Abstract
Let I be a matroidal ideal of degree d of a polynomial ring \(R=K[x_1,\ldots ,x_n]\), where K is a field. Let \({\text {astab}}(I)\) and \({\text {dstab}}(I)\) be the smallest integers m and n, for which \({\text {Ass}}(I^m)\) and \({\text {depth}}(I^n)\) stabilize, respectively. In this paper, we show that \({\text {astab}}(I)=1\) if and only if \({\text {dstab}}(I)=1\). Moreover, we prove that if \(d=3\), then \({\text {astab}}(I)={\text {dstab}}(I)\). Furthermore, we show that if I is an almost square-free Veronese type ideal of degree d, then \({\text {astab}}(I)={\text {dstab}}(I)=\lceil \frac{n-1}{n-d}\rceil \).
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Acknowledgements
We would like to deeply grateful to the referee for the careful reading of the manuscript and the helpful suggestions. The second author has been supported financially by Vice-Chancellorship of Research and Technology, University of Kurdistan under research Project No. 99/11/19299.
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Communicated by Mohammad Reza Koushesh.
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Mafi, A., Naderi, D. A Note on Stability Properties of Powers of Polymatroidal Ideals. Bull. Iran. Math. Soc. 48, 3937–3945 (2022). https://doi.org/10.1007/s41980-022-00721-z
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DOI: https://doi.org/10.1007/s41980-022-00721-z