On the multiplicity of hyperelliptic integrals

Abstract

Let I(t) = δ(t) ω be an Abelian integral, where H = y 2 − x n+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 reasoning goes as follows: we consider the analytic curve parameterized by the integrals along δ(t) of the n "Petrov" forms of H (polynomial 1-forms that freely generate the module of relative cohomology of H), and interpret the multiplicity of I(t) as the order of contact of γ(t) and a linear hyperplane of C n . Using the Picard-Fuchs system satisfied by γ(t), we establish an algebraic identity involving the wronskian determinant of the integrals of the original form ω along a basis of the homology of the generic fiber of H. The latter wronskian is analyzed through this identity, which yields the estimate on the multiplicity of I(t). Still, in some cases, related to the geometry at infinity of the curves {H = t} ⊆ C 2 , the wronskian occurs to be zero identically. In this alternative we show how to adapt the argument to a system of smaller rank, and get a nontrivial wronskian. 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 C n+1 . We observe that ord I(t) grows like an affine function with respect to deg ω.

A bound on the multiplicity of horizontal sections for a connection on a Riemann surface

“…Also, when studying the multiplicity of Abelian integrals associated to a hyperelliptic fibration in [8], we observe that odd degree Hamiltonians give rise to a Picard-Fuchs system with reducible monodromy, which is revealed by the presence of a vanishing cycle at infinity in the level curves of the polynomial. We build explicitly, by transformations on the Petrov frame, a quotient system and its corresponding minor determinant .…”

Meromorphic Connections on ℙ1 and the Multiplicity of Abelian Integrals

“…In particular, the works [3,17] have obtained the same upper bound for a linear system of differential equations using quite different methods. In [16], the author approaches the problem in the case of hyperelliptic integrals.…”