×

zbMATH — the first resource for mathematics

Residue formulae, vector partition functions and lattice points in rational polytopes. (English) Zbl 0926.52016
The authors develop residue formulae for a class of functions of several variables and apply these to various problems. In particular they obtain closed formulae for vector partition functions and for their continuous analogs. These imply an Euler-MacLaurin summation formula for vector partition functions and also for rational convex polytopes.
So the authors express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope.

MSC:
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
11P21 Lattice points in specified regions
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Michael Francis Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathematics, Vol. 401, Springer-Verlag, Berlin-New York, 1974. · Zbl 0297.58009
[2] Alexander I. Barvinok, Computing the volume, counting integral points, and exponential sums, Discrete Comput. Geom. 10 (1993), no. 2, 123 – 141. · Zbl 0774.68054
[3] M. Brion and M. Vergne, Lattice points in simple polytopes, J. Amer. Math. Soc. 10 (1997), 371-392. CMP 97:06 · Zbl 0871.52009
[4] M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. reine angew. Math. 482 (1997), 67-92. CMP 97:06 · Zbl 0862.14006
[5] Michel Brion and Michèle Vergne, Une formule d’Euler-Maclaurin pour les fonctions de partition, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 3, 217 – 220 (French, with English and French summaries). · Zbl 0869.65004
[6] Michel Brion and Michèle Vergne, Une formule d’Euler-Maclaurin pour les polytopes convexes rationnels, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 4, 317 – 320 (French, with English and French summaries). · Zbl 0870.52004
[7] Sylvain E. Cappell and Julius L. Shaneson, Genera of algebraic varieties and counting of lattice points, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 62 – 69. · Zbl 0847.14010
[8] Sylvain E. Cappell and Julius L. Shaneson, Euler-Maclaurin expansions for lattices above dimension one, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 7, 885 – 890 (English, with English and French summaries). · Zbl 0838.52018
[9] R. Diaz and S. Robins, The Ehrhart polynomial of a lattice \(n\)-simplex, Electronic Research Announcements of the AMS 2 (1996). CMP 96:17 · Zbl 0871.52008
[10] E. Ehrhart, Sur un problème de géométrie diophantienne linéaire. I. Polyèdres et réseaux, J. Reine Angew. Math. 226 (1967), 1 – 29 (French). · Zbl 0155.37503
[11] V. Ginzburg, V. Guillemin and Y. Karshon, Cobordism techniques in symplectic geometry, The Carus Mathematical Monographs,, Mathematical Association of America, to appear.
[12] V. Guillemin, Riemann-Roch for toric orbifolds, preprint (1995). · Zbl 0932.37039
[13] Masa-Nori Ishida, Polyhedral Laurent series and Brion’s equalities, Internat. J. Math. 1 (1990), no. 3, 251 – 265. · Zbl 0728.52008
[14] L. C. Jeffrey and F. C. Kirwan, Localization for non-abelian group actions, Topology 34 (1995), 291-327. CMP 95:08
[15] Jean-Michel Kantor and Askold Khovanskii, Une application du théorème de Riemann-Roch combinatoire au polynôme d’Ehrhart des polytopes entiers de \?^{\?}, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 5, 501 – 507 (French, with English and French summaries). · Zbl 0791.52012
[16] Tetsuro Kawasaki, The Riemann-Roch theorem for complex \?-manifolds, Osaka J. Math. 16 (1979), no. 1, 151 – 159. · Zbl 0405.32010
[17] A. V. Pukhlikov and A. G. Khovanskiĭ, The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, Algebra i Analiz 4 (1992), no. 4, 188 – 216 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 4, 789 – 812.
[18] P. McMullen, Transforms, diagrams and representations, Contributions to geometry (Proc. Geom. Sympos., Siegen, 1978) Birkhäuser, Basel-Boston, Mass., 1979, pp. 92 – 130. · Zbl 0445.52006
[19] Robert Morelli, A theory of polyhedra, Adv. Math. 97 (1993), no. 1, 1 – 73. · Zbl 0779.52016
[20] Bernd Sturmfels, On vector partition functions, J. Combin. Theory Ser. A 72 (1995), no. 2, 302 – 309. · Zbl 0837.11055
[21] M. Vergne, Equivariant index formulas for orbifolds, Duke Math. J. 82 (1996), 637-652. CMP 96:12
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.