Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians

Abstract

An explicit upper bound Z(3, n) 5n+15 is derived for the number of the zeros of the integral h → I (h) = H =h g(x, y) dx −f (x, y) dy of degree n polynomials f, g, on the open interval for which the cubic curve {H = h} contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals H <h x i y j dx dy, i, j = 0, 1, generating the module of complete Abelian integrals I (h) (over the ring of polynomials in h).

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

“…It should also be noticed that the Chebyshev property does not hold for some of the reversible cases [11]. Hence, one cannot expect that Theorem 1, as stated above, will also take place for the remaining reversible Hamiltonians which correspond to values a ∈ R \ (− 1 2 , 0), see Figure 1.…”

“…First we recall the normal form for all cubic Hamiltonians having a center. As in [10,11], introduce the following basic one-forms…”

“…As the one-form ω Z is of the second kind [11] then ω R is of second kind too (that is to say it has no residues). The Picard-Lefschetz formula implies that W δ 1 ,δ 2 (ω R , ω M ) is singlevalued in h on the complex plane C, and (26) implies that it has a single pole at h = h l .…”

Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields

“…The corresponding R[h] modules PH, A and its generators are computed in Section 6. We compute then the dimension of the real vector space An C A, formed by Abelian integrals (1) such that deg(P),deg(Q) ^ n. Part of these results are already known to the specialists [12], others were used without justification or were erroneously stated (see Remark 5 after Theorem 4 in Section 7). They can be used either as an illustration of Theorem 2, or as a reference in further study of the following Hilbert-Arnold problem.…”

“…The Hilbert-Arnold problem (called "weakened 16th LUBOMIR GAVRILOV 615 Hilbert problem" in [4], p. 313, and [17]) is It was proved recently [12] that Z(3,n) <5n+15 but the exact value of Z(3, n) is still unknown (even for n = 2!). For special Hamiltonians H the numbers Z(H,n) are computed in [22], [23], [24], [8] and [9].…”