Zero-generic initial ideals
Let \(K\) be a field, \(I\) a homogeneous ideal in \(K[x_1, \ldots, x_n]\) and \(>\) a monomial ordering. Let \(\mathrm{gin}_>(I)\) be the generic initial ideal. If \(K\) is infinite, there exists a non-empty Zariski open set \(U\subseteq \mathrm{GL}_n(K)\) of coordinate changes such that \(\mathrm{gin}_>(I)=\mathrm{in}_>(\varphi(I))\) for all \(\varphi\in U\). In order to obtain the same properties in characteristic \(p\) as in characteristic zero for the generic initial ideal the definition is modified. The zero-generic initial ideal of \(I\) with respect to \(>\) is defined by \(\mathrm{gin}_0(I)=(\mathrm{gin}_>((\mathrm{gin}_>(I))_\mathbb{Q}))_K\). If \(K\) has characteristic \(0\) then obviously \(\mathrm{gin}_0(I)=\mathrm{gin}_>(I)\). It is proved that \(\mathrm{gin}_0(I)\) is endowed with many interesting properties. Especially it satisfies Green's Crystallization Principle.
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