MOSCOWMATHEMATICALJOURNAL Volume3,Number3,July–September2003,Pages1167–1195 THE CENTER PROBLEM FOR THE ABEL EQUATION, COMPOSITIONS OF FUNCTIONS, AND MOMENT CONDITIONS Y.YOMDIN To V. I. Arnold on his 65th birthday Abstract. An Abel differential equation y0 = p(x)y2+q(x)y3 is said to have a center at a pair of complex numbers (a, b) if y(a) = y(b) for everysolutiony(x)withtheinitialvaluey(a)smallenough. Thisnotion is closely related to the classical center-focus problem for plane vector fields. Recently, conditions for the Abel equation to have a center have R R been related to the composition factorization of P = p and Q = q on the one hand and to vanishing conditions for the moments m = i,j R PiQjq on the other hand. We give a detailed review of the recent results in each of these directions. 2000 Math. Subj. Class. Primary: 30E99,30C99;Secondary: 34C99. Key words and phrases. Poincar´e center-focus problem, Abel differential equation,compositionoffunctions,generalizedmoments. 1. Introduction Consider the system of differential equations ( x˙ =−y+F(x, y), (1.1) y˙ =x+G(x, y) withF(x, y)andG(x, y)vanishingattheorigintogetherwiththeirfirstderivatives. Thesystem(1.1)hasacenterattheoriginifitsallsolutionsaroundzeroareclosed. The classical center-focus problem is to find conditions on F and G necessary and sufficient for the system (1.1) to have a center at the origin. This problem, together with the closely related second part of Hilbert’s 16th problem (asking for the maximal possible number of isolated closed trajectories of (1.1) with F(x, y) and G(x, y) being polynomials of a given degree), resists all the attacks till now. Many deep partial results have been obtained (see [5], [6], [26], [56],[67]),butgeneralcenterconditionsarenotknownevenforF(x, y)andG(x, y) being polynomials of degree 3. ReceivedNovember20,2002;inrevisedformMay15,2003. SupportedbytheISFGrant264/02andbytheMinervaFoundation. (cid:13)c2003 Independent University of Moscow 1167 1168 Y.YOMDIN V. I. Arnold suggested several related problems, which turned out to be very productive(see[5],[6],[39],[40]). Inparticular,astudyoftheinfinitesimalversions of the above problems led to an important progress in understanding the analytic structure of Abelian integrals [35], [41]–[44], [52], [59], [63], [66]. The present paper follows, in a sense, this last direction: the infinitesimal ver- sionofthecenter-focusprobleminacertainsettingleadstotheMomentcondition, which is central to complex analysis. On the other hand, composition algebra emerges naturally, providing “morphisms” of the structures involved. This paper presents an overview of the recent results relating the center-focus problem, mo- ments, and compositions. The version of the center-focus problem considered below was suggested and studied, together with the corresponding version of the Hilbert problem, in the 1970s at S. Smale’s seminar (see [58], [45]). Consider an Abel differential equation y0 = p(x)y2 +q(x)y3. This equation is saidtohaveacenteratapairofcomplexnumbers(a, b)ify(a)=y(b)foritsevery solution y(x) with the initial value y(a) small enough. The problem is to give necessary and sufficient conditions for a point to be a center in terms of p and q. (In this form, the center-focus problem has been explicitly stated and studied in [4], [3]. The corresponding version of the 16th Hilbert problem, asking for the maximal possible number of solutions y to a real Abel equation satisfying y(a)=y(b), was posed by C. Pugh; see [45]). This version (to which the original one can be reduced in many cases) suggests importanttechnicalsimplifications. Still,itapparentlyreflectsthemaindifficulties oftheoriginalproblem. Italsostressesitsrelationtoclassicalanalysisandalgebra. Inparticular, the“one-sided”momentsm =RbPkq naturallyariseinstudying k a the infinitesimal (with respect to (cid:15)) version of the center problem for the equation y0 = p(x)y2+(cid:15)q(x)y3 [21], and the “double moments” m = RbPiQjq arise in a i,j R a R higher-order perturbation analysis of this equation; here P = p and Q= q. Another important ingredient of our approach (introduced and initially studied in[4],[3]),compositionalgebraandacompositionfactorizationofP andQ,appears naturally for the Abel equation setting as a description of the “morphisms” of the problem. In particular, it is easy to see that the following “composition condition” implies (a, b) being a center as well as vanishing of one-sided and double moments: P(x)=P˜(W(x)), Q(x)=Q˜(W(x)), (1.2) whereP˜,Q˜,andW arepolynomialsandW(a)=W(b). Oneofthecentralproblems discussedbelowconcernsthenecessityofthecompositionconditionfor(a, b)being a center and for vanishing of the moments. A specific form of the third-degree Abel differential equation is not essential for most of the constructions below. They can be naturally generalized to differential equations of the form y0 = p (x)y2+p (x)y3+···+p (x)yn for arbitrary n and, 1 2 n with minor modifications, to the power series on the right-hand side. So, in fact, we can study the center conditions “y(a) = y(b) for every solution y(x)” for any first-order differential equation y0 =f(x, y). CENTER-FOCUS, MOMENTS, AND COMPOSITIONS 1169 We believe that this problem is important in its own right. We plan to survey the corresponding results elsewhere, limiting the scope of the present paper to the third-degree Abel equation. This paper is intended to present a detailed review of the recent results in the above directions obtained in [9], [13], [17], [10], [11], [47], [48], [49], [50], [66]. It is organized as follows. In Section 2, we give a more accurate overview of the main ingredients of the problem, namely, the Abeldifferentialequationandcenterconditions, composition algebra,andmomentconditions. Wealsoexplaininmoredetailtherelationbetween the problem under consideration and the classical center-focus problem. InSection3,localcenterconditionsarediscussed. Herepisassumedtobefixed, as well as the degree of q. The main result is that, for a generic p, the center and composition conditions coincide locally with respect to q. In Section 4, we discuss the quantitative moment problem, which includes two parts: (a) Vanishing of how many moments is necessary to conclude that the compo- sition condition holds? (b) If the moments are not exactly zero but small, what is the “deviation” of P and Q from the composition condition? Both these questions are motivated by the local center conditions of Section 3. An explicit answer provides an explicit bound for “locality” in the local center conditions. We discuss some cases where answers can be given. In Section 5, moments of rational functions on a closed curve are discussed. The main result is a simple and constructive necessary and sufficient condition for vanishingofdoublemoments,whichisobtainedasacombinationofthecomposition approach with the general Wermer–Harwey–Lawson theorem. InSection6,wediscussthecenterproblemonaclosedcurve. Heretheresultsare more partial, since less is known about the one-sided moment problem. However, some important relations between the double moments and the center conditions are obtained by the methods of Section 5. An interesting example of the Abel equation with elliptic coefficients is also discussed. InSection7,someresultsconcerningCauchy-typeintegralsofalgebraicfunctions on curves (possibly with self-intersections) are presented. In particular, we give a necessaryandsufficientconditionforvanishingofsuchintegralsnearinfinity(which isequivalenttovanishingofthecorrespondingmoments)intermsoframificationsof the integrands with respect to the integration curves. We also give a local analytic description of the integrals near the singularities of the integrands. These results are closely related to vanishing conditions for the one-sided moments discussed in the addendum. The “formal” aspects of the center-focus and moment problems and of their mutual relations are discussed in Section 8. This discussion is very preliminary, although we believe that an investigation in this direction may be important. In the addendum, written by F. Pakovich, some new results describing the van- ishing problem for one-sided polynomial moments on an interval are presented. 1170 Y.YOMDIN 2. Centers, Compositions, Moments, and the Relation to the Classical Center-Focus Problem 2.1. Centers for the Abel equations. We consider the Abel differential equa- tion dy =p(x)y2+q(x)y3, (2.1) dx where p(x) = P0(x) and q(x) = Q0(x) are meromorphic functions in a complex variable x. In what follows, P and Q are mostly polynomial, rational, or elliptic functions. Let γ be a curve in C avoiding the poles of P and Q and joining two points a, b ∈ C. The points a and b are called conjugated with respect to (2.1) along the curve γ if y(a) = y(b) for any solution y(x) to (2.1) analytically continued from a to b along γ with sufficiently small initial value y(a). In this case, we shall also say that (2.1) has a center at (a, b) along γ. The condition on p and q under which (2.1) has a center will be called the center condition. For a = b, this means that the solutions to (2.1) do not ramify on the closed curve γ. In this case, we shall say that (2.1) has a center along γ. Notethat,asy(a)goestozero,thesingularitiesofy(x)tendtothesingularitiesof P andQ. Hencethedefinitionofthecenterfor(2.1)dependsonlyonthehomotopy class of γ in C with all the singularities of P and Q removed. In particular, if a closedcurveγ iscontainedinasimplyconnecteddomainDandP andQareregular on D, then (2.1) has a center along γ. This is the most basic (and, essentially, the only apparent) sufficient condition for (2.1) to have a center. 2.2. Compositions. One of the main tools in our approach to studying center conditions is changes of the independent variable x in the equation (2.1). If such a change of variable transformes (2.1) into an equation having a center, then (2.1) itselfhasacenter. Abasicfactisthat,inmanycases(butnotalways),anecessary and sufficient condition for the center is a possibility to transform (2.1) into an equation having a center for “apparent reasons” (like the one given above). Of course, changes of independent variables is one of the most natural and clas- sical tools in the study and classification of ordinary differential equations. In par- ticular, from the classical analysis of algebraic solutions to the Fuchsian equations, it is known that certain equations of this type whose all solutions are algebraic can be reduced by a change of the independent variable to Gauss hypergeometric equations of a special type (see [7], [30] and the references cited therein). In our case of the first-order Abel equation (2.1), the transformation under a change of the independent variable takes an especially simple form [4], [3]. Let P(x)=P˜(W(x)), Q(x)=Q˜(W(x)). (2.2) Then the substitution of w =W(x) as an independent variable brings (2.1) into dy =p˜(w)y2+q˜(w)y3, (2.3) dw where p˜(w) and q˜(w) are the derivatives with respect to w of P˜(w) and Q˜(w), respectively. CENTER-FOCUS, MOMENTS, AND COMPOSITIONS 1171 Note that (2.1) can be rewritten as dy =y2dP +y3dQ. (2.4) Inthisnotation,thetransformationlawaboveexpressesthe“invarianceofthefirst differential”. Hence factorization of the equation (2.1) is equivalent to factorization of P and Qor,inotherwords,tofindingtheircommon“compositionrightfactors”W. This brings into the study of the center conditions the composition algebra of meromor- phic functions (in particular, of polynomials, rational functions, elliptic functions, etc.). The term “algebra” is used here a broad sense rather than to specify the algebraic structure of the composition, which is well known to be rather compli- cated and subtle (see [31], [51], [55]). In our approach to the center conditions, composition Algebra is one of the main (and most adequate, as the results below suggest) tools. Summarizing the relation of the factorization of the equation (2.1) to its center property,wedefinethefollowinghomotopycompositioncondition,which,according to the above considerations, is sufficient for (2.1) to have a center. Definition2.1. P,Q,andγ satisfythehomotopycompositionconditionifP(x)= P˜(W(x)) and Q(x)=Q˜(W(x)), where W maps C to a Riemann surface X in such a way that W(γ) is a closed curve δ in X (in particular, if a 6= b, then W(a) = W(b)), δ is contained in a simply connected domain D in X, and P˜ and Q˜ are regular in D. 2.3. Moments. Thelastmainingredientinourapproachisconsiderationofgen- eralized moments given by the following expression: Z b m = Pi(x)Qj(x)dQ(x), i, j =0, 1, .... (2.5) i,j a Theintegrationisalongapathγ connectingaandb. Variousspecialcasesarealso considered. Inourcomputations,especiallyimportantarethefollowing“one-sided” moments: Z b m = Pk(x)q(x)dx, k =0, 1, .... (2.6) k a The moments m and m (and their vanishing, called usually the moment i,j k condition)playacentralroleincomplexanalysis,especiallyinthecharacterization of the “boundary values” of complex functions and complex varieties, in the study of holomorphic and polynomial hulls in several complex variables, and in Banach algebras [60], [61], [2], [38]. For instance, if γ is closed, then the classical result of Wermer, Harwey, and Lawson [60], [61], [38] implies that vanishing of all the moments m is equivalent to the fact that the image of the curve γ under (P, Q) i,j bounds a compact analytic one-chain in C2. Vanishingofone-sidedmomentsisequivalenttoidenticalvanishingnearinfinity of the Cauchy-type integral Z P(b) g(z)dz I(t)= (2.7) z−t P(a) 1172 Y.YOMDIN along the path P(γ). Here z = P(x) and g(z) = Q(P−1(z)). Such integrals are studied by specific techniques and methods. This gives a complementary view of the moment problem as presented in [50] and in Section 7 below. Moments appear in the center problem for the Abel equation as follows. A necessary and sufficient condition for the equation to have a center is given by an infinitenumberofalgebraicequationsonthecoefficientsofP andQ. Eachofthese equations is given by vanishing of a certain expression containing iterated integrals of P and Q along γ (see Section 3.1 below). First of all, the first three of these equations (starting with the third one) turn out to be just the first three one-sided moments. Secondly, in general, the one-sided and double moments form part of the terms in the center equations. Thirdly, if we fix P, the center equations turn out to be polynomials of growing degrees in Q. The one-sided moments are exactly the linear (in Q) parts of these equations. In particular, for the parametric version of (2.1) dy =p(x)y2+(cid:15)q(x)y3, (2.8) dx the infinitesimal center conditions with respect to (cid:15) at (cid:15)=0 coincide with vanishing of the one-sided moments. Since all moments are integrals of one-forms, they are preserved by changes of the independent variable. Hence composition algebra as described above is equally relevant to the study of vanishing of moments. The result of Wermer, Harwey, and Lawsoncanbecombinedwithchangesoftheindependentvariabletoprovethatthe following homology composition condition is necessary and sufficient for vanishing of all the double moments on a closed curve γ (and sufficient for a nonclosed γ): Definition 2.2. P,Q,andγ satisfythehomologycompositionconditionifP(x)= P˜(W(x)) and Q(x)=Q˜(W(x)), where W maps C to a Riemann surface X in such a way that W(γ) is a closed curve δ in X (in particular, if a 6= b, then W(a) = W(b)), δ bounds a compact complex one-chain D in X, and P˜ and Q˜ are regular in D. (In contrast with the homotopy composition condition, we do not require here that D in X is simply connected. Hence, we get exactly the assumption of the Wermer–Harwey–Lawson theorem). As we shall see below, the homotopy composition condition is indeed stronger than the homology one. In particular, the homology composition condition does not generally imply (2.1) having a center. One of the main facts which make composition algebra a really working tool in the study of both the center and moment conditions is the following. For P and Q relatively prime in the compositional sense, the meromorphic mapping z → (P(z), Q(z)) of C to C2 is a generically one-to-one parametrization of its image which is an analytic curve Y in C2. Hence, Y is obtained from C minus the poles of (P, Q) by gluing together some finite subsets, and for most purposes C minus the poles of (P, Q) is a good approximation of Y. This approach is illustrated in more detail in [11] and in Section 5 below. CENTER-FOCUS, MOMENTS, AND COMPOSITIONS 1173 2.4. Relation to the classical center-focus problem. The centerproblemfor the Abel differential equation (2.1) is closely related to the classical center-focus problemforthehomogeneouspolynomialvectorfields(1.1)ontheplane(see, e.g., [56],[67]). LetF(x, y)andG(x, y)bepolynomialsinxandy ofdegreed. Consider the system of differential equations ( x˙ =−y+F(x, y), (2.9) y˙ =x+G(x, y). A solution x(t), y(t) to (2.9) is said to be closed if it is defined in the interval [0, t ] and x(0) = x(t ), y(0) = y(t ). The system has a center at the origin if its 0 0 0 all solutions around zero are closed. The classical center-focus problem is to find conditions on F and G which are necessary and sufficient for the system (2.9) to have a center at the origin. It was shown in [25] that system (2.9) with homogeneous F and G of degree d can be reduced to the trigonometric Abel equation dρ =p(θ)ρ2+q(θ)ρ3, θ ∈[0, 2π], (2.10) dθ where p(θ) and q(θ) are polynomials in sinθ and cosθ of degrees d+1 and 2d+2, respectively. Thus, (2.9) has a center if and only if all solutions ρ=ρ(θ) to (2.10) are periodic on [0, 2π], i.e., satisfy ρ(0)=ρ(2π). In its turn, the trigonometric Abel equation (2.10) can be transformed by an exponential substitution into the equation (2.1) with P and Q being Laurent poly- nomials on the unit circle S1. We investigate this specific situation in Section 6 below. However, much less can be said in this case than in the case of a polyno- mial Abel equation on the interval considered in Section 3. This is because our understanding of one-sided moments on a closed curve is at present insufficient. Althoughthecenterproblemonaninterval(or,ingeneral,onanon-closedcurve) statedinSection2.1abovedoesnotcorresponddirectlytotheclassicalcenter-focus problem, it is of interest by its own; this problem has been extensively investigated in [4], [26], [27], [28], [29], [45] and in other publications. It is a general belief that the center problem for the polynomial Abel equation (2.1) on the interval manifests all the main difficulties involved in the classical problem, while some technical details are possibly simplified. 3. Local Center Conditions In this section, which presents the results of [10], we restrict ourselves to the case of the polynomial Abel equation (2.1) on an interval. The main result is that, for a fixed generic P, the center and composition conditions coincide locally with respect to Q. Thisfollowsfromthefactthatthemomentandcenterconditionsturnouttobe very closely related, namely, the center equations can be considered as a nonlinear deformation of the moment equations preserving the composition subspace. More precisely, for a fixed P, vanishing of the moments m in (2.6) gives linear k equations on Q. On the other hand, the center conditions can be transformed into 1174 Y.YOMDIN asequenceofnonlinearequationsonQ(seeSection3.1below). Itturnsoutthatthe linear parts of the center equations are the moment equations. Here composition comesin: foraso-called“definite”P (seebelow),thesetofQsatisfyingthemoment equations is a linear subspace L defined by the composition condition. Now, the decisive observation is that the composition condition implies vanishing of each of the non-linear terms separately in the center equations. Hence, the linear parts of the center equations define the zero subspace L, while all the non-linear terms of these equations vanish on L. Itisasimplefactofcommutativealgebra(akindofNakayamalemma; see,e.g., [37, Chapter 4, Lemma 3.4]) that, locally, the set of zeroes of the center equation is L and that the local ideal generated by the center equations coincides with the ideal generated by the moment equations. To formulate the result more accurately, we have to adapt the general notions introduced above to our specific situation. In this section, we always assume that P(x)andQ(x)arepolynomialsinacomplexvariablex. Inthissetting,asufficient condition for the equation (2.1) to have a center at (a, b) (and along any curve γ, since P(x) and Q(x) are polynomials) is given by the following form of the composition condition: P(x)=P˜(W(x)), Q(x)=Q˜(W(x)), (3.1) where P˜, Q˜, and W are polynomials and W(a) = W(b). Indeed, if the compo- sition condition (3.1) is satisfied, then P(x), Q(x), and γ satisfy the homotopy composition condition of Definition 2.1 above, and hence (2.1) has a center. The composition conjecture (see [4], [19], [20], [26]) is that (3.1) is a necessary and sufficient condition for (2.1) to have a center. For P and Q of small degrees and of certain special forms, this conjecture has been proved in [4], [20], [21], [22], [23], [27], [12], [8], [9]. In the setting under consideration, the moment condition takes the form Z b m = Pk(x)q(x)dx=0, k =0, 1, .... (3.2) k a Definition 3.1. A polynomial P is called definite (with respect to (a, b)) if, for any polynomial Q, the moment condition (3.2) implies the composition condition (3.1). All polynomials P up to degree 5 are definite, as well as all indecomposable P (for every a 6= b), all P with P0(a) 6= 0, P0(b) 6= 0, etc. (see Section 7 and the addendum for a more detailed discussion). √ The Chebyshev polynomial T is not definite with respect to a = − 3/2 and √ 6 b= 3/2 (see [47] and the addendum). (There is an incorrect claim in [24] that any P is definite.) The following theorem is the main result of [10]. Theorem 3.1. For a fixed definite P and for a fixed degree d, there exists an (cid:15) = (cid:15)(P, d) > 0 such that, for any Q of degree d with k Q k≤ (cid:15), the center and composition conditions coincide. A sketch of the proof of Theorem 3.1 is given in Section 3.1. CENTER-FOCUS, MOMENTS, AND COMPOSITIONS 1175 Remark. Intherecentpaper[64],aninterestinganalysisofthecenterproblemfor the Abel differential equation is presented, which is partly similar to our approach. In particular, Theorem 5.6 of [64] is essentially a special case of our theorem 3.1 for P of degree 2. Although, formally, the statement of Theorem 5.6 of [64] is weaker (it does not guarantee the uniformity of the “locality size” with respect to the polynomials Q of a fixed degree), we believe that the proof of Theorem 5.6 of [64] essentially provides the uniform bound. The approach developed in [10] allows us to compute also the local Bautin ideal of (2.1) (i.e., the ideal in the local ring of polynomials in the coefficients of Q generated by the Taylor coefficients of the Poincar´e return mapping; see Section 3.1below). ForadefiniteP,thisidealturnsouttobegeneratedbythemomentsm k (considered as polynomials in the coefficients of Q). In particular, this implies an explicit bound on the “cyclicity” of the zero solution to (2.1), i.e., on the number of “periodic” solutions y (with y(a) = y(b)) which can bifurcate from the zero solution. Tobemoreprecise,foradefiniteP,wedefinethemomentBautinindexN(P, d) as follows. Definition 3.2. The moment Bautin index N(P, d) is the minimal number of the Z b momentsm = Pk(x)q(x)dxwhosevanishingimpliesthecompositioncondition k a (3.1) for any Q with degQ≤d. The moment Bautin index is explicitly known for degP ≤ 3 (see [22], [23], [8]). Under the assumption that all the roots of P are different, it is equal to [degQ/degP]+1. For real P and Q, N(P, d) can be expressed explicitly through the degree of P and d = degQ by using the standard techniques of moments and elimination of quantors in real algebraic geometry (see [24], [66]). In general, the existence of N(P, d) follows from the stabilization of decreasing sequence of linear subspaces (the zero sets of the sequence of the moment equations) in the space of the coefficients of Q. A natural conjecture is that N(P, d) always depends only on the degree of P and on d. We discuss the computation of N(P, d) in more detail in Section 4. Let a definite P and degQ = d be fixed, and let (cid:15)=(cid:15)(P, d)>0 be as defined in Theorem3.1. Thefollowingresultisobtainedin[10]bythetechniquesof[33],[34]. Theorem 3.2. There is a δ =δ(P, d)>0 such that, for any Q with kQk≤(cid:15), the number of solutions y to (2.1) satisfying y(a)=y(b) and |y(a)|≤δ does not exceed N(P, d). 3.1. Centerequations. Aclassicalapproachtodeterminingthecenter-focuscon- ditions for the Abel equation (2.1) is to seek its solutions in the form of the power series representing the Poincar´e first return map ∞ X y(x, y )=y + v (x, λ)yk, (3.3) a a k a k=2 where y(a, y ) = y is the initial value at the point a and λ = (λ , λ , ...) is the a a 1 2 (finite) set of the coefficients of p and q. We shall write v (x) for short. Then k 1176 Y.YOMDIN y(b) = y(b, y ) = y + P∞ v (b)yk and, hence, the condition y(b) ≡ y(a) is a a k=2 k a equivalent to v (b)=0 for k =2, 3, ..., ∞. k Itiseasytoshow(bysubstitutingoftheexpansion(3.3)intotheequation(2.1)) that v (x) satisfy the recurrence relations k  v (x)≡0, v01(x)≡1, v (0)=0, (3.4) n vn0(x)=p(x) X vi(x)vj(x)+q(x) X vi(x)vj(x)vk(x), n≥2. i+j=n i+j+k=n It was shown in [19] that, in fact, the recurrence relations (3.4) can be lin- earized; to be more precise the ideals I = {v , v , ..., v } are generated by k 1 2 k {ψ (x), ...ψ (x)}, where ψ (x) satisfy the linear recurrence relations 1 k k  ψ (x)≡0, ψ0(x)≡1, 1 (3.5) ψψn0((x0))==0−,(n−1)ψ (x)p(x)−(n−2)ψ (x)q(x), n≥2, n n−1 n−2 which are much more convenient than (3.4). Now, we see that each v (x) can be written as a sum of iterated integrals of the k form Z Z Z Z const· q p... p q (the order and the number of the integrands p and q vary). An explicit analysis of these expressions is not easy. Integration by parts can be used to simplify them, butitleadstoa“wordproblem”,whichhasbeenanalyzedonlypartly(andforthe recurrencerelation(3.4))in[28],[29]. UndertheassumptionthatP(a)=Q(a)=0, the first seven equations have the form (see, e.g., [24]) 0=P(b), 0=Q(b), 0=m , 0=m , 1 2 1Z b Z b 0=m − pQ2, 0=m −2 PpQ2, 3 2 4 a a Z b 1 Z b Z t 0=m − Q3p+23P3Qq−77 P2(t)q(t)dt Pq. 5 2 a a a Thefollowingtheoremsummarizestherelevantfactsaboutthecenterequations. Theorem 3.3. The degrees of the center equations considered as polynomials in the coefficients of Q grow linearly with slope 1. The linear terms of these equations 2 are the moments m . If the composition condition (3.1) is satisfied, then each term k in the center equations vanishes. The following result is essentially a version of the Nakayama Lemma from com- mutative algebra (see, e.g., [37, Chapter 4, Lemma 3.4]) adapted to our situation.