Jordan types with small parts for Artinian Gorenstein algebras of codimension three

We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determined by at most three parameters.


Introduction
The Jordan type of a graded Artinian algebra A and linear form ℓ is a partition determining the Jordan block decomposition for the (nilpotent) multiplication map by ℓ on A which is denoted by P ℓ,A = P ℓ . Jordan type determines the weak and strong Lefschetz properties of Artinian algebras. A graded Artinian algebra A is said to satisfy the weak Lefschetz property (WLP) if multiplication map by a linear form on A has maximal rank in every degree. If this holds for all powers of a linear form the algebra A is said to have the strong Lefschetz property (SLP). It is known that an Artinian algebra A has the WLP if there is a linear form ℓ where the number of parts in P ℓ is equal to the Sperner number of A, the maximum value of the Hilbert function h A . Also A has the SLP if there is a linear form ℓ such that P ℓ = h ∨ A the conjugate partition of h A see [11]. Jordan type of a linear form for an Artinian algebra captures more information than the weak and Strong Lefschetz properties. Recently, there has been studies about Jordan types of Artinian algebras also in more general settings, see [9][10][11] and their references. Studying Artinian Gorenstein algebras is of great interest among the researchers in the area. Gorenstein algebras are commutative Poincaré duality algebras [14] and thus natural algebraic objects to cohomology rings of smooth complex projective varieties. There has been many studies in the Lefschetz properties and Jordan types of Artinian Gorenstein algebras [2,4,5,7,15]. Gorenstein algebras of codimension two are complete intersections and they all satisfy the SLP. The list of all possible Jordan types of linear forms, not necessarily generic linear forms, for complete intersection algebras of codimension two is provided in [1].
In this article, we study the ranks of multiplication maps by linear forms on graded Artinian Gorenstein algebras that are quotients of polynomial ring S = k[x 1 , . . . , x n ] where k is a field of characteristic zero. In Section 3, we study such algebras with arbitrary codimension in terms of their Jordan types. We present an approach to determine the Jordan types of Artinian Gorenstein algebras using Macaulay duality. We assign a natural invariant to an Artinian Gorenstein algebra A providing the ranks of multiplication maps by a linear form ℓ in different degrees, called rank matrix, M ℓ,A , Definition 3.1. There is a 1-1 correspondence between rank matrices and so called Jordan degree types in Proposition 3.12. We provide necessary conditions for a rank matrix in Lemmas 3.6 and 3.7. We use this approach in Section 4 for Artinian Gorenstein algebras in polynomial rings with three variables. We give a complete list of rank matrices that occur for some Artinian Gorenstein algebra A and linear form ℓ where ℓ 3 = 0 and ℓ 2 = 0, see Theorems 4.2 and 4.4 for algebras with even and odd socle degrees respectively. As an immediate consequence in Corollary 4.6 we list rank matrices for linear forms where ℓ 2 = 0. In Theorem 4.8 we prove that the Jordan types of Artinian Gorenstein algebras with codimension three and linear forms ℓ where ℓ 4 = 0 is uniquely determined by the ranks of at most three multiplication maps, or equivalently, three mixed Hessians.

Preliminaries
Let S = k[x 1 , . . . , x n ] be a polynomial ring equipped with standard grading over a field k of characteristic zero. Let A = S/I be a graded Artinian ( its Krull dimension is zero) algebra where I is an homogeneous ideal. The Hilbert function of a graded Artinian algebra A = S/I is a vector of non-negative integers and we denote it by h A = (1, h 1 , . . . , h d ) where h A (i) = h i = dim k (A i ). The integer d is called the socle degree of A, that is the largest integer i such that h A (i) > 0. A graded Artinian algebra A is Gorenstein if h d = 1 and its Hilbert function is symmetric, i.e. h A (i) = h A (d − i) for 0 ≤ i ≤ d.
A famous result of F. H. S. Macaulay [13] provides a bound on the growth of Hilbert functions of graded Artinian algebras. F. H. S. Macaulay characterizes all vectors of non-negative integers that occur as Hilbert functions of standard graded algebras. Such a sequence is called an O-sequence.
Let R = k[X 1 , . . . , X n ] be the Macualay dual ring of S. Given a homogeneous ideal I ⊂ S the inverse system of I is defined to be a graded S-module M ⊂ R such that S acts on R by differentiation. For more details of Macaulay's inverse system see [3] and [8]. For graded Artinian Gorenstein algebras the inverse system is generated by only one form.  [12] it is known that an Artinian standard graded k-algebra A = S/I is Gorenstein if and only if there exists F ∈ R d , such that I = Ann S (F ). T. Maeno and J. Watanabe [14] described higher Hessians of dual generator F and provided a criterion for Artinian Gorenstein algebras having the SLP or WLP. i + Ann S (F )} i be a k-basis of A j . The entries of the j-th Hessian matrix of F with respect to B j are given by We note that when j = 1 the form Hess 1 (F ) coincides with the usual Hessian. Up to a non-zero constant multiple det Hess j (F ) is independent of the basis B j . By abusing notation we will write B j = {α (j) i } i for a basis of A j . R. Gondim and G. Zappalà [5] introduced a generalization of Hessians which provides the rank of multiplication maps by powers a linear form which are not necessarily symmetric.
i } i be k-bases of A j and A k respectively. The Hessian matrix of order (j, k) of F with respect to B j and B k is The rank of the j-th catalecticant matrix of F is equal to the Hilbert function of A in degree j, see [8,Definition 1.11].
We recall the definition of the Jordan degree type for a graded Artinian algebra and linear form.
Definition 2.5. [11, Definition 2.28] Let A be a graded Artinian algebra and ℓ ∈ A 1 . Suppose that P ℓ,A = (p 1 , . . . , p t ) is the Jordan type for ℓ and A, then there exist elements z 1 , . . . z t ∈ A, which depend on ℓ, such that {ℓ i z k | 1 ≤ k ≤ t, 0 ≤ i ≤ p k − 1} is a k-basis for A. The Jordan blocks of the multiplication map by ℓ is determined by the strings s k = {z k , ℓz k , . . . , ℓ p k −1 z k }, and A is the direct sum A = s 1 ⊕ · · · ⊕ s t . Denote by d k the degree of z k . Then the Jordan degree type, is defined to be the indexed partition S ℓ,A = (p 1d 1 , . . . , p td t ).

Rank matrices for Artinian Gorenstein algebras of linear forms
Throughout this section let S = k[x 1 , . . . , x n ] be a polynomial ring with n ≥ 2 variables equipped with standard grading over a filed k of characteristic zero. We let A = S/ Ann(F ) be a graded Artinian Gorenstein algebra with dual generator F ∈ R = k[X 1 , . . . , X n ] that is a homogeneous polynomial of degree d ≥ 2.
Definition 3.1. Let A = S/ Ann(F ) be an Artinian Gorenstein algebra with socle degree d. For linear form ℓ ∈ A define the rank matrix, M ℓ,A , of A and ℓ to be the upper triangular square matrix of size d + 1 with the following i, j-th entry We note that when i = 0 the algebra A (0) coincides with A.
Remark 3.3. By the definition of higher and mixed Hessians for every 0 ≤ i < j we have that We show that for every 0 ≤ i ≤ d the vector diag(i, M ℓ,A ) is the Hilbert function of some Artinian Gorenstein algebra. We denote the Macaulay inverse system module of A = S/ Ann(F ) by F .
Proof. By the definition of rank matrix M ℓ,A we have that the entries on the i-th diagonal of M ℓ,A are exactly the ranks of multiplication map by ℓ i on A in various degrees. Using Macaulay duality for every 0 ≤ j ≤ ⌊ d−i 2 ⌋ we get the following rk ×ℓ i : Note that the socle degree of We have that h A = (1, 3, 6, 7, 6, 3, 1). Consider ℓ = x 1 , then Then the rank matrix is as follows rk Hess (2,3) x 1 = 5, and rk Hess (2,2) Te following two lemmas provide conditions on every rank matrix M ℓ,A . First we set a notation. For a vector v of positive integers of length l denote by v + the vector of length l + 1 obtained by adding zero to vector v, that is v + = (0, v).
Proof. Using Macaulay duality, for every j ≥ 1 we have We conclude that h A (i) − (h A (i+1) ) + is the Hilbert function of ( ℓ i • F / ℓ i+1 • F ), and hence it is an O-sequence.
Lemma 3.7. For every i, j ≥ 1, the following inequality holds which shows that ϕ is also injective. Using Lemma 3.6 we get that h for every i, j ≥ 1 that implies the desired inequality.
Remark 3.8. The above lemma shows that for every i, j ≥ 1 the following inequality holds rk Hess As a consequence of the above lemmas, we provide necessary conditions for an upper triangular square matrix of size d + 1 with non-negative integers to occur for an Artinian Gorenstein algebra A and linear form ℓ ∈ A 1 .
In fact, for submatrix 3 1 5 4 the condition (iii) is not satisfied.
Definition 3.11 (Jordan degree type matrix). Let A = S/ Ann(F ) be an Artinian Gorenstein algebra and ℓ ∈ A a linear form. Assume that M ℓ,A is the rank matrix of A and ℓ. We define the Jordan degree type matrix, J ℓ,A , of A and ℓ to be the upper triangular matrix with the following non-negative entries Proposition 3.12. There is a 1-1 correspondence between the two matrices M ℓ,A and J ℓ,A associated to a pair (A, ℓ).
Proof. We use Equation (3.2) to provide an algorithm to obtain J ℓ, Then define the upper triangular matrix J ℓ,A where its entry i, j for every 0 ≤ i ≤ j is equal to We obtain M ℓ,A from J ′ ℓ,A in two steps. First we get the matrix J ′ ℓ,A from J ℓ,A . For each 0 ≤ i ≤ j, we have the following We illustrate the procedure provided in Proposition 3.12 for the Artinian Gorenstein algebra given in Example 3.5 with the rank matrix M ℓ,A . Using Equations (3.4) and (3.5) we get the following matrices.
Define the decreasing sequence d : , and recall that the second difference sequence of d is denoted by ∆ 2 d and its i-th entry is given by Proposition 3.14. Let A = S/ Ann(F ) be an Artinian Gorenstein algebra with socle degree d ≥ 2 and let ℓ ∈ A be a linear form. Then the Jordan type partition of ℓ for A is given by Proof. The Jordan type partition of ℓ for A is equal to the dual partition of the following partition Since for each 0 ≤ i ≤ d the rank of the multiplication map ×ℓ i : Thus the rank of ×ℓ i : A j −→ A j+i is equal to dim k (S/ Ann(ℓ i • F )) j and therefore we have So (3.8) is equal to the following partition The dual partition to the above partition is the Jordan type partition of A and ℓ as we claimed.
Example 3.15. Consider the Artinian Gorenstein algebra given in Example 3.5 and linear form ℓ = x 1 . The Jordan degree type matrix of A and ℓ is equal to the following matrix We have that P ℓ,A = (3, . . . , 3 9 ). In order to obtain the Jordan degree type S ℓ,A we recall the Definition 2.5 and note that the degree of each part in P ℓ,A is equal to the row index of the corresponding entry in Remark 3.16. Equation (3.3) may be expressed in terms of the mixed Hessians.
This recovers a result by R. Gondim and B. Costa [2,Theorem 4.7] determining Jordan types of Artinian Gorenstein algebras and linear forms using the ranks of mixed Hessians.

Jordan types of Artinian Gorenstein algebras of codimension three
In this section we consider graded Artinian Gorenstein quotients of S = k[x, y, z] where char(k) = 0. For an Artinian Gorenstein algebra A = S/ Ann(F ) with dual generator F ∈ R = k[X, Y, Z] of degree d ≥ 2 and a linear form ℓ we explain how we find the rank matrix M ℓ,A , and as a consequence the Jordan type P ℓ,A .
Let L 1 , L 2 , L 3 be linear forms in the dual ring R = k[X, Y, Z] such that ℓ • L 1 = 0 and ℓ • L 2 = ℓ • L 3 = 0. By linear change of coordinates we may assume that L 1 = X, L 2 = Y and L 3 = Z. Then F can be written in the following form In general G d−i could be a zero polynomial for some i.

4.1.
Jordan types with parts of length at most four. We will provide the list of all possible rank matrices M ℓ,A such that A is an Artinian Gorenstein algebra and ℓ is a linear form in A where ℓ 3 = 0. Assuming ℓ 3 = 0 implies that M ℓ,A has at most three non-zero diagonals. Consequently, we provide a formula to compute the Jordan type partitions for Artinian Gorenstein algebras and linear forms ℓ such that ℓ 4 = 0, which are Jordan types with parts of length at most four.
Consider Artinian Gorenstein algebra A = S/ Ann(F ) with socle degree d ≥ 2 and linear form ℓ such that ℓ 3 = 0. Without loss of generality we assume that ℓ = x and that F is in the following form In order to make the computations simpler, we choose the coefficients of the terms in F in a way that the entries of the catalecticant matrices of F are either zero or one.
We first consider the case when ℓ 3 = 0 but ℓ 2 = 0. Therefore, we assume that We determine all rank matrices that occur for such algebras and linear forms ℓ where ℓ 3 = 0. Equivalently, we determine all possible Hilbert functions for A, A (1) and A (2) . The rank matrices are slightly different for even and odd socle degrees, as excepted, thus we treat these cases separately. We first prove our result for Artinian Gorenstein algebras with even socle degree d ≥ 2. Later, in similar cases for odd socle degrees we refer to the relevant proof given for even socle degrees.
We will show in the theorems bellow that the rank matrix, M ℓ,A , for ℓ 3 = 0 and A is determined by three of its entries. These entries are exactly the maximum values in nonzero diagonals of M ℓ,A , that are maximum values of h A , h A (1) and h A (2) . The maximum value of the Hilbert function of an Artinian Gorenstein algebra is obtained in the middle degree. We denote by r, s and t the maximum value for the Hilbert function of h A (2) , h A (1) and h A respectively. We first provide all possible triples (r, s, t).
Lemma 4.1 (Even socle degree). There exists an Artinian Gorenstein algebra A with even socle degree d ≥ 2 and linear form ℓ ∈ A 1 where ℓ 2 = 0 but ℓ 3 = 0, such that if and only if Proof. We prove the statement by analysing the catalecticant matrices in the desired degrees.
In each case we first determine all possible ranks for each catalecticant matrix and then for each possible value we provide polynomials G d−2 , G d−1 and G d as in (4.1) giving the certain ranks.
The maximum value of the Hilbert function of A occurs in degree d 2 and it is equal to rk Cat F ( d 2 ). Pick the following monomial basis for Ad Then the catalecticant matrix of F with respect to B d 2 is equal to Which is equal to Since any Artinian algebra of codimension two has the SLP the rank of the j-th Hessian matrices of polynomials G d−2 , G d−1 and G d are equal to the ranks of their j-th catalecticant matrices. By linear change of coordinates, we may assume that z is the strong Lefschetz element for Artinian Gorenstein algebra k[y, z]/ Ann(G d−2 ). This implies that the lower right square submatrices of the catalecticant matrices of G d−2 in all degrees have maximal rank. Likewise, we may assume that y is the strong Lefschetz element for the Artinian Gorenstein algebra k[y, z]/ Ann(G d−1 ) which means that the upper left square submatrices of the catalecticant matrices of G d−1 in different degrees are all full rank.
We assume that the ranks of the lower right submatrices of Cat are equal to r and setting c d−r−1 = 1 and c i = 0 for every i = d − r − 1 provides the desired property. So

Now in order to obtain possible values for
Since the socle degree of A (1) is equal to d−1 that is an odd integer, we get that For every s ∈ [2r, 2r + rk B], we have rk B = s − 2r. We may assume that the upper left submatrix of B has rank s − 2r. Setting G d−1 = 0 provides that rk B = s − 2r = 0.
For every t ∈ [2s − r, 2s − r + rk A], we have that rk A = t − 2s + r. We may assume that the rank of the upper left submatrix of A is equal to t − 2s + r. For G d = 0 we get rk A = t − 2s + r = 0. Setting a t−3r−1 = 1 and a i = 0 for every i = t − 3r − 1 provides that rk A = t − 2s + r = 0. In other words, we choose G d as the following This implies that the Hilbert function of h A (2) has the maximum possible value up to degree d 2 − 1 and since the socle degree of A (2) is even and is equal to d − 2 we have So setting c d 2 −1 = 1 and c i = 0 for every i = d 2 − 1, or equivalently,

)! provides the desired ranks for the catalecticant matrices Cat
.

Since rk Cat
the rank of the above matrix is maximum possible and is equal to d − 1. This means that for every choice of polynomial G d−1 in this case we have In order to find possbile values for h A ( d 2 ), note that the rank of Cat F ( d 2 ) is at most equal to Note that setting G d−2 as (4.7), G d−1 = 0 and G d equal to the following (4.9) 2 . provides the desired ranks for the catalecticant matrix Cat F ( d 2 ) in (4.3).
We now prove that the rank matrix of A, or equivalently, Hilbert functions of A, A (1) and A (2) are completely determined by the maximum values of h A (2) , h A (1) and h A . We then provide all rank matrices for each possible combination of integers (r, s, t) listed in Lemma 4.1.
Theorem 4.2 (Even socle degree). Let A be an Artinian Gorenstein algebra with even socle degree d ≥ 2 and ℓ ∈ A 1 such that ℓ 2 = 0 and ℓ 3 = 0. Then Hilbert functions of A, A (1) and A (2) are completely determined by (r, ). More precisely, r + 1 ≤ i ≤ d 2 , -otherwise, i.e., t > 3r we have (4.14) Proof. We first show (1). Since the Hilbert function of Artinian Gorenstein algebras are symmetric it is enough to determine it up to the middle degree. We have that So A (2) is an Artinian Gorenstein algebra with codimension at most two and the maximum value of h A (2) is equal to r. The Hilbert function of A (2) increases by exactly one until it reaches r and it stays r up to the middle degree, d 2 − 1. So we get h A (2) as we claimed. The assumption on r implies that h A (2)

We now determine the Hilbert function of A. By assumption we have (h
On the other hand, (h A − (h A (1) ) + )(1) ≤ 2 and by Lemma 3.6 we conclude that h A − (h A (1) ) + is the Hilbert function of some algebra with codimension at most two. So for every 0 • Suppose that r = t − s, then s = 2r and t = 3r. Since we have r ≤ d 2 − 1 and the Hilbert function of an algebra with codimension two is unimodal, we get and thus h A (r) ≥ r + h A (1) (r − 1) = r + 2(r − 1) + 1 = 3r − 1.
Thus we have two possible values for h A (r), that is either equal to 3r − 1 or 3r. Clearly, h A (i) = 3r for every r Since the Hilbert function of an Artinian algebra with codimension two is unimodal and t − s ≤ d 2 + 1 we have that Therefore, If s − r = t − s, assuming r = s − r implies that s = 2r and t = 3r which contradicts the assumption that r < t − s. So we have r ≤ s − r − 1. Using (4.15) we get that We conclude that h A (i) = t, for every t − s ≤ i ≤ d 2 . We now prove (2). Notice that On the other hand, for every d ≥ 6 we have that Which implies that if and only if Proof. The maximum value of the Hilbert function of A occurs in degree d−1 2 and it is equal to the rank of the following catalecticant matrix .
First assume that r ∈ [1, d−1 2 − 1] and note that the socle degree of A (2) is odd, then we have that h A (2) 2 ) = r. We may assume that the ranks of the lower right submatrices of 2 ) are equal to r. Setting c d−r−1 = 1 and c i = 0 for every i = d − r − 1, or equivalently setting G d−2 as the following provides the desired property The Hilbert function of To prove (1), we use the same argument that we used to prove Lemma 4.1 part (1). Therefore, the following choice of G d−1 and G d completes the proof of part (1).
This forces the following submatrix of Cat G d−1 ( d−1 2 − 1) to have maximal rank, that is equal to d−1 2 − r.
Setting bd−1 2 +r = 1 and b i = 0 for every i = d−1 2 + r, or equivalently, Since B and B ′ both have maximal ranks for every choice of G d , we conclude , is a square matrix of size d. On the other hand For the polynomial G d−2 as in (4.20) we get that the last column of the above matrix is zero. So setting  ( d−1 2 ) is equal to the maximum possible. So, In fact, with this choice the last column of Cat F ( d−1 2 ) becomes non-zero and linearly independent from the previous columns. Theorem 4.4 (Odd socle degree). Let A be an Artinian Gorenstein algebra with odd socle degree d ≥ 3 and ℓ ∈ A 1 such that ℓ 2 = 0 and ℓ 3 = 0. Then Hilbert functions of A, A (1) and A (2) are completely determined by (r, s, t) = (h A (2) -If t = 3r then there are two possible Hilbert functions for A Proof. We first prove part (1). The assumptions on r and s imply that Therefore, applying Theorem 4.2 part (1) for d − 1 completes the proof of (1). Now we show (2). First note that h A (2) is the same as the previous case. By the assumption we have that In both cases we get that So for every i ∈ [0, d−1 2 ] we have (h A − (h A (1) ) + )(i) = i + 1, and therefore h A (i) = i + 1 + h A (1) (i − 1) which implies the desired Hilbert function for A.
To prove (3) we get h A (2) by replacing r by d−1 2 in the previous case. By the assumption we have that To obtain h A we observe that  Proof. In Lemmas 4.1 and 4.3 we provide the complete list of possible values for the maximum of the Hilbert function of any Artinian Gorenstein algebras A where ℓ 3 = 0. In fact, for each maximum value we produce a dual generator F for A. On the other hand, in Theorems 4.2 and 4.3 we prove that for each possible maximum value the Hilbert function of A is uniquely determined by the maximum value in all the cases except when r ∈ [1, ⌊ d 2 ⌋−1] and t = 3r for every d ≥ 4, in which we have two possibilities for h A . We show that both Hilbert functions provided for h A occur for some Artinian Gorenstein algebra A.
First assume that d ≥ 6, r ∈ [2, ⌊ d 2 ⌋ − 1] and t = 3r. This implies that s = 2r. Fixing h A (r) to be either 3r − 1 or 3r we provide a degree d polynomial satisfying (4.1) as the dual generator for Artinian Gorenstein algebra A such that h A (⌊ d 2 ⌋) = 3r. Pick the following monomial basis for A r B r = {x r , x r−1 y, x r−1 z, x r−2 y 2 , . . . , y r , y r−1 , z, . . . , z r } so (4.26) Cat This is equal to (4.27) Using what we have shown in Lemmas 4.1 and 4.3 part (1), setting c d−r−1 = 1 and all other coefficients in the polynomial F to be zero, or equivalently, where the ranks of Cat F (r − 2), Cat F (r − 1) and Cat F (r), or equally, h A (2) (r − 2), h A (2) (r − 1) and h A (2) (r) are given in Theorems 4.2 and 4.4.
In order to provide a polynomial F as the dual generator of A where h A (⌊ d 2 ⌋) = 3r = h A (r) = 3r, we set c d−r−1 = a d−r = 1 and all other coefficients to be zero, so We observe that setting a d−r = 1 in the matrices (4.3) and (4.17), the number of linearly independent columns does not increase and is equal to 3r. On the other hand, setting a d−r = 1 increases the number of linearly independent columns of Cat F (r) in (4.27) by one. In fact, by setting a d−r = 1 the last column of (4.27) becomes non-zero and not included in the span of the previous columns. Thus, the number of linearly independent columns in (4.27) is equal to 3r, so h A (r) = 3r. Now assume that d ≥ 4 and r = 1. We use the same argument as the previous case for the following matrix.
As an immediate consequence of above results we get the complete list of possible rank matrices for Artinian Gorenstein algebras and linear forms such that ℓ 2 = 0. We may assume that ℓ = 0, since otherwise multiplication map by ℓ is trivial. So we have A (1) = 0 and A (i) = 0, for all i ≥ 2. We denote by r and s the maximum values for h A (1) and h A respectively. Corollary 4.6. There exists an Artinian Gorenstein algebra A with socle degree d ≥ 2 and ℓ ∈ A 1 where ℓ = 0 and ℓ 2 = 0, such that if and only if is even. Moreover, the Hilbert functions of A and A (1) are completely determined by (r, s) as the following Proof. The proof is immediate by considering rank matrices with two non-zero diagonals given by h A (1) and h A (2) provided in Theorems 4.2 and 4.4.
Remark 4.7. The above threorems provide complete lists of rank matrices, M ℓ,A , for Artinian Gorenstein algebras A of codimension two and three satisfying ℓ 3 = 0. In fact, there might exists a linear form ℓ ′ = ℓ such that ℓ ′ = 0. and t = rk Hess Then using Theorems 4.2 and 4.4, we get the ranks of multiplication maps by ℓ, ℓ 2 and ℓ 3 on A in various degrees from the rank matrix of A (1) in terms of r, s, t. Then using Proposition 3.14, we get P ℓ,A , as we claimed in (4.32). Moreover, we proved in Theorems 4.2 and 4.4 that the rank matrix of A (1) is uniquely determined in terms of r, s and t except when t = 3r. In this case, there are two possible rank matrices for A (1) that is determined uniquely in terms of r and Hess (r,d−r−1) ℓ (F ). Now suppose that ℓ 3 = 0 and ℓ 2 = 0. Ranks of multiplication maps on A by ℓ and ℓ 2 are uniquely determined by r and s in Corollary 4.6, where r = rk Hess , and s = rk Hess Therefore, Proposition 3.14 implies that P ℓ,A is equal to (4.33).
Assume that ℓ 2 = 0 and ℓ = 0 and that r = rk Hess Then, Corollary 4.6 provides the rank of multiplication map by ℓ, by providing h A (1) which implies the desired Jordan type in this case. Remark 4.9. One may use 4.2 and 4.4 to get the Jordan degree types with parts of length at most four, similar to the above theorem. Thus, such Jordan degree type is also determined uniquely by at most the ranks of three mixed Hessians.
Based on computations in Macaulay2, for large number of cases up to socle degree nine, we have no example of Artinian Gorenstein algebras over polynomial rings with three variables that the necessary conditions given in Lemmas 3.6 and 3.7 are not sufficient. So we pose the following conjecture. u v w z we have that w + v ≥ u + z.

Acknowledgment
The author would like to thank Mats Boij for many helpful discussions and Anthony Iarrobino for his comments on the first draft of this paper. Experiments using the algebra software Macaulay2 [6] were essential to get the ideas behind some of the proofs. This work was supported by the grant VR2013-4545.