Finite numbers of initial ideals in non-Noetherian polynomial rings

Abstract:

In this article, we generalize the well-known result that ideals of Noetherian polynomial rings have only finitely many initial ideals to the situation of ideals in the polynomial ring $R=\mathbb {K}[x_{i,j} | 1\leq i\leq c,j\in \mathbb {N}]$ that are invariant under the action of the monoid $\mathrm {Inc}(\mathbb {N})$ of strictly increasing functions on $\mathbb {N}$. This monoid acts on $R$ by shifting the second variable index. We show that for every $\mathrm {Inc}(\mathbb {N})$-invariant ideal, the number of initial ideals with respect to term orders that are compatible with this $\mathrm {Inc}(\mathbb {N})$-action is finite. The article also addresses the question of how many such term orders exist. We give a complete list of the $\mathrm {Inc}(\mathbb {N})$-compatible term orders for the case $c=1$ and show that there are infinitely many for $c >1$.