A Note on Stability Properties of Powers of Polymatroidal Ideals

Let I be a matroidal ideal of degree d of a polynomial ring R = K [ x 1 , … , x n ] , where K is a field. Let astab ( I ) and dstab ( I ) be the smallest integers m and n , for which Ass ( I m ) and depth ( I n ) stabilize, respectively. In this paper, we show that astab ( I ) = 1 if and only if dst...

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Bibliographic Details
Published in:Bulletin of the Iranian Mathematical Society 2022-12, Vol.48 (6), p.3937-3945
Main Authors: Mafi, Amir, Naderi, Dler
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:Let I be a matroidal ideal of degree d of a polynomial ring R = K [ x 1 , … , x n ] , where K is a field. Let astab ( I ) and dstab ( I ) be the smallest integers m and n , for which Ass ( I m ) and depth ( I n ) stabilize, respectively. In this paper, we show that astab ( I ) = 1 if and only if dstab ( I ) = 1 . Moreover, we prove that if d = 3 , then astab ( I ) = dstab ( I ) . Furthermore, we show that if I is an almost square-free Veronese type ideal of degree d , then astab ( I ) = dstab ( I ) = ⌈ n - 1 n - d ⌉ .
ISSN:1017-060X
1735-8515
DOI:10.1007/s41980-022-00721-z