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

Lubomir Gavrilov, Iliya D. Iliev
2003 Transactions of the American Mathematical Society  
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 where α i are real and Σ ⊂ 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 ∈ Σ, such that the
more » ... ∈ Σ, such that the Abelian integral I(h) can have at least [ 3 2 g] − 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.
doi:10.1090/s0002-9947-03-03432-9 fatcat:dahelehqhjdndmuf6nrw7pug6u