Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation
aa r X i v : . [ m a t h . C A ] A p r UNIFORM UPPER BOUNDS FOR THE CYCLICITY OF THEZERO SOLUTION OF THE ABEL DIFFERENTIAL EQUATION
DMITRY BATENKOV AND GAL BINYAMINI
Abstract.
Given two polynomials
P, q we consider the following question:“how large can the index of the first non-zero moment e m k = R ba P k q be,assuming the sequence is not identically zero?”. The answer K to this questionis known as the moment Bautin index, and we provide the first general upperbound: K q + 3(deg P − . The proof is based on qualitativeanalysis of linear ODEs, applied to Cauchy-type integrals of certain algebraicfunctions.The moment Bautin index plays an important role in the study of bifurca-tions of periodic solution in the polynomial Abel equation y ′ = py + εqy for p, q polynomials and ε ≪
1. In particular, our result implies that for p satisfy-ing a well-known generic condition, the number of periodic solutions near thezero solution does not exceed 5 + deg q + 3 deg p . This is the first such bounddepending solely on the degrees of the Abel equation. Introduction
Polynomial moments and their Bautin index.
Throughout this paper
P, Q ∈ C [ z ] will denote a pair of polynomials, and p, q their respective derivatives.We will denote the degrees of P, Q (resp. p, q ) by d P , d Q (resp. d p , d q ). We also fixtwo points a, b ∈ C .Two related types of moment sequences corresponding to this data have been con-sidered in the literature, m k = m k ( P, Q ) := Z ba P k ( z ) Q ( z ) p ( z )d z, k = 0 , , , . . . (1) e m k = e m k ( P, q ) := Z ba P k ( z ) q ( z )d z, k = 0 , , , . . . (2)These two sequences are related through a simple formula due to [10] (see § §
2. The latter appears more directly in the study of perturbationsof the Abel equation, as explained in § Definition 1.
We define the vanishing index N ( P, Q, a, b ) to be the first index k such that m k ( P, Q ) = 0 , or ∞ if no such k exists. We define the moment Bautin G.B. was supported by the Banting Postdoctoral Fellowship and the Rothschild Fellowship. index N ( d P , d Q , a, b ) to be N ( d P , d Q , a, b ) := sup deg Q d Q , deg P d P N ( P,Q,a,b ) < ∞ N ( P, Q, a, b ) + 1 , (3) i.e. the least k ∈ N with the property that N ( P, Q, a, b ) > k implies N ( P, Q, a, b ) = ∞ for any P, Q with deg P d P and deg Q d Q .We define e N ( P, q, a, b ) and e N ( d P , d q , a, b ) analogously. Remark 2.
The moments m k ( P, Q ) are polynomials in the coefficients of P, Q .Let R denote the ring of polynomials in these coefficients and I k ⊂ R denote theideal by m , . . . , m k . Then N ( d P , d Q , a, b ) defined above is the first index for whichthe chain {√ I k } k ∈ N stabilizes. In particular, from noetherianity it follows that thisindex is well-defined (finite). An analogous remark holds for e N ( d P , d q , a, b ) . The moment Bautin index has been studied in various special cases, motivatedprimarily by its relation to perturbations of the Abel equation (see § d P = 2 ,
3. We refer the reader to [3, 5] and references therein for details. However,to our knowledge no general bound has been available. Our main result is thefollowing general bound for the moment Bautin index.
Theorem 1.
For any d P , d Q ∈ N we have N ( d P , d Q , a, b ) d Q + 3( d P − . (4) Similarly, for any d P , d q we have e N ( d P , d q , a, b ) d q + 3( d P − . (5)It is shown in § N ( P, Q, a, b ) is essentially equivalent to the order ofthe zero at t = ∞ of the moment generating function H ( t ) for the momentsequence { m k } . It turns out [10] that H ( t ) admits an analytic expressionas a Cauchy type integral for the algebraic function Q ( P − ( z )).(2) The Cauchy type integral above satisfies a (non-homogeneous) linear dif-ferential equation of Fuchsian type [7].The problem of estimating N ( P, Q, a, b ) is thus reduced to the study of the orderof zero at t = ∞ of solutions of certain Fuchsian differential equations. A detailedanalysis of the Fuchsian differential operator involved, and elementary considera-tions concerning its monodromy, allow us to give an a-priori upper bound for thisorder of zero, thus proving Theorem 1.1.2. Perturbations of the Abel equation.
The classical Hilbert’s 16th problemasks for bounding the number of limit cycles, i.e. isolated closed trajectories, of the
OUNDS FOR CYCLICITY OF ZERO SOLUTION OF ABEL EQUATION 3 polynomial vector field d x d t = − y + F ( x, y ) , d y d t = x + G ( x, y ) . (6)The closely related Poincar´e Center-Focus Problem asks for explicit conditions onthe polynomials
F, G in order for the system (6) to have a center. These problemsremain widely open, although during the years many partial results have beenobtained (see [6] for an exposition).An alternative context for the study of the problems above is provided by the Abeldifferential equation, y ′ = p ( x ) y + q ( x ) y , x ∈ [ a, b ] ⊂ R , (7)where p, q can be polynomials, trigonometric polynomials or even analytic functions[8]. A periodic solution in this context corresponds to solution y ( x ) satisfying y ( a ) = y ( b ), and a center corresponds to an Abel equation where every solutionwith a sufficiently small initial condition is periodic. The Abel equation analogueof the Hilbert 16th problem, known as the Smale-Pugh problem, is to bound thenumber of periodic solutions of (7) in terms of the degrees of p and q . It is generallybelieved that some (but not all) of the essential difficulties in the study of (6) canbe observed in (7), even when one restricts to the case of polynomial coefficients.On the other hand, the polynomial Abel equation allows for several importanttechnical simplifications, and significant progress has been achieved for the Center-Focus in this context using tools from polynomial composition algebra and algebraicgeometry [4, 11].The Smale-Pugh problem for the polynomial Abel equation remains open. Itsinfinitesimal version, first suggested in [5], is as follows: Problem 1.
How many periodic solutions can a small perturbation y ′ = p ( x ) y + εq ( x ) y , x ∈ [ a, b ] (8) of the “integrable” equation y ′ = p ( x ) y have? This is an Abel equation analog of the “Infinitesimal Hilbert 16th problem” forwhich an explicit bound was obtained in [1]. Following [5], in this paper we focusour attention on the periodic solutions bifurcating from the zero solution of (8).The unperturbed equation ( ε = 0) is a center if and only if R ba p ( x )d x = 0. Thus wemay choose the primitive P such that P ( a ) = P ( b ) = 0. As in the classical case,the study of the bifurcation of periodic solutions as well as the center conditionsfor the perturbation (8) begins with the study of the first variation of the Poincarmap.For technical reasons it is customary to consider the “reverse” map from time x = b to time x = a . Namely, let G ( y ) : ( C , → ( C ,
0) denote the germ of the analyticmap assigning to each initial condition y b the value G ( y b ) = η ( a ), where η is asolution of (8) satisfying η ( b ) = y b . We may view G as a germ of an analyticfunction in the coefficients of the polynomials p, q and ε as well. Fixed points of G DMITRY BATENKOV AND GAL BINYAMINI correspond to periodic solutions, and the identical vanishing of G ( y ) correspondsto a center. An explicit computation [5, Proposition 4.1] gives the expansiondd ε (cid:12)(cid:12) ε =0 G ( y ) = − y Z ba q ( x )1 − yP ( x ) d x = ∞ X k =0 e m k y k +3 . (9)As in the classical study of perturbation of Hamiltonian planar systems, it followsfrom this variational computation that the number of periodic solutions bifurcatingfrom the zero solution of (8) is bounded by the order of zero of the right hand side,i.e. e N ( P, q, a, b ) + 3, assuming that this number is finite. On the other hand, if thefirst variation vanishes identically then one must in general consider higher ordervariations in ε , further complicating the study of bifurcating periodic solutions.A surprising feature of the Abel equation (8) is that for many polynomials p , thevanishing of the first variation (9) automatically implies the identical vanishing ofthe Poincar map. Toward this end we recall the following definition. Definition 3 ([3]) . The polynomials
P, Q are said to satisfy the composition con-dition (PCC) on [ a, b ] if there exists a polynomial W ( x ) with W ( a ) = W ( b ) , andpolynomials ˜ P, ˜ Q such that P ( x ) = ˜ P ( W ( x )) , Q ( x ) = ˜ Q ( W ( x )) . A polynomial P is called “definite” (w.r.t a, b ), if for any polynomial Q , vanishingof all the moments e m k ( P, q ) implies PCC for P, Q . Definite polynomials are ubiquitous. In the deep works [9, 11] all counter-exampleshave been classified and shown to admit a rigid algebraic structure.Whenever the polynomials
P, Q satisfy the PCC, the corresponding Abel equa-tion (7) automatically admits a center, as can be seen by a simple change of variableargument. We thus see that for a definite polynomial P , the vanishing of all mo-ments e m k ( P, q ) implies the identical vanishing of the Poincar map G ( y ). Therefore,in a sense the bifurcation of periodic solutions in (8) is fully controlled by the firstvariation (9). More formally, the following holds. Theorem 2 ([3]) . Let P be a definite polynomial, and fix the parameters a, b, d q .Then for any k q k ≪ with deg q d q , the number of periodic solutions of (7) with | y ( a ) | ≪ is at most e N ( d P , d q , a, b ) + 3 . As a Corollary of Theorem 1 we therefore have the following first general estimatefor the number of limit cycles near the zero solution for an Abel equation (7) with k q k small. Corollary 4.
Under the conditions of Theorem 2, the number of periodic solutionsis bounded by d q + 3 d p . Organization of the paper. In § { m k } , { e m k } which turn out to be Cauchy-typeintegrals. We thus reduced the study of the corresponding vanishing indices to thestudy of the order of zero of these generating functions at infinity. In § g ( z )satisfies a linear ODE L g = 0 then the corresponding Cauchy-type integral I ( t ) OUNDS FOR CYCLICITY OF ZERO SOLUTION OF ABEL EQUATION 5 satisfies a non-homogeneous linear ODE L I = R , where R is a rational functionof known degree. Subsequently, in § § Polynomial moments and generating functions
Recall the notations of § { m k } , { e m k } as follows: H ( t ) = ∞ X k =0 m k t − ( k +1) H ( t ) = Z ba Q ( z ) p ( z ) t − P ( z ) d z, (10) e H ( t ) = ∞ X k =0 e m k t − ( k +1) e H ( t ) = Z ba q ( z ) t − P ( z ) d z. (11)Clearly, ord ∞ H ( t ) = N ( P, Q, a, b ) + 1 ord ∞ e H ( t ) = e N ( P, q, a, b ) + 1 . (12)In particular, we have the following. Proposition 5.
We have N ( d P , d Q , a, b ) = sup H ( t ) ord t = ∞ H ( t ) . (13) where the supremum is taken over all pairs P, Q with respective degrees bounded by d P , d Q and H ( t ) denotes the corresponding moment generating function. It turns out that H ( t ) and e H ( t ) are related by a simple formula, which impliesin particular that the study of their orders of vanishing at t = ∞ are essentiallythe same [10, Claim, p.40]. We repeat the argument of [10] in order to obtain anexplicit description of relation between these orders. Lemma 6.
The condition e m = 0 is equivalent to Q ( a ) = Q ( b ) . Moreover, underthis condition we have e m k +1 = − ( k + 1) m k for k ∈ N . In particular, we have e N ( P, q, a, b ) N ( P, Q, a, b ) + 1 . (14) Proof.
Derivating under the integral sign we haved H ( t )d t = − Z ba Q ( z ) p ( z )( t − P ( z )) d z = − Z ba Q d (cid:18) t − P ( z ) (cid:19) = − (cid:2) Q ( z ) t − P ( z ) (cid:3) ba + Z ba q ( z ) t − P ( z ) d z = Q ( a ) t − P ( a ) − Q ( b ) t − P ( b ) + e H ( t ) (15)Comparing the t − coefficient we see that e m = 0 if and only if Q ( a ) = Q ( b ), andunder this condition e m k +1 = − ( k + 1) m k as claimed. (cid:3) The moment generating function (10) has the form of a Cauchy integral. Indeed,choose the curve of integration γ ′ from a to b in (10) to be some smooth curve DMITRY BATENKOV AND GAL BINYAMINI avoiding the critical values of P ( z ) (except perhaps at the endpoints). Then setting γ = P ( γ ′ ) and substituting w = P ( z ) in (10) we obtain H ( t ) = Z γ Q ( P − ( w )) t − w d w (16)where P − ( w ) denotes the branch of P − lifting γ to γ ′ .3. Cauchy-type integrals and linear differential operators
Let L be a scalar differential operator, L = c r ( z ) ∂ r + · · · + c ( z ) , c , . . . , c r ∈ C [ z ] . (17)Let γ ⊂ C be a smooth curve, and assume that γ does not pass through the singularpoints of L , except perhaps at its endpoints. Finally let g be a solution of L g = 0defined on γ , and assume further that g is bounded on γ (including at the possiblysingular endpoints). We denote by p + , p − the endpoints of γ .Then we define the Cauchy-type integral I ( t ) = Z γ g ( z ) z − t d z (18)It is classically known that I ( t ) is a holomorphic functions defined on C \ γ , andmoreover that the boundary values I + and I − of I ( t ) on γ from above and belowrespectively satisfy I + − I − = g | γ . Moreover I ( t ) can be analytically continuedalong any path avoiding the endpoints of γ .Kisunko [7] proved the following (under the extra mild assumption that g is holo-morphic at the endpoints of γ ). Proposition 7.
We have L I ( t ) = R ( t ) where R ( t ) is a rational function havingpoles of order at most r at p + , p − and no other poles on C .Sketch of proof. By the classical properties of I ( t ) mentioned above, L I ( t ) is a(possibly multivalued) analytic function on C \ { p + , p − } with ramifications p + , p − ,and the difference between the two branches near the branch cut at γ is g . But since L g = 0, the boundary values of L I + and L I − agree, so L I is in fact a univaluedholomorphic function defined on C \ { p + , p − } . We will show that it has poles oforder at most r at p + , p − and at most a pole at ∞ .Since g | γ is bounded, we may derive under the integral sign and write L I ( t ) = r X k =0 ( − k k ! c k ( t ) Z γ g ( z )( z − t ) k +1 d z (19)We now show that L I ( t ) admits polynomial growth of order at most r at p + (andthe same arguments work for p − ). It is enough to consider each of the integralsin (19) separately. Moreover, we may assume that γ is a small piece of a smoothcurve near p + (because the integral over the rest of γ is analytic at p + ). Choose acoordinate system where p + = 0.Let M denote an upper bound for | g ( z ) | on γ . Let t be a point in a puncturedneighborhood of p + . Since g ( z ) admits analytic continuation along any curve in OUNDS FOR CYCLICITY OF ZERO SOLUTION OF ABEL EQUATION 7 the punctured neighborhood, may deform γ without changing L I ( t ) so that forsome positive constants C, D independent of t ,(1) For every z ∈ γ , we have | z − t | > C | t | and also | z − t | > C | z | .(2) On γ we have | d z | D d | z | .(3) Write g = γ + γ where γ is the part of γ which lies in { z : | z | < t } and γ is the rest. Then the length of γ is at most D | t | , and the length of γ is at most D .We now estimate (cid:12)(cid:12) Z γ g ( z )( z − t ) k +1 d z (cid:12)(cid:12) Z γ M ( C | t | ) − k − | d z | + Z γ M D ( C | z | ) k +1 d | z | length( γ ) M ( C | t | ) − k − + (cid:20) − M DC k +1 k | z | − k (cid:21) ··· | t | O ( | t | − k ) (20)proving the claim.Finally, it is easy to see that I ( t ) and its derivatives have a zero at t = ∞ , andsince the coefficients of L are polynomial it follows that I ( t ) has at most a pole at ∞ as well. (cid:3) A differential operator for Q ( P − )Let V denote the linear space spanned by the d P branches of the algebraic function g ( z ) := Q ( P − ( z )). We denote r := dim V , and note that r may be strictly smallerthan d P . Denote by p , . . . , p s the critical values of P .4.1. The operator L . By a theorem of Riemann [6, Theorem 19.7], there existsa linear r -th order differential operator L , with polynomial coefficients, L = c r ( z ) ∂ r + · · · + c ( z ) , c , . . . , c r ∈ C [ z ] (21)whose solution space coincides with V . Moreover, L is uniquely determined by therequirement that c r , . . . , c do not share a non-trivial common factor. We recall theconstruction of L .Recall that the Wronskian W ( f , . . . , f n ) of a tuple of functions is defined to be W ( f , . . . , f n ) := det f · · · f n ∂f · · · ∂f n ... ∂ n − f · · · ∂ n − f n (22)Now let g , . . . , g r denote r branches of g ( z ) which span V . Then clearly for any f ∈ V we have W ( g , . . . , g r , f ) = 0. We define the operator e L given by e L ( f ) = W ( g , . . . , g r , f ) W r = (cid:2) ∂ r + r − X k =0 e c k ( z ) ∂ k (cid:3) f where e c k = W k ( g , . . . , g r ) W r ( g , . . . , g r )(23)where W i are the minors obtained when expanding the Wronskian W ( g , . . . , g r , f )along the last column. If the monodromy of g along a closed curve γ induces thelinear automorphism M γ : V → V then the corresponding monodromy along γ of DMITRY BATENKOV AND GAL BINYAMINI each W k is given by multiplication by det M γ . In particular, the coefficients e c k areunivalued functions.4.2. The divisors [ W k ] . Let k = 0 , . . . , r and z ∈ C P . Choose any local rep-resentative of the functions g , . . . , g r . Since these functions have at most a finiteramification and moderate growth at z , we may expand W k = ∞ X j = − N a k,j ( z − z ) j/q (24)where q and N are some natural numbers. Suppose that a k,j is the first non-zerocoefficient among the a k,j . Then we say that the fractional order of W k at z isord z W k := j /q . This notion is well-defined: indeed, the monodromy of W k alongany curve is given by multiplication by a non-zero constant and hence does notchange the order. We define the fractional divisor [ W k ] of W to be[ W k ] := X z ∈ C P ord z W k ( z )[ z ] . (25)This sum is locally-finite, and hence finite. Moreover it is clear that [ e c k ] = [ W k ] − [ W r ]. In particular, e c k admits finitely many singularities of finite order. Since wehave already seen that e c k is univalued, it is in fact a rational function.We can also write the divisor [ W k ] in terms of residues. Indeed, since the mon-odromy of W k along any curve is given by multiplication by a constant, the one-form d Log W k is a univalued one-form. It is easy to verify in local coordinates thatit in fact has only finitely many poles, all of first order, and[ W k ] = X z ∈ C P Res z (d Log W k )[ z ] . (26)For any divisor D = P n i [ z i ] we denote D z i = n i anddeg D = X n i , D + = X n i > n i [ z i ] , D − = − X n i n i [ z i ] . (27)In particular, it follows from the above that deg[ W k ] = 0.4.3. An estimate for deg[ W r ] + . Our next goal is to estimate deg[ W r ] + . Since[ W r ] is principal, it will suffice to estimate deg[ W r ] − . Recall that W r = W ( g , . . . , g r )where g k = Q ( P − k ( z )) and P − ( z ) , . . . , P − r ( z ) denote r different branches of P − ( z ). If z ∈ C is not a critical value of P then these functions are all holo-morphic around z , and hence [ W r ] z is non-negative.Let the critical value p i have exactly m i < d P preimages, and write b i := d P − m i for the number of critical points (counted with multiplicities) over p i . Then at most2 b i of the branches g k may be ramified at p i . We expand the determinant defining W r and note that: • since g k is bounded, its order is non-negative; • differentiation can decrease the order by at most 1; • differentiation cannot decrease the order below zero for holomorphic g k . OUNDS FOR CYCLICITY OF ZERO SOLUTION OF ABEL EQUATION 9
We thus conclude thatord p i W r > ( − r + 1) + · · · + ( − r + ν ) , where ν = min( r, b i ) . Since b + · · · + b s = d P −
1, it is not hard to see that the maximal value for thefollowing sum is obtained when b i = 1 for i = 1 , . . . , s , and in any case s X i =1 ord p i W r > − (2 d P − d P − . (28)It remains to estimate the order of W r at ∞ . Choose a coordinate w around ∞ suchthat P ( w ) = w − d P . Then any branch of Q ( P − ( w )) has the Puiseux expansion Q ( P − ( w )) = Q ( w − /d P ) = αw − d Q /d P + · · · , α = 0 (29)where · · · denote higher order terms. Moreover, the derivative ∂ z = − w ∂ w in-creases the order of zero at w = 0 by at least one. Expanding the determinantdefining W r we see that ord ∞ W r > − rd Q d P + r ( r − . (30)In conclusion, we havedeg[ W r ] + = deg[ W r ] − d Q rd P + (2 d P − d P − − r ( r − . (31)4.4. An estimate for deg c r . We wish to derive an estimate for the number ofsingularities of L , or more specifically for deg c r . By definition, c r is a polynomialand [ c r ] + is the least common upper bound for [ e c ] − , . . . , [ e c r − ] − in (23). Recallthat [ e c k ] = [ W k ] − [ W r ].We first note that L is a Fuchsian operator. Indeed, since the solutions of L , beingalgebraic functions, have moderate growth at each singularity, this follows from atheorem of Fuchs [6, Theorem 19.20]. Thus by definition the order of [ c r ] at anypoint p ∈ C cannot exceed r . We will apply this to the points p , . . . , p s .Let now z ∈ C and z
6∈ { p , . . . , p s } . Then the branches g , . . . , g r are holomorphicat z , so z is not a point of [ W ] − , . . . , [ W r − ] − . In other words, z can only be apoint of [ e c ] − , . . . , [ e c r − ] − if it comes from [ W r ] + . Thus we see that[ c r ] + s X i =1 r [ p i ] + [ W r ] + (32)Using (31) and noting that s d P − r d P , we have the following Proposi-tion. Proposition 8.
The following estimate holds, deg c r = deg[ c r ] + d Q rd P + 3( d P − − r ( r − . (33) Estimate for the order of H ( t ) at infinity Recall from § H ( t ) denotes the moment generating function (10), which canbe represented (around t = ∞ ) as a Cauchy-type integral (16) of the algebraicfunction g ( z ) = Q ( P − ( z )). Recall from § V denotes the linear span of thebranches of g ( z ) with r := dim V and L the differential operator (21) satisfying V = ker L . Proposition 9. If H ( t ) then L H ( t ) .Proof. Assume that L H ( t ) ≡
0. Then H ( t ) ∈ V . Moreover, H ( t ) is holomorphicat t = ∞ , and in particular it is invariant under the monodromy around infinity M ∞ and hence also under the operator T ∞ : V → V, T ∞ := 1 d P d P − X k =0 M k ∞ . (34)Recall that g ( z ) = Q ( P − ( z )) and P − ( z ) has cyclic monodromy at ∞ . It followsthat the image of T ∞ is one-dimensional and spanned byIm T ∞ = C { S } , S ( t ) := X w : P ( w )= t Q ( w ) . (35)Moreover, S ( t ) is a polynomial: for instance, it is has no poles on C and moderategrowth at ∞ . We conclude that H ( t ) is a polynomial. Finally, H ( t ) has a zeroat t = ∞ by definition, and since it is also a polynomial it follows that H ( t ) ≡ (cid:3) Let D = t∂ t denote the Euler operator, and recall that it also gives the Euleroperator at t = ∞ (up to a sign). The following proposition describes the behaviorof L around infinity. Proposition 10.
We may write L ( t ) = u ( t ) b L b L := D r + b c r − D r − + · · · + b c , (36) where b c r − , . . . , b c are rational functions, holomorphic at t = ∞ , and u ( t ) is arational function satisfying ord ∞ u > − (cid:20) d Q rd P + 3( d P − − r ( r + 1)2 (cid:21) . (37) Proof.
The existence of an expression (36) is a direct consequence of the fact that L is a Fuchsian operator at t = ∞ (see [6, Proposition 19.18]). Using Proposition 8we have ord ∞ u = r − deg c r > − (cid:20) d Q rd P + 3( d P − − r ( r + 1)2 (cid:21) , (38)as claimed. (cid:3) Finally we have the following estimate.
Lemma 11. If H ( t ) then ord ∞ H ( t ) d Q rd P + 3( d P − − r ( r − . (39) OUNDS FOR CYCLICITY OF ZERO SOLUTION OF ABEL EQUATION 11
Proof.
Using Proposition 7 we have L H ( t ) = R ( t ), where R ( t ) has at most twopoles of order r in C . Moreover, by Proposition 9 R ( t ) is non-zero. It follows thatord ∞ R ( t ) r . Using Proposition 10 we haveord ∞ ( b L H ( t )) = ord ∞ R ( t ) − ord ∞ u ( t ) r + d Q rd P + 3( d P − − r ( r + 1)2 d Q rd P + 3( d P − − r ( r − . It remains only to note that the application of b L cannot decrease the order of zero,and the claim follows. (cid:3) Finally we complete the proof of our main result.
Proof of Theorem 1. If H ( t ) ∞ H ( t ) d Q + 3( d P − , (40)and the claim for N ( d P , d Q , a, b ) follows by Proposition 5. The claim for e N ( d P , d q , a, b )then follows from Lemma 6, noting that d Q = d q + 1. (cid:3) References [1] G. Binyamini, D. Novikov, and S. Yakovenko. On the number of zeros of Abelian integrals.
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Department of Computer Science, Technion - Israel Institute of Technology, Haifa32000, Israel
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