Discrete volumes of lattice polyhedra via vector analysis
Tóm tắt
Pick’s theorem relates the number of lattice points to the area for a lattice polygon. Diaz and Robins gave a proof of Pick’s theorem by using the Weierstrass
$$\wp $$
-function and complex analysis. As an analogue to lattice convex polyhedra, Reeve’s theorem is known as a solid version of Pick’s theorem. In this paper, we study counting lattice points on a lattice polyhedron by using vector analysis, and we extend Reeve’s theorem to nonconvex polyhedral complexes.
Tài liệu tham khảo
Beck, M., Robins, S.: Computing the Continuous Discretely. Springer, Berlin (2007)
Brianchon, C.J.: Théorème nouveau sur les polyèdres. J. Ecol. Polytech. 15, 317–319 (1837)
Diaz, R., Robins, S.: Pick’s formula via the Weierstrass \(\wp \)-function. Am. Math. Mon. 102(5), 431–437 (1995)
Ehrhart, E.: Polynômes arithmétiques et Méthode des Polyèdres en Combinatoire, Birkhäuser (1977)
Macdonald, I.G.: Polynomials associated with finite cell-complexes. J. Lond. Math. Soc. 4(2), 181–192 (1971)
Pick, G.A.: Geometrisches zur Zahlenlehre. Sitzenber Lotos 19, 311–319 (1899)
Reeve, J.E.: On the volume of lattice polyhedra. Proc. Lond. Math. Soc. 7(3), 378–395 (1957)
Varberg, D.E.: Pick’s theorem revisited. Am. Math. Mon. 92, 584–587 (1985)