arXiv subjects
Publications and source records attributed to Maorong Ge.
It is well known that Thom-Boardman symbols are realized by non-increasing sequences of nonnegative integers. A natural question is whether the converse is also true. In this paper we answer this question affirmatively, that is, for any non-increasing sequence of nonnegative integers, there is a map-germ with the prescribed sequence as its Thom-Boardman symbol.
In [7] we obtained a formula for the Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring by relating it to the Hilbert depth of powers of the irrelevant maximal ideal. In this paper, we prove that these two Hilbert depth formulas are equivalent to each other. Our result reveals that there is a strong connection between these two classes of seemingly unrelated monomial ideals. We conjecture that their Stanley depths are equivalent as well.
In this paper, we obtain explicit formulas for the Hilbert series and Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring.
In this paper, we partially confirm a conjecture, proposed by Cimpoeaş, Keller, Shen, Streib and Young, on the Stanley depth of squarefree Veronese ideals $I_{n,d}$. This conjecture suggests that, for positive integers $1 \le d \le n$, $\sdepth (I_{n,d})= \lfloor \binom{n}{d+1}/\binom{n}{d} \rfloor+d$. Herzog, Vladoiu and Zheng established a connection between the Stanley depths of quotients of monomial ideals and interval partitions of certain associated posets. Based on this connection, Keller, Shen, Streib and Young recently developed a useful combinatorial tool to analyze the interval partitions of the posets associated with the squarefree Veronese ideals. We modify their ideas and prove that if $1 \le d \le n \le (d+1) \lfloor \frac{1+\sqrt{5+4d}}{2}\rfloor+2d$, then $\sdepth (I_{n,d})= \lfloor \binom{n}{d+1}/\binom{n}{d} \rfloor+d$. We also obtain $ \lfloor \frac{d+\sqrt{d^2+4(n+1)}}{2} \rfloor \le \sdepth(I_{n,d}) \le \lfloor \binom{n}{d+1}/\binom{n}{d} \rfloor+d$ for $n > (d+1) \lfloor \frac{1+\sqrt{5+4d}}{2}\rfloor+2d$. As a byproduct of our construction, We give an alternative proof of Theorem $1.1 $ in $[13]$ without graph theory.