Lattice Size of Width One Lattice Polytopes in ℝ³

2022/07/26 by Alajmi, Abdulrahman, Soprunova, Jenya
#11H06 #52B20 #52C05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.13124

Abstract

The lattice size ls_Δ(P) of a lattice polytope P is a geometric invariant, which was formally introduced in relation to the problem of bounding the total degree and the bi-degree of the defining equation of an algebraic curve, but appeared implicitly earlier in geometric combinatorics. In this paper, we show that for an empty lattice polytope P⊂ℝ³ there exists a reduced basis of ℤ³ which computes its lattice size ls_Δ(P). This leads to a fast algorithm for computing ls_Δ(P) for such P. We also extend this result to another class of lattice width one polytopes P⊂ℝ³. We then provide a counterexample demonstrating that this result does not hold true for an arbitrary lattice polytope P⊂ℝ³ of lattice width one.

Related

• Lattice Size in Higher Dimension
• Lattice Size and Generalized Basis Reduction in Dimension 3
• Lattice Size of Plane Convex Bodies
• The lattice size of a lattice polygon
• Enumeration of lattice polytopes by their volume