 − r 2 = 0 with arbitrary r > 0 invariant. Therefore this vector field has infinitely many invariant circles, and these fields are not considered in Theorem 1.1.
In short a quadratic vector field can have at most one concentric invariant circle. Nontrivial real solutions different from the unit circle that we obtain by solving equation (2.4) are This is, we obtain the invariant circle f (x, y) = x 2 + (y + 1) 2 − r 2 = 0 for the systemẋ . This is, we obtain the invariant circle f (x, y) = x 2 + (y − 1) 2 − r 2 = 0 for the systeṁ This is, we obtain the invariant circle f (x, y) = x 2 + (y − q) 2 − r 2 = 0 for the systemẋ (2.7) Thus, systems (2.5), (2.6) and (2.7) possess two invariant circles. Therefore we have proved that the maximum number of invariant circles for a quadratic vector field is two.
Proof of Corollary 1.2. The proof follows immediately from the proof of Proposition 1.2.

Invariant n-dimensional spheres
In this section we shall prove Theorem 1.2 and Corollary 1.3 which contain our results for polynomial vector fields in C n+1 .
Since the number of factors (x 1 − a 1 ) 2 + (x 2 − a 2 ) 2 + · · ·+ (x n+1 − a n+1 ) 2 − r 2 = 0 of the polynomial ε W (X ) is at most its degree divided by 2, by Proposition 1.1 it follows the number of statement (a) of the theorem. Now we consider a set of concentric invariant n-dimensional spheres. Doing an appropriate change of coordinates we can consider that the equations of these spheres have the form x 2 1 + x 2 2 + · · · + x 2 n+1 − r 2 = 0. Then all these spheres can be written in the form f = 0 with f ∈ W , where W is the C-vector subspace of C[x 1 , x 2 , · · · , x n+1 ] generated by 1, x 2 1 , x 2 2 , . . . , x 2 n+1 . So, by Proposition 1.1 if f = 0 is one of these spheres, then f is a factor of Since m 1 ≥ · · · ≥ m n+1 and from the definition of determinant it follow that the degree of the polynomial ε W (X ) is the degree of one of the polynomials of the determinant (3.2) corresponding to a permutation of n + 1 of their elements with maximal degree, which is given by the expression X n+1 (x 2 1 )X n (x 2 2 )X n−1 (x 2 3 ) · · · X 2 (x 2 n )X (x 2 n+1 ).
Proof of Corollary 1.3. The proof is obtained by a simple calculation using Theorem 1.2.