Log-concavity of level Hilbert functions and pure O-sequences

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arXiv:2210.09447v2 [math.AC] 7 Sep 2023

LOG-CONCAVITY OF LEVEL HILBERT FUNCTIONS

AND PURE O-SEQUENCES

FABRIZIO ZANELLO

Abstract. We investigate log-concavity in the context of level Hilbert functions and pure

O-sequences, two classes of numerical sequences introduced by Stanley in the late Seventies

whose structural properties have since been the object of a remarkable amount of interest

in combinatorial commutative algebra. However, a systematic study of the log-concavity of

these sequences began only recently, thanks to a paper by Iarrobino.

The goal of this note is to address two general questions left open by Iarrobino’s work:

1) Given the integer pair (r, t), are all level Hilbert functions of codimension rand type t

log-concave? 2) How about pure O-sequences with the same parameters?

Iarrobino’s main results consisted of a positive answer to 1) for r= 2 and any t, and for

(r, t) = (3,1). Further, he proved that the answer to 1) is negative for (r, t) = (4,1).

Our chief contribution to 1) is to provide a negative answer in all remaining cases, with

the exception of (r, t) = (3,2), which is still open in any characteristic. We then propose a

few detailed conjectures speciﬁcally on level Hilbert functions of codimension 3 and type 2.

As for question 2), we show that the answer is positive for all pairs (r, 1); negative for

(r, t) = (3,4); and negative for any pair (r, t) with r≥4 and 2 ≤t≤r+ 1. Interestingly,

the main case that remains open is again (r, t) = (3,2). Further, we conjecture that, in

analogy with the behavior of arbitrary level Hilbert functions, log-concavity fails for pure

O-sequences of any codimension r≥3 and type tlarge enough.

1. Introduction and main definitions

This note is inspired by the recent paper [20], where Iarrobino began investigating the

log-concavity of the Hilbert functions of Gorenstein and level artinian graded algebras, in

response to a question posed by Chris McDaniel. Surprisingly, despite the substantial amount

of literature devoted to understanding the algebraic, combinatorial, and geometric properties

of these Hilbert functions, before [20] little attention was paid to log-concavity, an important

numerical property with applications to several mathematical ﬁelds (see, e.g., the classical

surveys [10, 32]). Recall that a sequence (an)n∈Nis said to be log-concave if ai−1ai+1 ≤a2

for all indices i.

2020 Mathematics Subject Classiﬁcation. Primary: 13D40; Secondary: 05E40, 13E10, 13H10.

Key words and phrases. Hilbert function; pure O-sequence; log-concavity; unimodality; level algebra;

Gorenstein algebra; Interval Conjecture.

2 FABRIZIO ZANELLO

Iarrobino’s main results are that log-concavity holds for all Gorenstein Hilbert functions

of codimension 3, while it fails in codimension 4 (see below, as well as [20], for the relevant

terminology). Also, log-concavity holds for all level Hilbert functions of codimension 2.

Thus, two general and natural questions that arise from [20] are: 1) Given the integer pair

(r, t), are all level Hilbert functions of codimension rand type tlog-concave? 2) How about

pure O-sequences (i.e., monomial level Hilbert functions) with the same parameters?

The goal of this note is to make progress on both questions. In Section 2, we investigate

part 1). We provide a full characterization of the pairs (r, t) for which the answer is positive,

with the sole exception of (r, t) = (3,2), which remains open in any characteristic. Note that

this is also one of the main unsolved cases when it comes to unimodality. We then propose

and discuss in detail ﬁve open problems speciﬁcally on level Hilbert functions of codimension

3 and type 2, which we hope will be the object of future research in this area.

In Section 3, we address part 2). Our main results are a positive answer for all pairs (r, 1);

negative for (r, t) = (3,4); and negative for all pairs (r, t) such that r≥4 and 2 ≤t≤r+ 1.

In all instances where log-concavity fails, it will be possible to exhibit some elegant, inﬁnite

families of pure O-sequences with the relevant parameters. Also interesting, the main case

left open is again (r, t) = (3,2). Further, we conjecture that, in analogy with the situation

in the arbitrary level case, log-concavity fails for pure O-sequences of any given codimension

r≥3 and type tlarge enough.

We now brieﬂy recall the main deﬁnitions. We refer to [20] and to standard texts such

as [33] (for level Hilbert functions) and [5] (for pure O-sequences) for any basic facts or

unexplained terminology.

We consider standard graded artinian algebras A=R/I, where R=k[x1,...,xr] is a poly-

nomial ring over an inﬁnite ﬁeld k(a priori, we make no assumptions on the characteristic

of k), I⊂Ris a homogeneous ideal, and all xihave degree one. Assuming without loss of

generality that Icontains no nonzero linear forms, we deﬁne ras the codimension of A.

Recall that the Hilbert function of Ais given by hi= dimkAi, for all i≥0. It is a standard

fact that Ais artinian if and only if its Hilbert function is zero in all large degrees. Hence,

we may identify the Hilbert function of Awith its h-vector h(A) = (1, h1,...,he), where e

is the largest index such that he>0, and is called the socle degree of A(or of h(A)).

The socle of Ais the annihilator of the maximal homogeneous ideal (x1, . . . , xr). We deﬁne

the socle-vector of Aas s= (0, s1,...,se), where siis the dimension over kof the socle in

degree i. We then say that Ais level if s= (0,...,0, t), i.e., if its socle is concentrated in

the last degree. The integer t=se=heis called the Cohen-Macaulay type (or simply type)

of A. Finally, Ais Gorenstein if Ais level of type t= 1.

LOG-CONCAVITY OF LEVEL HILBERT FUNCTIONS AND PURE O-SEQUENCES 3

We say that a Hilbert function h= (1, h1=r, h2,...,he=t) is level (of codimension r

and type t) if there exists a level algebra Awith h(A) = h.

Apure O-sequence is the Hilbert function of a monomial level algebra. Equivalently, in

combinatorial terms, h= (1, h1=r, h2,...,he=t) is a pure O-sequence (of codimension r

and type t) if there exist t(monic) monomials of degree ein rvariables having a total of hi

monomial divisors in degree i, for all i.

For instance, (1,3,4,2) is a pure O-sequence of codimension 3, type 2, and socle degree 3,

generated by the monomials xyz and xz2. Indeed, these two monomials have a total of four

monomial divisors in degree 2 (xy, xz, yz, z2), and three in degree 1 (x, y, z).

Level Hilbert functions and pure O-sequences — as well as much of the combinatorial

commutative algebra that we know today — ﬁnd their inception in Richard Stanley’s seminal

work of the late Seventies [30, 31]. These two topics have since constituted an unusually

active and successful research area of commutative algebra, to which this author also devoted

at least a decade of his mathematical career. We refer to [28] and its references for a

nonexhaustive list of contributions to the theory of level Hilbert functions since Richard’s

original manuscripts, and to the AMS Memoir [5] (and references thereof) speciﬁcally for

pure O-sequences. We hope that this brief paper, including the open problems we present

in the next two sections, can stimulate further work in this area.

2. Log-concavity of arbitrary level Hilbert functions

The goal of this section is to characterize the pairs (r, t) such that all level Hilbert functions

of codimension rand type tare log-concave. We will achieve this with the sole exception of

(3,2), which remains open in any characteristic.

We ﬁrst need to review (Macaulay’s) inverse systems, also known as Matlis duality. For two

good, in-depth introductions to this theory, we refer the reader to [13, 21]. For simplicity’s

sake, in the brief description that follows we assume that the characteristic of the base ﬁeld

kis zero (or larger than the socle degree of the algebra), but the relevant parts of the theory,

including Lemma 1 and Proposition 2 below, can be made characteristic-free using so-called

divided powers in lieu of diﬀerentials.

Given a polynomial ring R=k[x1,...,xr], consider the graded R-module S=k[y1,...,yr],

where xiacts on Sby partial diﬀerentiation with respect to yi. This establishes a bijective

correspondence between artinian algebras R/I and ﬁnitely generated R-submodules Mof

S, where I= Ann(M)⊂Ris the annihilator of M, and M=I−1is the submodule of S

annihilated by I(see [13], Remark 1, p. 17).

If R/I has socle-vector s= (0, s1,...,se), then Mis (minimally) generated by sihomoge-

neous forms of degree i, for all i= 1,...,e. Further, the Hilbert function of R/I is given by

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arXiv:2210.09447v2[math.AC]7Sep2023LOG-CONCAVITYOFLEVELHILBERTFUNCTIONSANDPUREO-SEQUENCESFABRIZIOZANELLOAbstract.Weinvestigatelog-concavityinthecontextoflevelHilbertfunctionsandpureO-sequences,twoclassesofnumericalsequencesintroducedbyStanleyinthelateSeventieswhosestructuralpropertieshavesincebeenth...

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