Skip to main content
Log in

Minimal Polygons with Fixed Lattice Width

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

We classify the unimodular equivalence classes of inclusion-minimal polygons with a certain fixed lattice width. As a corollary, we find a sharp upper bound on the number of lattice points of these minimal polygons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Barile, M., Bernardi, D., Borisov, A., Kantor, J.-M.: On empty lattice simplices in dimension 4. Proc. Amer. Math. Soc. 139(12), 4247–4253 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Castryck, W., Cools, F.: Linear pencils encoded in the Newton polygon. Int. Math. Res. Not. IMRN 2017(10), 2998–3049 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Castryck, W. Cools, F.: The lattice size of a lattice polygon. J. Combin. Theory Ser. A 136, 64–95 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Castryck, W., Cools, F., Demeyer, J., Lemmens, A.: Computing graded Betti tables of toric surfaces. Accepted in Trans. Amer. Math. Soc., arXiv:1606.08181 (2016)

  5. Draisma, J., McAllister, T.B., Nill, B.: Lattice-width directions and Minkowski’s \(3^d\)-theorem. SIAM J. Discrete Math. 26(3), 1104–1107 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fejes Tóth, L., Makai Jr., E.: On the thinnest non-separable lattice of convex plates. Stud. Sci. Math. Hungar. 9(1974), 191–193 (1975)

  7. Haase, C., Ziegler, G.M.: On the maximal width of empty lattice simplices, European J. Combin. 21(1), 111–119 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lubbes, N., Schicho, J.: Lattice polygons and families of curves on rational surfaces. J. Algebraic Combin. 34(2), 213–236 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sebő, A.: An introduction to empty lattice-simplices. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.E. (eds.) Integer Programming and Combinatorial Optimization (Graz, 1999), pp. 400–414. Lecture Notes in Comput. Sci., 1610, Springer, Berlin (1999)

Download references

Acknowledgements

The second author is supported by the Flemish Research Council (FWO). We also want to thank the anonymous referees for their useful remarks.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alexander Lemmens.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cools, F., Lemmens, A. Minimal Polygons with Fixed Lattice Width. Ann. Comb. 23, 285–293 (2019). https://doi.org/10.1007/s00026-019-00431-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-019-00431-0

Keywords

Mathematics Subject Classification

Navigation