Modules of the Abelian integrals and the Picard$ndash$Fuchs systems*

Novikov, D.

doi:10.1088/0951-7715/15/5/305

Cited by 7 publications

(4 citation statements)

References 22 publications

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“…Its actual degree depends on the choice of the integrands. It is shown by L. Gavrilov in [6] and D. Novikov in [11] that one can plug the Petrov forms into the period matrix P and get det P = c • (t − t 1 ) . .…”

Section: Main Resultmentioning

confidence: 99%

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On the multiplicity of hyperelliptic integrals

Moura¹

2004

Nonlinearity

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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.

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Section: Main Resultmentioning

confidence: 99%

“…As shown by Gavrilov [6] and Novikov [11], one can plug the Petrov forms into the period matrix F and obtain for det F a polynomial of degree n (actually coinciding with P (t) up to a multiplicative constant).…”

Section: Resultsmentioning

confidence: 99%

On the multiplicity of hyperelliptic integrals

Moura¹

2004

Nonlinearity

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“…Elementary proofs of (1.18) and of the latter implication were obtained separately by Yu. S. Ilyashenko (his proof is represented in 2.4) and D. Novikov [12].…”

Section: -A Generic Complex Polynomial H Is Said To Be Ultramorse Ifmentioning

confidence: 98%

An explicit formula for period determinant

Glutsyuk¹

2006

Ann. inst. Fourier

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“…With various degrees of generality, completeness and precision it appeared in many sources, among them the recent paper [Gav98]. Other sources, together with the complete demonstrations, can be found in [Nov02].…”

Section: Nondegeneracy Of the Period Matrixmentioning

confidence: 99%