论文标题
宽度的晶格大小一个晶格多塔，$ \ mathbb {r}^3 $
Lattice Size of Width One Lattice Polytopes in $\mathbb{R}^3$
论文作者
论文摘要
晶格polytope $ p $的晶格尺寸$ \ operatorname {ls_δ}（p）$是几何不变的，它是针对整体程度和界限的总体程度和双度方程式的正式引入的，但在代数曲线的定义方程式中，但在较早的基础上隐含地隐含着隐含的基因组合图。在本文中，我们表明，对于一个空的晶格polytope $ p \ subset \ mathbb {r}^3 $，存在$ \ mathbb {z}^3 $的简化基础，该基础计算其晶状体大小$ \ propatateOrnAme {ls_δ}（p）$。这导致了用于计算此类$ p $的快速算法。我们还将此结果扩展到另一类的晶格宽度一个polytopes $ p \ subset \ mathbb {r}^3 $。然后，我们提供了一个反例，证明该结果对于晶格宽度的任意晶格polytope $ p \ subset \ mathbb {r}^3 $不满。
The lattice size $\operatorname{ls_Δ}(P)$ 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 $P\subset\mathbb{R}^3$ there exists a reduced basis of $\mathbb{Z}^3$ which computes its lattice size $\operatorname{ls_Δ}(P)$. This leads to a fast algorithm for computing $\operatorname{ls_Δ}(P)$ for such $P$. We also extend this result to another class of lattice width one polytopes $P\subset\mathbb{R}^3$. We then provide a counterexample demonstrating that this result does not hold true for an arbitrary lattice polytope $P\subset\mathbb{R}^3$ of lattice width one.
