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Lattice points in convex polyhedra. (Point entiers dans les polyèdres convexes.) (French) Zbl 0667.52011

Consider a convex polyhedron \(P\) in a real vector space all of whose vertices are points of a lattice \(M\). The author describes the lattice points of \(P\), i.e. \(P\cap M\), and also enumerates the lattice points of a multiple \(nP\) of \(P\). He introduces the characteristic function \(F(P)\) of \(P\) by first associating with each point \(Q\) of \(M\) a monomial \(x_ 1^{m_ 1}\ldots x_ d^{m_ d}\), where \(Q=(m_ 1,\dots, m_ d)\), the coordinates shown being in a basis of \(m\) over the integers \(\mathbb Z\). \(F(P)\) is just the sum of these monomials associated with the lattice points of \(P\). He similarly defines the characteristic function of a convex polyhedral rational cone \(C\) as the sum of the above monomials associated with the lattice points of \(C\). Now let \(S\) be the set of vertices of \(P\). For all \(s\in S\), let \(C_ s\) be the (convex polyhedral rational) cone generated by the points \(-s+p,\) for \(p\in P\). One can view \(C_ s\) as the cone tangent to \(P\) at \(s\), translated so that its vertex is at \(0\).
The main theorem is: \[ F(P)=\sum_{s\in S}x^ sF(C_ s). \]
It is established using the connections between convex polyhedra and toric varieties.
The main result is applied to yield:
1) the precise number of lattice points of \(nP\), as a polynomial in \(n\);
2) the “reciprocity law” linking the number of lattice points in \(nP\) to their number in the relative interior of \(nP\);
3) the integral of the exponential of a linear form on \(P\), which is used to show the existence of “generalized mixed volumes” for compact convex sets.
Reviewer: H.Herda

MSC:

11H06 Lattices and convex bodies (number-theoretic aspects)
12L10 Ultraproducts and field theory
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References:

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