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

Abstract

Let P ( x ) P(x) be a real polynomial of degree 2 g + 1 2g+1 , H = y 2 + P ( x ) H=y^2+P(x) and δ ( h ) \delta (h) be an oval contained in the level set { H = h } \{H=h\} . We study complete Abelian integrals of the form \[ I ( h ) = ∫ δ ( h ) ( α 0 + α 1 x + … + α g − 1 x g − 1 ) d x y , h ∈ Σ , I(h)=\int _{\delta (h)} \frac {(\alpha _0+\alpha _1 x+\ldots + \alpha _{g-1}x^{g-1})dx}{y}, \;\;h\in \Sigma , \] where α i \alpha _i are real and Σ ⊂ R \Sigma \subset \mathbb {R} is a maximal open interval on which a continuous family of ovals { δ ( h ) } \{\delta (h)\} exists. We show that the g g -dimensional real vector space of these integrals is not Chebyshev in general: for any g > 1 g>1 , there are hyperelliptic Hamiltonians H H and continuous families of ovals δ ( h ) ⊂ { H = h } \delta (h)\subset \{H=h\} , h ∈ Σ h\in \Sigma , such that the Abelian integral I ( h ) I(h) can have at least [ 3 2 g ] − 1 [\frac 32g]-1 zeros in Σ \Sigma . Our main result is Theorem 1 in which we show that when g = 2 g=2 , exceptional families of ovals { δ ( h ) } \{\delta (h)\} exist, such that the corresponding vector space is still Chebyshev.