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A Riemann-Roch theorem for integrals and sums of quasipolynomials over virtual polytopes. (English. Russian original) Zbl 0798.52010

St. Petersbg. Math. J. 4, No. 4, 789-812 (1993); translation from Algebra Anal. 4, No. 4, 188-216 (1992).
This paper is a direct continuation of the authors’ paper [Algebra Anal. 4, No. 2, 161–185 (1992; Zbl 0791.52010)]. The authors study special measures of convex chains, namely, integrals and lattice sums of quasi-polynomials and give detailed computation of these measures. A striking connection between integration and lattice summation is obtained: the main result is a “Riemann-Roch theorem”, connecting (through a “Todd operator”) the lattice sum and the integral of the same quasi-polynomial over a family of convex chains. A method of computing the number of lattice points of a polyhedron using a Riemann-Roch theorem was given in the second author’s paper [Funct. Anal. Appl. 11, 289–296 (1978); translation from Funkts. Anal. Prilozh. 11, No. 4, 56–64 (1977; Zbl 0445.14019)]. A short presentation of the algebro-geometric origin of the main result is given: in fact, the usual Riemann-Roch theorem for a smooth toric variety is a special case of the above “Riemann-Roch theorem”.

MSC:

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C40 Riemann-Roch theorems
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