Complete hyperelliptic integrals of the first kind and their non-oscillation

Let P(x) be a real polynomial of degree 2g + 1, H = y 2 + P(x) and δ(h) be an oval contained in the level set {H = h}. We study complete Abelian integrals of the form I(h) = ∫ δ(h) (α 0 + α 1 x +... + α g-1 x g-1 /y)dx y, h ∈ Σ, where α i are real and E ⊂ R is a maximal open interval on which a continuous family of ovals {δ(h)} exists. We show that the g-dimensional real vector space of these integrals is not Chebyshev in general: for any g > 1, there are hyperelliptic Hamiltonians H and continuous families of ovals δ(h) ⊂ {H = h}, h E Σ, such that the Abelian integral I(h) can have at least [3/2g] - 1 zeros in Σ. Our main result is Theorem 1 in which we show that when g = 2, exceptional families of ovals {δ(h)} exist, such that the corresponding vector space is still Chebyshev.

[1]  J. M. Boardman,et al.  Singularties of differentiable maps , 1967 .

[2]  Egbert Brieskorn,et al.  Die monodromie der isolierten singularitäten von hyperflächen , 1970 .

[3]  V. Arnold,et al.  Ordinary Differential Equations , 1973 .

[4]  N. A'campo.  Le groupe de monodromie du déploiement des singularités isolées de courbes planes I , 1975 .

[5]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[6]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[7]  G. S. Petrov,et al.  Number of zeros of complete elliptic integrals , 1984 .

[8]  P. Griffiths,et al.  Geometry of algebraic curves , 1985 .

[9]  G. S. Petrov,et al.  The chebyshev property of elliptic integrals , 1988 .

[10]  Vladimir I. Arnold,et al.  Some unsolved problems in the theory of differential equations and mathematical physics , 1989 .

[11]  Alexandru Dimca,et al.  Singularities and Topology of Hypersurfaces , 1992 .

[12]  Vladimir I. Arnold.  Sur quelques problèmes de la théorie des systèmes dynamiques , 1994 .

[13]  Isochronicity of plane polynomial Hamiltonian systems , 1997 .

[14]  Iliya D. Iliev,et al.  Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians , 1998 .

[15]  Lubomir Gavrilov,et al.  Petrov modules and zeros of Abelian integrals , 1998 .

[16]  R. Roussarie,et al.  Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem , 1998 .

[17]  Lubomir Gavrilov.  Nonoscillation of Elliptic Integrals Related to Cubic Polynomials with Symmetry of Order Three , 1998 .

[18]  Lubomir Gavrilov,et al.  Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields , 1999 .

[19]  G. Petrov.  TANGENTIAL HILBERT PROBLEM FOR PERTURBATIONS OF HYPERELLIPTIC HAMILTONIAN SYSTEMS , 1999 .

[20]  Lubomir Gavrilov,et al.  The infinitesimal 16th Hilbert problem in the quadratic case , 2001 .

[21]  J. Llibre,et al.  Chebyshev property of complete elliptic integrals and its application to abelian integrals , 2002 .