Pukhlikov, A. V.; Khovanskii, A. G. 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”. Reviewer: Vasile Brînzănescu (Bucureşti) Cited in 3 ReviewsCited in 56 Documents 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 Keywords:pointed cones; polytopes; Todd operator; Riemann-Roch theorem; convex chains; integrals; lattice sums; quasipolynomial Citations:Zbl 0791.52010; Zbl 0445.14019 PDFBibTeX XMLCite \textit{A. V. Pukhlikov} and \textit{A. G. Khovanskii}, St. Petersbg. Math. J. 4, No. 4, 789--812 (1992; Zbl 0798.52010); translation from Algebra Anal. 4, No. 4, 188--216 (1992)