Gr\"obner basis and Krull dimension of Lov\'asz-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 $[n]=\{1,\ldots, n\}$. If $S=\mathbb{K}[x_1, \ldots, x_n, y_1, \ldots, y_n]$ is a polynomial ring, then the graded ideal \[ L_\Gamma^\mathbb{K}(2) = \left( x_{i}x_{j} + y_{i}y_{j} \colon \quad \{i, j\} \in E(\Gamma)\right) \subset S,\] is called the Lov\'{a}sz-Saks-Schrijver ideal, LSS-ideal for short, of $\Gamma$ with respect to $\mathbb{K}$. In the present paper, we compute a Gr\"obner 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.