Resumen de On stability properties of powers of polymatroidal ideals

Shokoufe Karimi, Amir Mafi

• Let R=K[x_1,\ldots ,x_n] be the polynomial ring in n variables over a field K with the maximal ideal \mathfrak {m}=(x_1,\ldots ,x_n). Let {\text {astab}}(I) and {\text {dstab}}(I) be the smallest integer n for which {\text {Ass}}(I^n) and {\text {depth}}(I^n) stabilize, respectively. In this paper we show that {\text {astab}}(I)={\text {dstab}}(I) in the following cases:

  (i) I is a matroidal ideal and n\le 5.

  (ii) I is a polymatroidal ideal, n=4 and \mathfrak {m}\notin {\text {Ass}}^{\infty }(I), where {\text {Ass}}^{\infty }(I) is the stable set of associated prime ideals of I.

  (iii) I is a polymatroidal ideal of degree 2.

  Moreover, we give an example of a polymatroidal ideal for which {\text {astab}}(I)\ne {\text {dstab}}(I). This is a counterexample to the conjecture of Herzog and Qureshi, according to which these two numbers are the same for polymatroidal ideals.