Lexsegment ideals of Hilbert depth 1

Abstract

Let I ⊂ S = K[x 1 , . . . , xn] be a lexsegment ideal, generated by monomials of degree d. The main aim of this paper is to characterize when the Hilbert depth of I will be 1, in the standard graded case. In addition to this, we will give an estimate of depth of squarefree monomial ideals, generalizing a result of Popescu [Pop12]. We will also show that Stanley conjecture holds for squarefree stable ideals, in the multigraded case.

How to compute the Hilbert depth of a graded ideal

“…where I n,d is the squarefree Veronese ideal. As pointed out in section 4 of [Sh2], these two results are equivalent, since H m d (t) = (1 − t) d−1 H I n+d−1,d (t). By comparing Hilbert depth and Stanley depth, it is natural to ask if sdepth(m d ) = ⌈ n d+1 ⌉ holds for d ≥ 2.…”

“…Except some special cases, this conjecture remains open. (For details, see the introduction in [Sh2]. )…”

“…Note that for simplicity, hdepth(I) in this paper is the same as Hdepth 1 (I) in [BKU1] and hdepth 1 (I) in [Sh2]. Also, in the rest of this paper, by a graded ideal, we mean a Z-graded ideal.…”

“…In [Sh2], some conjectures were made about the Hilbert depth and Stanley depth of a lex ideal generated by monomials of the same degree. In Section 3 of this paper we will give some counterexamples to these conjectures.…”

“…The author was originally interested in finding some counterexamples to the conjectures in [Sh2] and in using the Hilbert depth to help study the Stanley depth of squarefree Veronese ideals. After computing many examples, Algorithm 2.16 was first developed.…”