an:00024744
Zbl 0760.39001
Loeser, Fran??ois; Sabbah, Claude
Finite difference equations and determinants of integrals of multiform functions
FR
Comment. Math. Helv. 66, No. 3, 458-503 (1991).
00158084
1991
j
39A10 33E20 15A15 58J52
finite difference equations; determinants of integrals of multiform functions; finite difference systems
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.
A.Buium (Bucure??ti)
Zbl 0695.33004; Zbl 0699.33004; Zbl 0339.35021