We develop techniques for the verification of the Chebyshev property of Abelian integrals based on paper by Birth of canard cycles. These techniques are a combination of theoretical results, analysis of asymptotic behavior of Wronskians, and rigorous computations based on interval arithmetic. We apply this approach to tackle a conjecture formulated by Dumortier and Roussarie in [F. Dumortier, R. Roussarie, Birth of canard cycles, Discrete Contin. Dyn. Syst. 2 (2009) 723–781], which we are able to prove for q ? 2. They investigate the number of limit cycles that appear near a slow-fast Hopf point, i.e., its cyclicity. Their main results show that this cyclicity is finite and, modulo the following conjecture, provide its precise upper bound. Conjecture. For each integer i ? 0, let us define J ¯ i ( h ) = ? ? h y 2 i ? 1 dx , where ? h ?{ A ( x )+ B ( x ) y 2 }= h with and B ( x ) = e- 2 x . then ( J ¯ 0 ,J ¯ 1 ,...,J ¯ n ) is an ECT-system on for n ? 0. Figure 1.7 shows Phase portrait of the associated Hamiltonian system with H ( x,y ) = A ( x )+ B ( x ) y 2 . To begin with, let f 0 ,f 1 ,...,f n ?1 be analytic functions on an interval I. Definition 1. ( f 0 ,f 1 ,...,f n ?1) is a Chebyshev CT-system on I if, for k = 1 , 2 ,...,n , any nontrivial Figure 1.7 linear combination ? 0 f 0 ( x ) + ? 1 f 1 ( x ) + ... + ? k 1 f k 1 ( x ) = 0 ? ? has at most k ? 1 isolated zeros on I. Definition 2. ( f 0 ,f 1 ,...,f n 1 ) is an extended Chebyshev system (ECT-system) on I if, for k = 1 , 2 ,...,n , any nontrivial linear combination ? 0 f 0 ( x ) + ? 1 f 1 ( x ) + ... + ? k- 1 f k - 1 ( x ) = 0 has at most k ? 1 isolated zeros on I counted with multiplicities. It is clear that if ( f 0 ,f 1 ,...,f n- 1 ) is an ECT-system on I, then it is a CT-system on I. ?
