Sum-integral interpolators and the Euler-Maclaurin formula for polytopes (Q2884392)

If \(P\) is a lattice polyhedron in a rational vector space \(V\) and \(f\) is a polynomial function defined on \(P\), then one is interested in nice expressions for the sum \(s_f(P)\) of all \(f(x)\) where \(x\) runs through all lattice points of \(P\). In particular, such a formula is called ``local Euler-Maclaurin'', if it expresses \(s_f(P)\) as a sum over all faces \(F\) of \(P\) of integrals of the form \(\int_F D(P,F)\cdot f\). Here \(D(P,F)\) is a differential operator of infinite order, but with constant coefficients that acts as a kind of a weight working on \(f\). The striking point is the locality, i.e.\ \(D(P,F)\) is supposed to depend only on the (conical) shape of \(P\) in a neighborhood of a general point of \(F\).NEWLINENEWLINEThose formulae are known to exist (McMullen), but the operators are not uniquely determined. The choice can become canonical if one uses some additional structure like a scalar product on \(V\) or the presence of a complete flag in \(V^*\). In the present paper, the authors obtain a solution of a similar problem involving exponential sums arising as the kernels of the Laplace transformation. Here, the weight becomes a true function \(\mu(P,F)\) called interpolator, and knowing \(\mu\) does even imply solutions of the above Euler-Maclaurin problem and its inverse.NEWLINENEWLINEMoreover, the interpolator \(\mu\) is effectively computable, and it becomes canonical if understood as being parametrized by the flag variety on \(V^*\).