Gröbner basis and Krull dimension of Lovász-Saks-Sherijver ideal associated to a tree

Abstract

Let $\mathbb{K}$ be a field and $n$ be a positive integer. Let $\Gamma =([n],E)$ be a simple graph, where $[n]=\{1,\dots ,n\}$. If $S=\mathbb{K}[x_1,\dots ,x_n,y_1,\dots ,y_n]$ is a polynomial ring, then the graded ideal \[ L_Γ^\mathbb{K}(2) = \left( x_{i}x_{j} + y_{i}y_{j} \colon \quad \{i, j\} \in E(Γ)\right) \subset S,\] is called the Lovász-Saks-Schrijver ideal, LSS-ideal for short, of $\Gamma$ with respect to $\mathbb{K}$. In the present paper, we compute a Gröbner basis of this ideal with respect to lexicographic ordering induced by $x_1>\cdots >x_n>y_1>\cdots >y_n$ when $\Gamma =T$ is a tree. As a result, we show that it is independent of the choice of the ground field $\mathbb{K}$ and compute the Hilbert series of $L_T^{\mathbb{K}}(2)$. Finally, we present concrete combinatorial formulas to obtain the Krull dimension of $S/L_T^{\mathbb{K}}(2)$ as well as lower and upper bounds for Krull dimension.

23 pages, 1 figure