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Journal of Geometry

, 110:16 | Cite as

Discrete volumes of lattice polyhedra via vector analysis

  • Yasuzo NishimuraEmail author
Article
  • 16 Downloads

Abstract

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.

Keywords

Lattice polyhedron Pick’s theorem Gauss divergence theorem 

Mathematics Subject Classification

Primary 52B20 52B10 

Notes

References

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Education, Humanities and Social SciencesUniversity of FukuiFukuiJapan

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