Finite difference equations and determinants of integrals of multiform functions. (Equations aux différences finies et déterminants d’intégrales de fonctions multiformes.) (French) Zbl 0760.39001

The aim of the paper is to give a general formula for a determinant whose entries are integrals of the form \[ \int_ \gamma f_ 1^{s_ 1} \dots f_ p^{s_ p}\omega, \] where \(f_ i\) are complex polynomials in \(n\) variables, \(\omega\) is an algebraic \(n\)-form and \(\gamma\) are suitable \(n\)-cycles. This generalizes previous work of A. N. Varchenko [Izv. Akad. Nauk SSSR 53, No. 6, 1206-1235 (1989; Zbl 0695.33004) and 54, No. 1, 146-158 (1990; Zbl 0699.33004)] who considered the case \(\deg f_ i=1\).
The starting point of the theory contained in the present paper is a construction of K. Aomoto [J. Fac. Sci., Univ. Tokyo, Sect. I A 22, 271-297 (1975; Zbl 0339.35021)] relating the above integrals to certain finite difference systems.


39A10 Additive difference equations
33E20 Other functions defined by series and integrals
15A15 Determinants, permanents, traces, other special matrix functions
58J52 Determinants and determinant bundles, analytic torsion
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