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Analysis reveals the index of ass-stability is consistently 1, while v-stability varies widely.
In this paper, we compare the index of ass-stability astab(I) and the index of v-stability vstab(I) of powers of a graded ideal I. We prove that astab(I)=1(I) for any graded ideal I in a 2-dimensional polynomial ring, and that vstab(I) can be any positive integer in this situation. Moreover, given any integers a,b≥1, we construct a graded ideal I in a $3(a+1)$-dimensional polynomial ring such that (astab(I),vstab(I))=(a,b).
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Ficarra et al. (2025) studied this question.
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