A criterion for inclusion in LT(I)

Fix a monomial order on R = F[x_1, \ldots, x_n] and suppose G = \{g_1, \ldots,g_m\} is a Gröbner basis for the ideal I in R. Prove that h \in LT(I) if and only if h is a sum of monomials each divisible by some LT(g_i).


Since G is a Gröbner basis, we have LT(I) = (LT(g_1), \ldots, LT(g_m)). Thus LT(I) is a monomial ideal with monomial generators LT(g_i). By this previous exercise, h \in LT(I) if and only if each term in h is divisible by some LT(g_i).

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