GROEBNER BASIS AND KRULL DIMENSION OF LOVASZ-SAKS-SHERIJVER IDEAL ASSOCIATED TO A TREE

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)ight) 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.