${\mathbb{K}}[{\pmb{x}}]$, the ideal generated by $P$ is defined as $\langle P\rangle\triangleq\Big\{\sum_{i=1}^n f_ih_i\mid n\in \mathbb{N},\ f_i\in P,\ h_i\in {\mathbb{K}}[{\pmb{x}}]\Big\}.$

3.  The product of two ideals $I$ and $J$ is defined as $$I\times J=\langle\{f\cdot g\mid f\in I, g\in J\}\rangle.$$

The ideal generated by $P$ is actually the minimal one of ideals that contain $P$. When $P=\{f_1,f_2,\cdots,f_n\}$ is a finite set, we simply write $\langle P\rangle$ as $\langle f_1,f_2,\cdots,f_n\rangle$. Given two polynomial sets $P$ and $Q$, we define $P\cdot Q\triangleq\{f\cdot g\mid f\in P, g\in Q\}$. Obviously, $\langle P\rangle\times\langle Q\rangle=\langle P\cdot Q\rangle$.

Every ideal $I\subseteq {\mathbb{K}}[{\pmb{x}}]$ is finitely generated, that is, $I=\langle f_1,f_2,\cdots,f_n\rangle$ for some $f_1,f_2,\cdots,f_n\in {\mathbb{K}}[{\pmb{x}}]$. Here $\{f_1,f_2,\cdots,f_n\}$ is called a basis of $I$.

We define the degree of an ideal $I$ as $$\operatorname{gdeg}(I)=\min\{\deg(P)\mid P\ \text{is a basis of}\ I\}.$$ Note that an ideal may have different bases. However, using the Buchberger’s algorithm under a fixed monomial ordering, a unique (reduced) Gröbner basis of $I$, denoted by $\operatorname{GB}(I)$, can be computed from any other basis. We also simply write $\operatorname{GB}(\langle P\rangle)$ as $\operatorname{GB}(P)$ for any basis $P$. An important property of Gröbner basis is that the remainder of any polynomial $f$ on division by $\operatorname{GB}(P)$, written as $\operatorname{Rem}(f,\operatorname{GB}(P))$, satisfies that $$f\in\langle P\rangle\Leftrightarrow\operatorname{Rem}(f,\operatorname{GB}(P))=0.$$

The 