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Let $I = (f_1,\dots,f_r) \subset \mathbb{Z}[x_1, \dots, x_n]$. Further, let $J = (f_1, \dots, f_m)$ and $K = (f_{m+1}, \dots, f_r)$. Suppose that we know a Grobner basis for $I = (g_1, \dots, g_s)$ and $J = (h_1, \dots, h_t)$. Could we use this to find a Grobner basis for $K$? Also, is there any relationship between that basis of $J$ and $I$?

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