Extension of the Lasserre–Avrachenkov theorem on the integral of multilinear forms over simplices

摘要

The Lasserre–Avrachenkov theorem on integration of symmetric multilinear forms over simplices establishes a method (called LA) for integrating homogeneous polynomials over simplices. Although the computational complexity of LA is generally much higher than that of the other known methods (e.g. Grundmann–Moller formula), it is still useful in deriving closed-form expressions for the value of such integrals. However, LA cannot be directly applied for nonhomogeneous polynomials. It is shown in this paper that Lasserre–Avrachenkov theorem holds for a wider class of symmetric forms, to be called quasilinear forms. This extension can substantially facilitate derivation of a closed-form expression (not computation) for integral of some nonhomogeneous polynomials (such as ∏j=1qbj+∑i=1nci,jxi) over simplices.