Homework 5

Fix an integer $1 \leq \ell \leq n$ and define the term order $\prec_\ell$ on $k[x_1,\ldots,x_n]$ as follows: $x^{u} \prec_\ell x^{v}$ if and only if $\sum_{i=1}^{\ell}u_i < \sum_{i=1}^{\ell}v_i$ or $\sum_{i=1}^{\ell}u_i = \sum_{i=1}^{\ell}v_i$ and $x^{u} \prec x^{v}$ where $\prec$ is the graded reverse lexicographic order. 

a) Let $I$ be an ideal in $k[x_1,\ldots,x_n]$ and $G$ a Gröbner basis for $I$ with respect to $\prec_\ell$ and the variable ordering $x_1 > x_2 > \cdots > x_n$.  Show that $G_\ell = G \cap k[x_{\ell +1},\ldots,x_n]$ is a Gröbner basis for $I_\ell$ with respect to the graded reverse lexicographic order on $k[x_{\ell +1},\ldots,x_n]$. 

b) Consider the ideal $I = \langle t^2 + x^2 + y^2 + z^2, t^2 + 2x^2 - xy -z^2, t + y^3 -z^3 \rangle \subset \mathbb{Q}[t,x,y,z]$.  Find a Gröbner basis for $I_1 = I\cap \mathbb{Q}[x, y, z]$ with respect to the graded reverse lexicographic term order. 

c) (Bonus challenge!) Find a Gröbner basis for $I_1$ with respect to the lexicographic order.  Which Gröbner basis do you prefer?