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

Abstract

Let $K$ be a field and n be a positive integer. Let $\Gamma =\left(\right[n],E)$ be a simple graph, where $\left[n\right]=\{1,\dots ,n\}$. If $S=K[x_1,\dots ,x_n,y_1,\dots ,y_n]$ is a polynomial ring, then the graded ideal$L_{\Gamma }^K\left(2\right)=(x_i{x}_j+y_i{y}_j:\{i,j\}\in E(\Gamma \left)\right)\subset S,$ is called the Lovász-Saks-Schrijver ideal, LSS-ideal for short, of Γ with respect to $K$. 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 $K$ and compute the Hilbert series of $L_T^K\left(2\right)$. Finally, we present concrete combinatorial formulas to obtain the Krull dimension of $S/L_T^K\left(2\right)$ as well as lower and upper bounds for Krull dimension.