On minimal Gorenstein Hilbert functions

We conjecture that a class of Artinian Gorenstein Hilbert algebras called full Perazzo algebras always have minimal Hilbert function, fixing codimension and length. We prove the conjecture in length four and five, in low codimension. We also prove the conjecture for a particular subclass of algebras that occurs in every length and certain codimensions. As a consequence of our methods we give a new proof of part of a known result about the asymptotic behavior of the minimum entry of a Gorenstein Hilbert function.


Introduction
Gorenstein algebras appear as cohomology rings in several categories.For instance, real orientable manifolds, projective varieties, Kahler manifolds, convex polytopes, matroids, Coxeter groups and tropical varieties are examples of categories for which the ring of cohomology is an Artinian Gorenstein K-algebra.The fundamental point is that these algebras can be characterized as algebras satisfying Poincaré duality, see [18].
We deal with standard graded Artinian Gorenstein K-algebras over a field of characteristic zero.A natural and classical problem consists in understanding their possible Hilbert function, sometimes also called Hilbert vector.When the codimension of the algebra is less than or equal to 3, all possible Hilbert vectors were characterized in [21]; in particular, they are unimodal, i.e. they never strictly increase after a strict decrease.While it is known that non unimodal Gorenstein h-vectors exist in every codimension greater than or equal to 5 (see [4,5,6]), it is open whether non unimodal Gorenstein h-vectors of codimension 4 exist.For algebras with codimension 4 having small initial degree the Hilbert vector is unimodal (see [20,17]).
Consider the family AG K (r, d) of standard graded Artinian Gorenstein K-algebras of socle degree d and codimension r.By Poncaré duality, the Hilbert function of A ∈ AG K (r, d) is a symmetric vector Hilb(A) = (1, r, h 2 , . . ., h d−2 , r, 1), that is h k = h d−k .There is a natural partial order in this family given by: (1, r, h 2 , . . ., h d−2 , r, 1) (1, r, h2 , . . ., hd−2 , r, 1), if h i ≤ hi , for all i ∈ {2, . . ., d − 2}.The maximal Hilbert functions are associated to compressed algebras and completely described in [13].In fact the Hilbert vector of a compressed Gorenstein algebra is a maximum in AG K (r, d).On the other hand, classifying minimal Hilbert functions is a hard problem.We do not know in general if there is a minimum.Moreover, given two comparable Gorenstein Hilbert functions, it is not true that any symmetric vector between them is Gorenstein.Some partial results in this direction were obtained in [23] and called the interval conjecture.
The first example of a non-unimodal Gorenstein h-vector was given by Stanley (see [21,Example 4.3]).He showed that the h-vector (1,13,12,13,1) is indeed a Gorenstein h-vector.In [14] the authors showed that Stanley's example is optimal, i.e. if we consider the h−vector (1,12,11,12,1), it is not Gorenstein.We say that a vector is totally non unimodal if A totally non-unimodal Gorenstein Hilbert vector exists for every socle degree d ≥ 4 when the codimension r is large enough.It is related to a conjecture posed by Stanley and proved in [15,16] and also a consequence of our Proposition 2.3, see Corollary 2.4.
From Macaulay-Matlis duality, every standard graded Artinian Gorenstein Kalgebra can be presented by a quotient of a ring of differential operators by a homogeneous ideal that is the annihilator of a single form in the dual ring of polynomials.full Perazzo algebras are associated with full Perazzo polynomials, they are the family that we will study in detail.Perazzo polynomials are related to Gordan and Noether theory of forms with vanishing Hessian (see [19,Chapter 7] and [10]).In [10] the author introduced the terminology Perazzo algebras to denote the Artinian Gorenstein algebra associated to a Perazzo polynomial.In [9, ?] the authors study the Hilbert vector and the Lefschetz properties for Perazzo algebras in codimension 5.In [8] the authors study full Perazzo algebras focusing on socle degree 4, showing that they have minimal Hilbert vector in some cases.In this paper we deal with codimension greater than 13 and we are more interested in full Perazzo algebras.In the case of socle degree 4 we recall the known results.
We now describe the contents of the paper in more detail.In the first section we recall the basics on Macaulay-Matlis duality, see 1.1.In the next subsection we recall the classical bounds for Hilbert functions given by Macaulay, Gotzman and Green summarized in 1.4.
In the second section we recall the definition of full Perazzo algebras and we pose the full Perazzo Conjecture (see Conjecture 2.6).A full Perazzo polynomial of type m and degree d is a bigraded polynomial of bidegree (1, d − 1) given by f = x j M j where {M j |j = 1, . . ., m+d−2 d−1 } is a basis for K[u 1 , . . ., u m ] (d−1) .The associated Artinian Gorenstein algebra is called full Perazzo algebra (see Section 2 for more details).
Conjecture.Let H be the Hilbert vector of a full Perazzo algebra of type m ≥ 3 and socle degree d ≥ 4 and let r = r(m, d) its codimension.Then H is minimal in the family of Hilbert vectors of Artinian Gorenstein algebras of codimension r and socle degree d, that is, if Ĥ is a comparable Artinian Gorenstein Hilbert vector such that Ĥ H, then Ĥ = H.
In the third section we prove special cases of the Conjecture in socle degree 4 and we try to fill the gaps in order to classify all possible Hilbert functions up to codimension 25 (see Theorem 3.6, Corollary 3.7 and Proposition 3.8).In socle degree 5 we prove the Conjecture for m ∈ {3, 4, 5, 6, 7, 8, 9, 10} (see Theorem 3.15) and a stronger version of the conjecture for m = 3 (see Corollary 3.16).
In the fourth section we prove our main result that the full Perazzo Conjecture is true for arbitrary socle degree d ≥ 4 and type m = 3.
Theorem.Every full Perazzo algebra with socle degree d ≥ 4 of type m = 3 has minimal Hilbert function.
In the last section we give a new proof of part of a result originally proved in [16], concerning the asymptotic behavior of the minimum entry of a Gorenstein Hilbert function (see Theorem 5.2).

Preliminaries
Most of the background material presented here can be found in [8].
1.1.Macaulay-Matlis duality.In this section we recall some basic results from Macaulay-Matlis duality for Artinian Gorenstein algebras over a field K of characteristic zero.We recall that in characteristic zero we can use a differential version of Macaulay-Matlis duality.
Let A = K[X 1 , . . ., X n ]/I = d i=0 A i be a standard graded Artinian K−algebra with A d = 0.The Hilbert function of A can be described by the vector Hilb(A) = (1, h 1 , . . ., h d ) where h i = dim A i .We say that A is Gorenstein if dim A d = 1 and for every i = 1, . . ., d, the natural pairing given by multiplication In this context, d is the socle degree of the algebra and assuming I 1 = 0, n is the codimension of A.
Let us regard the polynomial algebra R = K[x 1 , . . ., x n ] as a module over the algebra More generally, given any Q submodule M of R we define the ideal of Q: On the other side we have the notion of inverse system.Given I ⊂ Q be an ideal, we define the inverse system I −1 which is a Q submodule of R: By Macaulay-Matlis duality we have a bijection: From the Theory of Inverse Systems, we get the following characterization of standard Artinian Gorenstein graded K-algebras.A proof of this result can be found in [18,Theorem 2.1].
Theorem 1.1.(Double annihilator Theorem of Macaulay) Let R = K[x 1 , . . ., x n ] and let Q = K[X 1 , . . ., X n ] be the ring of differential operators.Let A = d i=0 A i = Q/I be an Artinian standard graded K-algebra.Then A is Gorenstein if and only if there exists f ∈ R d such that A ≃ Q/ Ann(f ).
In the sequel we always assume that char(K) = 0, A = Q/I, I = Ann Q (f ) and I 1 = 0.
We want to stress that the bijection given by Macaulay-Matlis duality preserves bigrading, that is, there is a bijection: and codimension r = m + n if we assume, without loss of generality, that I 1 = 0. Remark 1.2.With the previous notation, all bihomogeneous polynomials of bidegree (1, d − 1) can be written in the form and we assume that I 1 = 0, so codim A = m+n.
1.2.Classical Bounds of Hilbert function.We recall some classical bounds for the growth of the Hilbert function of Artinian K−algebras.The three main results are due to Macaulay, Gotzmann and Green; before stating them, we need to recall the following definition: An expansion of type (1) always exists and is unique (see, e.g., [7,Lemma 4.2.6]).Following [7], we define for any integers a and b, where we set s c = 0 whenever s < c or c < 0.  Recall that when A is Artinian and Gorenstein, then its Hilbert function is a finite, symmetric O-sequence.

Minimal Gorenstein Hilbert functions
We recall the construction of full Perazzo algebras, introduced in [8].
Definition 2.1.Let K[x 1 , . . ., x n , u 1 , . . ., u m ] be the polynomial ring in the n variables x 1 , . . ., x n and in the m variables u In this case f is called full Perazzo polynomial of type m and degree d.The associated algebra is a full Perazzo algebra of socle degree d and codimension m + τ m .
Proposition 2.3.Let A be a full Perazzo algebra of type m ≥ 2 and socle degree d.Then for k = 0, . . ., In particular, its Hilbert function is totally non-unimodal for r >> 0.
Proof.Using the bigrading of A and considering that the polynomial f has degree 1 in the variables x 1 , . . ., x τm , fixed k = 0, . . ., ⌊ d 2 ⌋, we have the following decomposition: To verify that the Hilbert vector is asymptotically totally non unimodal it is enough to see that as a function of Corollary 2.4.For every d ≥ 4 there is a positive integer r 0 such that for all r ≥ r 0 there is an Artinian Gorenstein algebra with socle degree d and codimension r having a totally non unimodal Hilbert vector.
Proof.Let m be large enough in order to guarantee that the Hilbert vector of the full Perazzo algebra A = Q/ Ann(f ), of type m and socle degree d has a totally non unimodal Hilbert vector.For every r > m ] and consider the algebra It is easy to see that the Hilbert vector of A ′ is given by h ′ k = h k + s for k = 0, d, therefore, it is totally non-unimodal and the result follows.
Let d ≥ 4, r ≥ 3. Consider the family AG(r, d) of standard graded artinian Gorenstein K-algebras of socle degree d and codimension r.In this section we will consider K, a fixed field of characteristic 0. We know that the Hilbert function of A ∈ AG(r, d) is a symmetric vector Hilb(A) = (1, r, h 2 , . . ., h d−2 , r, 1), with h i = h d−i by Poincaré duality.
Consider the family of length d symmetric vectors of type (1, r, h 2 , . . ., h d−2 , r, 1), where h i = h d−i .There is a natural partial order in this family If h i ≤ hi , for all i ∈ {2, . . ., d − 2}.This order can be restricted to AG(r, d) which becomes a poset.Definition 2.5.Let r, d be fixed positive integers and let H be a length d + 1 symmetric vector (1, r, h 2 , . . ., h d−2 , r, 1).We say that H is a minimal Artinian Gorenstein Hilbert function of socle degree d and codimension r if there is an Artinian Gorenstein algebra such that Hilb(A) = H and H is minimal in AG(r, d) with respect to .To be precise, if Ĥ is a comparable Artinian Gorenstein Hilbert vector such that Ĥ H, then Ĥ = H.
We now present the full Perazzo Conjecture.
Conjecture 2.6.Let H be the Hilbert vector of a full Perazzo algebra of type m and socle degree d.Then H is minimal in AG(r, d).

Minimal Gorenstein Hilbert functions in low socle degree
In this section we study Gorenstein Hilbert functions of algebras with socle degree 4 and 5. Part of the results in socle degree 4 can be found in [8].
Proposition 3.4.Let m ≥ 3. We have that Proof.For r = m + m+2 3 there exists the full Perazzo Algebra.It realizes the Gorenstein sequence .
Proof.Let A be a Gorenstein algebra with Hilbert function (1, r, h, r, 1) and let L be a general linear form.Using the same argument as in Proposition 3.1 in [14], we get that the Hilbert function of A/(L) is of the type By the theorems of Green and of Macaulay we have s ≤ (r (3) ) −1 0 and Proof.By Theorem 3.6 and by Theorem 2.5 in [14], (1, 25, 21, 25, 1) is a Gorenstein sequence, so δ(25) ≥ 4. We have to prove that (1, 25, 20, 25, 1) is not a Gorenstein sequence.Indeed 25 (3) = 6  3 + 3 2 + 2 1 , so (25 (3) ) −1 0 = 12.Since u = 5 we have that Proof.For r = 26, we have to prove that (1, 26, 21, 26, 1) is not a Gorenstein sequence.Indeed, let A = R/I be a Gorenstein algebra with Hilbert function (1,26,21,26,1), L be a general linear form.We set J = (I ≤3 ), J = (J, L)/(L) and S = R/(L).By the theorems of Green and of Macaulay and repeating the above method, R/ J has Hilbert function (1,25,8,13).As R/ J has maximal growth from degree 2 to degree 3 and J has no new generators in degree 4, by Gotzmann's theorem we get h R/ J (t) = t+2 2 + t.Therefore, J is the saturated ideal, in all degrees ≥ 2 of the union of a plane and a line in P 24 .It follows that, up to saturation, J is the ideal of a scheme T given by the union in P 25 of a 3−dimensional linear variety, a plane and m points (possibly embedded).Hence, 50 ≤ h R/ J (4) ≤ 45, which is absurd.Now, for r = 27, following the same argument as above, we prove that the sequence (1, 27, 22, 27, 1) is not Gorenstein.In this case we conclude 50 ≤ h R/ J (4) ≤ 46.
The case m = 3 will be dealt with in general in the next section.We can assume m ≥ 4.

A family of minimal Gorenstein Hilbert functions
Proof.First of all, consider d > 2k − 1.In this case, Now there are only two other cases to consider: 1) They are similar, we will do the details for d = 2k.In this case, we have: Indeed we know that the k-binomial expansion of k+1 2 has two blocks The first block consists of binomials of type s+1 s and the second one of type s s .Therefore: If k − j + 1 > k − 2, then j ≤ 2, but the cases j = 1 and j = 2 are not possible.In fact, suppose, j = 2, since: It is absurd for k > 2. The case j = 1 is analogous, the result follows.Proof.We want to show that the Hilbert vector of the full Perazzo algebra is a minimal Gorenstein Hilbert vector.Let Ĥ = (1, ĥ1 , ĥ2 , ĥ3 , . . ., ĥd−1 , 1) be a comparable Artinian Gorenstein Hilbert vector Ĥ H of length d + 1 and ĥ1 = h 1 .We will proceed in steps to show that Ĥ = H.Consider, on the contrary, one of the following situations: (1) For some k ∈ {2, . . ., ⌊d/2⌋ − 1}, ĥk < h k ; (2) For d = 2q, suppose that ĥt = h t for all t < q and ĥq < h q ; (3) For d = 2q + 1, suppose that ĥt = h t for all t < q and ĥq < h q .
We will show that all of these situations give rise to a contradiction.(1).Let A = Q/I with I = Ann(f ) be a standard graded Artinian Gorenstein K-algebra such that for some k ∈ {2, . . ., ⌊d/2⌋ − 1}.Suppose that k is minimal satisfying this property, that is, for t < k we get ĥt = h t , by the comparability hypothesis.Let L ∈ A 1 be a generic linear form and let S = Q/(L).We get the following exact sequence: . By Green's theorem we have By Lemma 4.1, we have We consider only the case h ′ d−k+1 = d, the other cases are similar.We have We recall that . Case d = 2q is even.Suppose that ĥt = h t for all t < q and ĥq < h q .Let L ∈ Q be a generic linear form and S = Q/(L).We have the following exact sequence: is also Gorenstein.In the middle we get the following diagram: 2 ) q ) −1 0 ≤ q + 2 + q − 2 = 2q.We study the case h ′ q+1 = 2q, the other cases are similar.We have a It is a contradiction.
(3).If d = 2q + 1 is odd.Suppose that ĥt = h t for all t < q and ĥq < h q .By the same argument: We consider only the case h ′ q+2 = 2q + 1.We have We have where the terms q−1 q−1 + • • • + 3 3 are q − 3.By Macaulay's theorem we have , the last terms are q − 3.
By Macaulay's theorem we have It is a contradiction.The result follows.

Asymptotic behavior of the minimum
In this section we give a new proof of part of Theorem 3.6 in [16].Let P m = m + m+d−2 d−1 be the codimension of a full Perazzo algebra of type m.Denote by µ d,k (r) the minimal entry in degree k of a Gorenstein h-vector with codimension r and socle degree d.Since in both sides the limit exists and are the same, the result follows.

Theorem 1 . 4 .
Let A = R/I be a standard graded K-algebra, and L ∈ A a general linear form (according to the Zariski topology).Denote by h d the degree d entry of the Hilbert function of A and by h ′ d the degree d entry of the Hilbert function of A/(L).Then: (Macaulay):
Consider the family of full Perazzo algebras of type m = 3 and socle degree d ≥ 4. Its Hilbert function is given by h k = k+2 k + 2+2q−k 2q−k , for k ≤ ⌊d/2⌋ and by symmetry we get h d−k = h k .Lemma 4.1.Let k ≤ ⌊d/2⌋.Then we have:

Theorem 4 . 2 .
Every full Perazzo algebra with socle degree d ≥ 4 of type m = 3 has minimal Hilbert function.
n, linearly independent and algebraically dependent polynomials in the variables u 1 , . . ., u m .The associated algebra is called a Perazzo algebra, it has codimension m + n and socle degree d.