On the ideal of orthogonal representations of a graph in R2
On the ideal of orthogonal representations of a graph in R2
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DOI：
10.1016/j.aam.2015.09.009
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发表时间：
2015-10
期刊：
影响因子：
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通讯作者：
J. Herzog;Antonio Macchia;S. Madani;V. Welker
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文献类型：
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作者：
J. Herzog;Antonio Macchia;S. Madani;V. Welker
在本文中，我们从代数角度研究了在 d= 2 的情况下 R d 中简单图 G 的正交表示。由 Lovász 引入的图的正交表示是从顶点集到 R d 的映射，其中不相邻的顶点被发送到正交向量。我们展示了由表达该条件的方程生成的理想的代数性质，并推导了 d= 2 和 R 被任意场替换时各种正交嵌入的几何性质。特别是，当理想是激进的时，我们进行分类，并提供简化的初级分解 if− 1∉ K。这导致将正交嵌入的多样性描述为素理想定义的变量的并集。特别是，这适用于激励情况 K= R。
In this paper, we study orthogonal representations of simple graphs G in R d from an algebraic perspective in case d= 2. Orthogonal representations of graphs, introduced by Lovász, are maps from the vertex set to R d where non-adjacent vertices are sent to orthogonal vectors. We exhibit algebraic properties of the ideal generated by the equations expressing this condition and deduce geometric properties of the variety of orthogonal embeddings for d= 2 and R replaced by an arbitrary field. In particular, we classify when the ideal is radical and provide a reduced primary decomposition if− 1∉ K. This leads to a description of the variety of orthogonal embeddings as a union of varieties defined by prime ideals. In particular, this applies to the motivating case K= R.