On the multiplicity of hyperelliptic integrals

Abstract

Let be an Abelian integral, where H = y2 − xn+1 + P(x) is a hyperelliptic polynomial of Morse type, δ(t) a horizontal family of cycles in the curves {H = t}, and ω a polynomial 1-form in the variables x and y. We provide an upper bound on the multiplicity of I(t), away from the critical values of H. Namely: ord I(t) ⩽ n − 1 + (n(n − 1)/2) if deg ω < deg H = n + 1. The multiplicity of I(t) is regarded as the order of contact between a hyperplane in Cn and an integral curve of the Picard–Fuchs system. We observe that for Hamiltonians of even degrees, the above bound can be yet improved, since the corresponding monodromy representation is reducible. For a form ω of arbitrary degree, we are led to estimating the order of contact between γ(t) and a suitable algebraic hypersurface in Cn+1. We observe that ord I(t) grows like an affine function with respect to deg ω.