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

Loeser, François; Sabbah, Claude

Comment. Math. Helv. 66, No. 3, 458-503 (1991).

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.

Reviewer: A.Buium (Bucureşti)

MSC:

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

Keywords:

finite difference equations; determinants of integrals of multiform functions; finite difference systems

Citations:

Zbl 0695.33004; Zbl 0699.33004; Zbl 0339.35021