\begin{document} \title{ Properties of polymatroidal ideals } \author {Amir Mafi} \address{Amir Mafi, Department of Mathematics, University of Kurdistan, P. O. Box:416, Sanandaj, Iran.} \email{a\_mafi@ipm.ir} \subjclass[2010]{13A1s, 13B30, 13C15.} \keywords{Polymatroidal ideals, normal ideals, depth and associated stability numbers.} \begin{abstract} Consider a polynomial ring $R=K[x_1,\ldots,x_n]$ over a field $K$ and let $I$ be apolymatroidal ideal of degree $d$. In this presentation, we delve into fundamental concept related to polymatroidal ideals within the realm of combinatorial combinatorial commutative algebra. Our focus includes properties of astab (associated prime stability) and dstab (depth stability) of polymatroidal ideals. additionally, we explore Cohen-Macaulay and sequentially Cohen-Macaulay matroidal ideals. In the end we give some examples such that the astab and dstab are unrelated and also we give some known problems and conjectures about this subject. For each unexplained notion or terminology, we refer the reader to \cite{HH2}, \cite{SM} . \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{} \bibitem{BH} S. Bandari and J. Herzog, {\it Monomial localizations and polymatroidal ideals}, Eur. J. Comb., {\bf 34}(2013), 752-763. \bibitem{B1} M. Brodmann, {\it Asymptotic stability of $\Ass(M/{I^nM})$}, Proc. Am. Math. Soc., {\bf 74}(1979), 16-18. \bibitem{B} M. Brodmann, {\it The asymptotic nature of the analytic spread}, Math. Proc. Cambridge Philos. Soc., {\bf 86}(1979), 35-39. \bibitem{C} H. J. Chiang-Hsieh, {\it Some arithmetic properties of matroidal ideals}, Comm. Algebra, {\bf 38}(2010), 944-952. \bibitem{CH} A. Conca and J. Herzog, {\it Castelnuovo-Mumford regularity of products of ideals}, Collect. Math., {\bf 54}(2003), 137-152. \bibitem{GS} D. R. Grayson and M. E. Stillman, {\it Macaulay 2, a software system for research in algebraic geometry}, Available at {http://www.math.uiuc.edu/Macaulay2/}. \bibitem{HH} J. Herzog and T. Hibi, {\it Discrete polymatroids}, J. Algebraic Combin., {\bf 16}(2002), 239-268. \bibitem{HH1} J. Herzog and T. Hibi, {\it Cohen-Macaulay polymatroidal ideals}, Eur. J. Comb., {\bf 27}(2006), 513-517. \bibitem{HH2} J. Herzog and T. Hibi, {\it Monomial ideals}, GTM., vol.260, Springer, Berlin, (2011). \bibitem{HM} J. Herzog and A. Mafi, {\it Stability properties of powers of ideals in regular local rings of small dimension}, Pacific J. Math., {\bf 295}(2018), 31-41. \bibitem{HQ} J. Herzog and A. Qureshi, {\it Persistence and stability properties of powers of ideals}, J. Pure and Applied Algebra, {\bf 219}(2015), 530-542. \bibitem{HRV} J. Herzog, A. Rauf and M. Vladoiu, {\it The stable set of associated prime ideals of a polymatroidal ideal}, J. Algebraic Combin., {\bf 37}(2013), 289-312. \bibitem{HV} J. Herzog and M. Vladoiu, {\it Squarefree monomial ideals with constant depth function}, J. Pure and Appl. Algebra, {\bf 217}(2013), 1764-1772. \bibitem{HT} L. T. Hoa, N. D. Tam, {\it On some invariants of a mixed product of ideals}, Arch. Math., {\bf 94}(2010), 327-337. \bibitem{KM} Sh. Karimi and A. Mafi, {\it On stability properties of powers of polymatroidal ideals}, Collect. Math.,{\bf 70}(2019), 357-365. \bibitem{MN} A. Mafi and D. Naderi, {\it A note on stability properties of powers of polymatroidal ideals}, Bull. Iranian. Math. Soc., {\bf 48}(2022), 3937-3945. \bibitem{MS} A. Mafi and H. Saremi, {\it Strong persistence and associated primes of powers of monomial ideals}, Comm. Algebra, {\bf 51}(2023), 859-863. \bibitem{SM} H. Saremi and A. Mafi, {\it Unmixedness and arithmetic properties of matroidal ideals}, Arch. Math., {\bf 114}(2020), 299-304. \bibitem{T} T. N. Trung, {\it Stability of associated primes of integral closures of monomial ideals}, J. Comb. Theory Ser. A {\bf 116}(2009), 44-54 \bibitem{SM} B. Sturmfels and E. Miller, {\it Combinatorial Commutative Algebra}, Graduate Texts in Mathematics, Springer, (2005). \end{thebibliography} \end{document}
