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Abdul Dissertation.pdf (364.27 KB)

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On The Lattice Size With Respect To The Standard Simplex in 3D.

Alajmi, Abdulrahman N
http://rave.ohiolink.edu/etdc/view?acc_num=kent1598893373379275

Abstract Details

2020, PHD, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
We study the lattice size lsΔ(P) of a lattice polytope P, which is defined to be the smallest integer dilation of the standard simplex Δ that contains P, up to affine unimodular maps. The motivation for studying this object comes from algebraic geometry and coding theory. An "onion's skins" algorithm for finding lsΔ(P) for a lattice polygon P ⊂ R² was developed by Castryck, Cools, and Schicho. Recently, Harrison, Soprunova, and Tierney provided a faster algorithm for computing the lattice size in dimension 2, and also explained a link between the lattice size and reduced bases [6]. In this dissertation, we work on computing lsΔ(P) in dimension 3. We provide a partial "onion skins" algorithm for computing the lattice size. We also compute explicitly the lattice size for many families of empty lattice 3-polytopes, that is, lattice polytopes whose only integer points are their vertices. For this, we use Howe's classification of empty lattice 3-polytopes. We next show that for lattice width one pyramids P ⊂ R³ there exists a reduced basis that computes lsΔ(P), which provides an algorithm for computing lsΔ(P). We also conjecture that this holds true for any lattice width one polytope P ⊂ R³.
Jenya Soprunova, Dr. (Advisor)
Ivan Soprunov, Dr. (Committee Member)
Artem Zvavitch, Dr. (Committee Member)
Feodor Dragan, Dr. (Committee Member)
Gerassimos Petratos, Dr. (Committee Member)
50 p.
Mathematics
lattice size, discrete geometry, polytopes, simplex,

Recommended Citations

Citations

  • Alajmi, A. N. (2020). On The Lattice Size With Respect To The Standard Simplex in 3D. [Doctoral dissertation, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1598893373379275

    APA Style (7th edition)

  • Alajmi, Abdulrahman. On The Lattice Size With Respect To The Standard Simplex in 3D. 2020. Kent State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1598893373379275.

    MLA Style (8th edition)

  • Alajmi, Abdulrahman. "On The Lattice Size With Respect To The Standard Simplex in 3D." Doctoral dissertation, Kent State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=kent1598893373379275

    Chicago Manual of Style (17th edition)

kent1598893373379275
112
© 2020, all rights reserved.
This open access ETD is published by Kent State University and OhioLINK.