I would like to ask the following. Given only the leading terms of an ideal $I$, namely the set $LT(I)$, is it possible to find a Groebner Basis of $I$? If not always, then when is it possible? We know that $\langle LT(I)\rangle = \langle LT(g_1), \dots, LT(g_n)\rangle$ for a Groebner basis $g_1, \dots, g_n$ but can we find exactly one basis $g_1, \dots, g_n$ given only the $LT(I)$?
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Added the groebner-bases tag as the commutative-algebra tag alone is fairly general. Also did some formatting such as replacing < with \langle.
Michael Albanese
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Can we find a Groebner Basis?
Sln
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