Finite difference equations and determinants of integrals of multiform functions (Q1179514)

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 \textit{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 \textit{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.