The 10th Seminar on Differential Equations and Dynamic Systems 6-7 November 2013, University of Mazandaran, Babolsar, Iran, pp xx-xx
Monodromy Group of Polynomials and Application in Differential Equations Razie Shafeii Lashkarian ∗ , Dariush Behmardi, Shahnaz Taheri Department of Mathematics, University of Alzahra, Tehran, Iran E-mail: razieh
[email protected],
[email protected] ,
[email protected] Abstract. A polynomial P is called definite with respect to [a, b] if for any polynomial q vanishing of the one sided moments implies (and hence is equivalent to) the composition condition. There is a relation between the geometry of the branches P −1 (z) and definiteness, this relation can be stated by the monodromy group of P . In this paper we recall the monodromy group of polynomials and some classes of definite polynomials and we give a new class of definite polynomials. Keywords: definite polynomials,composition condition, composition conjecture, Abel equation, moment, monodromy group of polynomials. 2010 MSC: 34C25 30E05 34M35.
1. Introduction Consider the planar system (1.1) ∗
x˙ = P (x, y) y˙ = Q(x, y)
Speaker. 1
2
where P and Q are analytic functions in some planar region Ω, vanishing at the origin together with their first order derivatives. If the origin is a center of linearized system, then it is a center or a focus of the system (1.1). Distinguishing between the centers and the foci of the nonlinear system (1.1) is an old open problem and has been remained unsolved even when P and Q are homogenous polynomials of degree three. (Poincar´ e-center-focus problem) By using an appropriate change of coordinates in some planar systems, the system converts to a scalar non-autonomous differential equation of the form (1.2)
z˙ =
m X
Ak (t)z k .
k=0
There is a one to one correspondence between the limit cycles surrounding the origin of (1.1) and the positive periodic solutions of (1.2), for more details see [6, 7, 9, 14]. A solution u(t) of the equation (1.2) is called periodic if for some ω ∈ R we have u(0) = u(ω). If m = 3, the equation (1.2) is called an Abel equation. The system (1.1) with homogeneous P, Q of degree d can be reduced to the Abel differential equation x˙ = A(t)x3 + B(t)x2 ,
(1.3)
where A(t), B(t) are polynomials in sin(t), cos(t) of degrees 2d + 1, d + 1 respectively. Consider the Abel equation (1.3) where A(t), R B(t) are continuous R functions, if there ˜ ˜ exists a periodic function u(t) such that A(t)dt = A(u(t)), B(t)dt = B(u(t)), ˜ are continuous too, then the origin is a center for (1.3) [3] (In this case where A˜ and B is said that A(t), B(t) satisfy the composition condition). The composition conjecture asks if the composition condition is necessary condition too. This conjecture was first appeared for trigonometric polynomials in [3], and it was answered negatively in [1]. The composition conjecture was considered later for polynomials A(t), B(t) and it was answered negatively too [10]. It is notable to construct classes for which the composition conjecture is true, see for example [2]. another interesting conjecture in this field is the composition conjecture for the moments or the moment vanishing problem [4]. This conjecture states that A(t), B(t) satisfy the composition condition if and only if all the first order one sided moments of them equals zero, where the one sided moments are defined by: Z b Z b k k B (t)A(t)dt, µk = A (t)B(t)dt, (1.4) mk = a
a
Rt
Rt
where A(t) = a A(s)ds, B(t) = a B(s)ds. There are examples of trigonometric polynomials which answer this conjecture negatively. It is not known any example of polynomials that both moment conditions (1.4) are true but the composition condition is not satisfied. In this paper we study some families of polynomials for which the composition condition and vanishing the moments are equivalent. In section 2 we give the definition of the definite polynomials and give a survey of the results which describe some families of definite polynomials. In section 3 we recall the notion of
3
the monodromy group of polynomials and in the final section by using this concept we give a new family of definite polynomials. 2. definite polynomials Definition 2.1. A polynomial P is called definite (with respect to a, b ∈ C) if for any polynomial q the vanishing of the one-sided moments mk , µk , 0 ≤ k < ∞ implies R (and hence is equivalent to the Polynomial Composition condition for P andQ = q
Definition 2.2. A polynomial P is called decomposable if there is a nontrivial composition representation P (x) = R (S(x)) with the degrees of both polynomials R and S greater than 1.
There are some classes of polynomials which are definite. We present some of them in the following. (1) If a and b are simple zeroes of P then P is definite with respect to [a, b], see [8]. (2) If all the zeroes of P except possibly a, b are simple then P is definite with respect to [a, b], see [5]. (3) if for any critical value cof P except possibly 0 the pre-image P −1 (c) contains exactly one critical point of P and P (a) = P (b) = 0 for some a, b then P is definite on [a, b], see [5]. (4) If P is indecomposable then P is definite with respect to [a, b] for any a, b at which P (a) = P (b) [12]. (5) Every P of a prime degree is indecomposable and hence is definite with respect to [a, b] for any a, b at which P (a) = P (b) [12]. (6) Every polynomial up to degree 5 is definite with respect to any closed interval [a, b]. There is a familiar example of non-definite polynomial. √
√
Example 2.3. Let a = − 23 , b = 23 and let P is Chebyshev polynomial T6 , then P is not definite with respect to [a, b] [10]. There is a strong relation between the branches P −1 (z) and to be definite. In the next section we recall this relation with the notion of monodromy group of the polynomials and we prove a result that generalize the last case (polynomials up to degree 5). 3. the monodromy group of polynomials In this section we introduce the monodromy group of a polynomial P (z), for more details see [11]. Let c be a noncritical value of P (z) and let c1 , c2 , ..., ck be all finite critical values. Join c with c1 , c2 , ..., ck by non-intersecting arcs γ1 , γ2 , ..., γk and make a star S. If it is necessary renumerate c1 , c2 , ..., ck in such a way that in a counterclockwise rotation around c every γs , 1 ≤ s ≤ k − 1, is followed by γs+1. Show the preimage of S under the map P (z) : C → C by λP . Indeed λP is a (n + 1)−colored
4
graph embedded into the Riemann sphere: for each 1 ≤ s ≤ k preimages of the point cs , are vertices of λP colored by the sth color, and preimages of the point c are vertices colored by white color, and edges of λP are preimages of the arcs γs , 1 ≤ s ≤ k. One can show easily that λP is connected and has no cycles. Therefore it is a planar tree. λP is called the cactus of P (z). The set of stars λP is in corresponding with the set of branches of P −1 (z). For any P (z) and a, b ∈ C let us define the extended cactus ˜ P as follows. Let c1 , ..., c˜ be all finite critical values of P (z) complemented by P (a) λ k or P (b) (or both of them) if P (a) or P (b) is not a critical value of P (z). Consider ˜ P = P −1 {S}. ˜ P is ˜ Clearly λ an extended star S˜ connecting c with c1 , ..., ck˜ and set λ a (k˜ + 1)− colored graph which is a planar tree. Thus there exists an oriented path ˜ P with starting point a and the ending point b. Γa,b ⊆ λ Example 3.1. Let P (z) = z 2 , a = (−1, 0), b = (1, 0), c = (0, 1), the only critical ˜ P is shown in fig.1. value of P (z) is (0, 0). The star S˜ and the cactus λ Now one can define the monodromy group GP of P (z) as below. GP is generated by the permutations gs ∈ Sn , 1 ≤ s ≤ k, where can be identified with the permutation gˆs , 1 ≤ s ≤ k, acting on the set of stars of λP in the following way. gˆs sends the star Si , 1 ≤ i ≤ n, to the ”next” one in a counterclockwise direction around its vertex of color s. For example for the cactus shown in fig.1 we have g1 = (1 2), g2 = (1)(2). R P Let Q(z) = q(z)dz and for each s, 1 ≤ s ≤ k˜ define φs (z) = ki=1 fs,i Q(Pi−1 (z)), where fs,i 6= 0 if and only if the path Γa,b passes through a vertex ν of the star Si colored by the sth color (we don’t take into account the stars Si for which Γa,b ∩ Si contains only the point ν). Furthermore if under a moving along Γa,b the vertex ν is followed by the center of Si then fs,i = −1 otherwise fs,i = 1. As an example for the cactus shown in fig.1 we have: φ1 (z) = Q(P1−1 (z)) − Q(P2−1 (z)), φ2 (z) = −Q(P1−1 (z)) + Q(P2−1 (z)). 4. The main results Our main results will be proved by the use of the following lemma and some theorems in [11]. Lemma 4.1. In Γa,b each middle vertex is followed by a white vertex and follows a white vertex too. Proof. Since each white vertex is the center of a star and the cactus has no cycles, thus each vertex either is a leaf that followed by the center of it’s star or is a common vertex between tow different stars. In the second case the vertex is followed by the center of the second star and follows the center of the first one, and the proof is complete. Theorem 4.2. [11] The composition condition holds if and only if the equality φs (z) ≡ ˜ 0 for any s, 1 ≤ s ≤ k. ˜ define the weight ω(s) of the sth color on Γa,b as a number of For each 1 ≤ s ≤ k, vertices ν ∈ Γa,b colored by the s−th color with the convention that vertices a, b are
5
counted with the coefficients 1/2. For example for the cactus shown in figure 2 we have w1 = w2 = 1. Theorem 4.3. [11] Let P (z), q(z) ∈ C[z], q(z) 6= 0, a, b ∈ C, a 6= b satisfy the composition condition. Suppose that there exists s, 1 ≤ s ≤ n ˜ , such that ω(s) = 1 on Γa,b . Then the one sided moments of P (z), q(z) are zero. Theorem 4.4. [11] Let P (z), q(z) ∈ C, q(z) 6= 0, a, b ∈ C, a 6= b. Suppose that at least one from points a and b is not a critical point of the polynomial P (z). Then the moment vanishing and the composition condition are equivalent. Theorem 4.5. Let P (z), q(z) ∈ C[z], q(z) = 6 0, a, b ∈ C, a 6= b, suppose that all one sided moments equals zero and l(Γa,b ) ≤ 5. Then the composition condition holds. Proof. By theorem 3.4 one can suppose that a, b are critical points of P (z). If l(Γa,b ) = 1 the only possible forms for Γa,b ) are sown in fig. 2, and since a, b are critical points of P (z) both of the forms are impossible (indeed by a similar argument it is impossible that l(Γa,b ) be an odd number). If l(Γa,b ) = 2, since P (a) = P (b) the only possible form is shown in fig. 2, and in this case we have φx (z) = −Q(Pi−1 )(z)+Q(Pi−1 )(z) ≡ 0 and by theorem 3.2 the result holds. If l(Γa,b ) = 4, Γa,b can be as shown in fig. 2. In the second form ω(y) = 1 and by theorem 3.3 the result holds. For the first form we have: φx (z) = −Q(Pi−1 )(z) + Q(Pi−1 )(z) − Q(Pj−1 )(z) + Q(Pj−1 )(z) ≡ 0, and by theorem 3.2 the proof is complete.
5. Figures Figures 1. the star and the cactus of polynomial P (z) = z 2 :
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Figures 2. l(Γa,b ) = 1, 2, 4:
References 1. M. A. M. Alwash, On a condition for a center of cubic non-autonomous equations, Proc. Roy. Soc. Edinburgh 113A (1989) 289-291. 2. M. A. M. Alwash, The composition conjecture for the Abel equations, Expos. Math. 27 (2009) 241-250. 3. M. A. M. Alwash, N. G. Lloyd, Non-autonomous equations related to polynomial two-dimensional systems, Proc. Roy. Soc. Edinburgh, 105A (1987) 129-152. 4. M. Briskin, Y. Yomdin, Tangential version of Hilbert 16th problem for the Abel equation, Moscow Math. J. 5(1) (2005) 23-53. 5. M. Blinov, M. Briskin, Y.Yomdin, Local center conditions for Abel equation and cyclicity of it’s zero solution, Journal dAnal. Math, to appear. Math., AMS, Providence, RI, 2005, 65-82. 6. M. Carbonell, J. Llibre, Limit cycles of a class of polynomial systems, J. Math. Anal. Appl. 109, 187-199, (1988). 7. L. A. Cherkas, Number of limit cycles of an autonomous second order system, J. Differential Equations, 5, 666-668, (1976). 8. C. Christopher, Abel equations: composition conjectures and the model problem, Bull. Lond. Math. Soc. 32(3), 332-338, (2000). 9. J. Devlin, N. G. Lloyd, J. M. Pearson, Cubic systems and Abel equations, J. Differential Equations 147, 435-454, (1998). 10. F. Pakovich, A counterexample to the Composition Conjecture, Proc. Amer. Math. Soc. 130 (2002) 3747-3749. 11. F. Pakovich, On polynomials orthogonal to all powers of a given polynomial on a segment, Bull. Sci. Math. 129(9) (2005) 749-774. 12. F. Pakovich, On the polynomial moment problem, Math. Research Letters, 10, 401-410, (2003). 13. F. Pakovich, N. Roytvarf, Y. Yomdin, Cauchy type integrals of algebraic functions, Israel J. Math., 144, 221291, (2004). 14. A. A. Panov The number of periodic solutions of polynomial differential equations, Math. Notes 64 (5), 622-628, (1998).
