Lattice Size in Higher Dimension

TL;DR

The paper addresses the problem of determining lattice sizes for a natural family of higher-dimensional lattice simplices $T_{p_1\dots p_d}$, extending previous results in low dimensions. It develops an elementary, self-contained approach centered on lattice width, $l_1$-size, and unimodular transformations, with key lemmas governing widths under $GL(d,\mathbb Z)$ actions. The main contributions are explicit formulas, namely $\operatorname{ls_\Delta}(T_{p_1\dots p_d})=k+3$ and $\operatorname{ls_\square}(T_{p_1\dots p_d})=k+2$ where $\alpha=\sum_{i=1}^{d-1} p_i$, $k=\left\lfloor\frac{p_d-2}{\alpha+1}\right\rfloor$, and under the condition $p_d\ge\alpha^2-\alpha$, realized by an explicit unimodular matrix; plus a detailed analysis of primitive directions and gcd constraints. This work extends lattice-size computation to arbitrary dimension for a natural simplex family and informs algorithmic strategies for higher-dimensional lattice problems.

Abstract

The lattice size of a lattice polytope is a geometric invariant which was formally introduced in the context of simplification of the defining equation of an algebraic curve, but appeared implicitly earlier in geometric combinatorics. Previous work on the lattice size was devoted to studying the lattice size in dimension 2 and 3. In this paper we establish explicit formulas for the lattice size of a family of lattice simplices in arbitrary dimension.

Figures (2)

• Figure 1: Lattice polygon $P$ with $\operatorname{w}_{(1,1)}(P)=4$.
• Figure 2:

Theorems & Definitions (16)

• Definition 1.1
• Example 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 3.1
• proof
• Lemma 3.2
• proof