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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 $I$?

We can assume that $<$ is a global monomial order, and in$_<(I)$ a radical ideal.

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    $\begingroup$ Depends on "<"? $\endgroup$
    – markvs
    Commented Mar 22, 2022 at 4:55
  • $\begingroup$ To be specific. Suppose $in_<(I)$ is a radical ideal. Let $P_1, \ldots, P_t$ be the minimal prime ideals of $I$. Then, $in_<(P_1), \ldots,in_<(P_t)$ are the minimal prime ideals of $in_<(i)$. Is this true? $\endgroup$
    – Wágner Badilla
    Commented Mar 22, 2022 at 19:18
  • $\begingroup$ There are two different definitions of "monomial order". Do you assume that $<$ is a well order? $\endgroup$
    – markvs
    Commented Mar 22, 2022 at 19:23
  • $\begingroup$ I assume that $<$ is a global monomial order. Thus, $<$ is a well order. $\endgroup$
    – Wágner Badilla
    Commented Mar 22, 2022 at 19:26
  • $\begingroup$ What is a "global monomial order"? $\endgroup$
    – markvs
    Commented Mar 22, 2022 at 19:37

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