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

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 Pi(q)(j-1) (b(j) + Sigma(n)(i-1)c(i,)x(ji))) over simplices. (c) 2009 Elsevier Inc. All rights reserved.