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Let $I \subseteq S=\mathbb{k}\left[x_{1}, \ldots, x_{d}\right]$ be an ideal, $<$ be a monomial order on $S$ and let $T=S[t]=\mathbb{k}\left[x_{1}, \ldots, x_{d}, t\right]$. There exists $\omega \in \m …

Let $S=K[x_1,\ldots, x_n]$ polynomial ring. Let $I \subseteq S$ an ideal and $<$ be a monomial order in $S$. Is it possible to describe the minimal primes of in$_<(I)$ from the minimal primes of $I$? …

Let $S=K[x_1,\ldots, x_n]$ polynomial ring. Let $I \subseteq S$ an ideal and $<$ be a (global) monomial order in $S$. If in$_<(I)$ a radical ideal, then in$_<(I)=$ in$_<(P_1) \;\cap$ in$_<(P_2)\cap \l …

Since $in(I)$ is a radical ideal, $I$ is a radical ideal. Thus, $I=\bigcap_{i=1}^{l} P_i$, and so, $in(I)=in\left( \bigcap_{i=1}^{l} P_i \right) \subseteq \bigcap_{i=1}^{l} in(P_i)$. Now, we show that …