WARING DECOMPOSITIONS OF SPECIAL TERNARY FORMS WITH DIFFERENT HILBERT FUNCTIONS

ELENA ANGELINI, LUCA CHIANTINI, AND ALESSANDRO ONETO

Abstract. We prove the existence of ternary forms admitting apolar sets of points of cardinality equal to the Waring rank, but having different Hilbert function and different regularity. This is done exploiting liaison theory and Cayley-Bacharach properties for sets of points in the projective plane.

1. Introduction

1.1. Waring decompositions. Let S = ⨁_{d∈ℤ} S_{d} = ℂ[x_{0}, …, x_{n}] be the standard graded polynomial ring in n variables with complex coefficients.

Definition 1.1 (Waring rank). Given a degree-d homogeneous polynomial f ∈ S_{d}, a Waring decomposition of f is an expression as sum of d-th powers of linear forms, i.e.,

(1.1)  f = ∑_{i=1}^{r} ℓ_{i}^{d}, ℓ_{i} ∈ S_{1}.

The minimal length r of such a decomposition is the Waring rank, denoted rk(f).

The study of Waring decompositions has its roots in the XIX century thanks to the works of Sylvester [40, 41], Clebsch [18], Richmond [36], Palatini [31, 32], Terracini [42], among many others. The question regained a lot of interest in the last decades. The celebrated theorem by Alexander and Hirschowitz in 1995 established the general Waring rank in any number of variables and degrees [1]. The attention to Waring decompositions has been motivated also by its connection to tensor decompositions; in this case, symmetric tensors. Indeed, tensor decompositions find a lot of uses in several applications, see for example [26, 27]. The computation of most tensor decompositions is known to be an NP-hard problem and the same is conjectured fo symmetric tensors [24]. Therefore, a theoretical study of related algebraic and geometric structures can be useful to shed lights on them.

1.2. Apolar sets of points. Waring decompositions have a geometric interpretation in terms of Veronese varieties. Consider the Veronese embedding given by

ν_{d} : ℙS_{1} ⟶ ℙS_{d}, [ℓ] ↦ [ℓ^{d}].

Definition 1.2 (Apolar sets of points). Given f ∈ S_{d} and a set of points A ⊂ ℙS_{1}, we say that A is apolar to f if [f] ∈ ⟨ν_{d}(A)⟩. If there is no subset A′ ⊂ A such that [f] ∈ ⟨ν_{d}(A′)⟩ we say that A is non-redundant. If the cardinality of A equals the Waring rank of f we say that A computes (the rank of) f.

2020 Mathematics Subject Classification. Primary: 14C20, 14N07; Secondary: 13A02, 13C40, 14N05, 15A69.

The authors are members of the Italian GNSAGA-INDAM.

Any Waring decomposition of a polynomial f ∈ S_{d} has a geometric counterpart given by a set of points A ⊂ ℙS_{1} apolar to f, i.e., the equation (1.1) is equivalent to say that [f] ∈ ⟨ν_{d}(A)⟩ for A = {[ℓ_{1}], …, [ℓ_{r}]} ⊂ ℙS_{1}. This geometric interpretation allows us to define a notion of rank with respect to any algebraic variety X ⊂ ℙ^{N}. Namely, the rank of any p ∈ ℙ^{N} with respect to X is the minimal cardinality of a set of points A ⊂ X such that p ∈ ⟨A⟩.

A more algebraic way to relate Waring decompositions with sets of points is through apolarity theory. By Macaulay [28] (see also [25, Lemma 2.14]), there is a one-to-one correspondence between artinian Gorenstein graded algebras with socle degree d and homogeneous polynomials of degree d. Let R = ⨁_{d∈ℤ} R_{d} = ℂ[y_{0}, …, y_{n}] be a second polynomial ring and consider the apolar action of R over S by partial derivatives, i.e., for any g ∈ R_{e}, f ∈ S_{d}, we define g ∘ f := g(∂_{0}, …, ∂_{n})f ∈ S_{d−e}. Given f ∈ S_{d}, the apolar ideal of f is the annihilator of f with respect to the apolar action, i.e.,

Ann(f) := {g ∈ R : g ∘ f = 0} ⊂ R.

Macaulay’s duality affirms that an artinian algebra A = R/J is Gorenstein of socle degree d if and only if J = Ann(f) for some f ∈ R_{d}. The Apolarity Lemma, see [25, Lemma 1.15], affirms that Waring decompositions of a polynomial f ∈ S_{d} correspond to sets of points A ⊂ ℙS_{1} such that I(A) ⊂ Ann(f), i.e., the equation (1.1) is equivalent to say that I(A) ⊂ Ann(f) for A = {[ℓ_{1}], …, [ℓ_{r}]} ⊂ ℙS_{1}. In other words, the study of Waring decompositions is equivalent to study ideals of sets of points in ℙS_{1}, i.e., radical homogeneous ideals of height one, contained in artinian Gorenstein ideals.

1.3. Hilbert function of apolar sets of points. Most of algebraic invariants of algebraic varieties are encoded by the Hilbert function. Given a homogeneous ideal I ⊂ R, let I_{d} = I ∩ R_{d} be its degree-d homogeneous part. The Hilbert function of R/I is the numerical function h_{R/I} : ℤ → ℤ with

(1.2)  h_{R/I}(d) := dim_{ℂ}[R_{d}/I_{d}] = dim_{ℂ} R_{d} − dim_{ℂ} I_{d}

If A ⊂ ℙS_{1} is a set of points, then we write h_{A} for the Hilbert function of R/I(A) where I(A) is the defining homogeneous ideal of A. The study of Hilbert functions of reduced sets of points has been successfully used to compute Waring ranks. For example, the Hilbert function of the apolar ideal of a given polynomial provides a lower bound for the Waring rank of the polynomial. In some cases, these were shown to be optimal, see e.g. [10, 9].

If a polynomial admits a unique minimal Waring decomposition, then we say that the polynomial is identifiable. It is worth to mention that the general homogeneous polynomial with rank r strictly smaller than the general one is identifiable except for the special cases (n, d, r) ∈ {(2, 6, 9), (3, 4, 8), (5, 3, 9)}, see [17]. In [5, 3, 4], Cayley-Bacharach properties and liaison theory of sets of points in the projective plane were exploited to provide a criterion to certify that a form is identifiable. On the other hand, when the polynomial is not identifiable, it is interesting to study the family of apolar sets of points computing the rank of the polynomial, see [35]. In [30], a classification of low rank polynomials is given in terms of the possible Hilbert functions of apolar sets of points computing the rank of the polynomials.

It is well-known that the Hilbert function of a set of points is strictly increasing until it reaches its cardinality. We call regularity of A the smallest integer i for which the Hilbert function of A stabilizes, i.e., such that h_{A}(i) = h_{A}(i + 1).

1.4. The question. In this short note, we address the following question.

Question 1.3 ([30, Question 2.11]). Let f ∈ S_{d}. Do all sets of points computing the rank of f share the same regularity? More generally, do they share the same Hilbert function?

The answer is known to be positive for binary forms (n = 2), because all sets of points with same cardinality on the projective line share the same Hilbert function, and for monomials x_{0}^{d_{0}} ⋯ x_{n}^{d_{n}} because all sets of points computing the rank are known to be complete intersections with generators of degrees d_{1} + 1, …, d_{n} + 1 if d_{0} ≤ d_{i} for all i, see [8]. In the present note, we prove that the answer to Question 1.3 is negative for ternary forms (n = 3) if the degree is sufficiently large. We provide two cases. In Section 3.1, we consider degree-10 ternary forms having rank 22, that is the general rank. Among them, we consider the special ones which admit a unique quintic in the apolar ideal. We show in Theorem 3.1 that the general such form has a decomposition lying on the quintic and one not contained in the quintic. We recall that the study of the connection between catalecticant matrices and the Hilbert function of certain decompositions was also used in [16] where a complete stratification of ternary 6-ics of rank nine is given.

Along with the proof, we notice that there exists degree-10 ternary forms of rank 22 having a unique minimal (irreducible and smooth) quintic in the apolar ideal whose minimal decompositions do not lie on such quintic, see Remark 3.7. Also among the well-known case of monomials we can find examples in which generators of minimal degree of the apolar ideal cannot be used to define a minimal decomposition, however, in that case, the minimal generators are not square-free and the fact that they have to be discarted is trivial. Our example underlines that the naïve approach of looking for minimal decompositions over reduced hypersurfaces of minimal degree among the ones in the apolar ideal might fail.

In Section 3.2, we consider degree-13 ternary forms having rank 30, that is strictly subgeneric. In Theorem 3.9, we show that there exist two sets of 30 points which compute the rank of a degree-13 ternary form and have different regularity. Our proofs follow the ideas of [5, 3, 4], by exploiting properties of Hilbert functions of points in the projective plane and, in particular, liaison theory. In Section 4, we show an explicit construction with the support of the algebra software Macaulay2 [23]. In the present work, we do not investigate the minimality of these examples because we believe that it would require a thorough analysis of several possible cases. In particular, there might be examples in smaller degrees. Note that up to degree 6 there are no examples thanks to [16], but degrees between 7 and 9 might be worth to be analyzed in future works.

2. Algebraic properties of points in ℙ^{n}

2.1. Hilbert function of points in ℙ^{n}. Let A ⊂ ℙ^{n} be a set of points of cardinality ℓ(A). We recalled in (1.2) the definition of the Hilbert function of A. The first difference is the numerical function given by

Dh_{A} : ℤ → ℤ, Dh_{A}(i) := h_{A}(i) − h_{A}(i − 1).

We recall some elementary properties of Hilbert functions of points in ℙ^{n} that will be used later. We refer to [14] for an extensive exposition including proofs.

Lemma 2.1. [14, Lemma 6.1, Proposition 6.3, Proposition 6.4] Let A ⊂ ℙ^{n} be a set of points of cardinality ℓ(A).

(1) h_{A}(j) ≤ ℓ(A), for all j;

(2) Dh_{A}(j) = 0, for j < 0;

(3) h_{A}(0) = Dh_{A}(0) = 1;

(4) Dh_{A}(j) ≥ 0, for all j;

(5) h_{A}(i) = ∑_{j=0}^{i} Dh_{A}(j);

(6) h_{A}(j) = ℓ(A), for all j ≫ 0;

(7) Dh_{A}(j) = 0, for j ≫ 0;

(8) ∑_{j=0}^{∞} Dh_{A}(j) = ℓ(A);

(9) for any A′ ⊂ A, h_{A′}(j) ≤ h_{A}(j) and Dh_{A′}(j) ≤ Dh_{A}(j) for any j;

(10) if Dh_{A}(j) ≤ j, then Dh_{A}(j) ≥ Dh_{A}(j + 1).

Finally, we denote

h_{A}^{1}(d) := ℓ(A) − h_{A}(d) = ∑_{i=d+1}^{∞} Dh_{A}(i).

The following proposition allows us to understand if, given two sets of points, there exists a homogeneous polynomial of given degree to which they are both apolar.

Proposition 2.2. [3, Proposition 2.19] Let A, B ⊂ ℙ^{n} be finite sets of points and let Z = A ∪ B. Then, for any d ∈ ℕ,

dim (⟨ν_{d}(A)⟩ ∩ ⟨ν_{d}(B)⟩) = ℓ(A ∩ B) − 1 + h_{Z}^{1}(d).

The latter proposition, together with a result by Davis [19], allows to show the following result that will be used to compute the rank of the forms constructed in the examples.

Proposition 2.3 ([3, Proposition 2.20]). Let f be a ternary form of degree d and let A be a non-redundant finite set computing the rank of f. Then, there is no other B ⊂ ℙ^{2} non-redundant finite set computing the rank of f with A ∩ B = ∅, ℓ(B) ≤ ℓ(A) and such that, if Z = A ∪ B:

(1) Dh_{Z}(j) = j + 1 for j ∈ {0, …, i − 1} and Dh_{Z}(i) ≤ i;

(2) Dh_{Z}(j_{0}) = Dh_{Z}(j_{0} + 1) = e < i for some j_{0} > i − 1.

2.2. Liaison for points in ℙ^{2}. We describe here some construction that will be used in Section 3 to construct the examples replying to Question 1.3. These are basic constructions from liaison theory; we refer to [34, 29] for a complete presentation of the theory.

Definition 2.4. Let A, B ⊂ ℙ^{2} be two sets of points. We say that A and B are linked if there exists a complete intersection Z such that I_{B} = I_{Z} : I_{A}. When A ∩ B = ∅, this is equivalent to say that Z = A ∪ B. We say that B is the residue of A with respect to Z.

A set of points in ℙ^{2} is arithmetically Cohen-Macaulay of codimension 2, hence a minimal free resolution is provided by the Hilbert-Burch Theorem, see [22, Section 3A]. If A and B are linked, then a Hilbert-Burch matrix of B can be recovered from one of A by the mapping cone construction, see [29, Proposition 5.2.10].

Let S(i) be the polynomial ring with the gradation shifted by i, i.e., S(d)_{i} := S_{d+i}. A similar notation is used for every graded S-module. If A and B are linked through a complete intersection Z of type (d_{1}, d_{2}), then we have the following free resolutions

0 → S(−d_{1} − d_{2}) → S(−d_{1}) ⊕ S(−d_{2}) → I_{Z} → 0
↓  ↓ ϕ  ↓
0 ⟶  F_{1}  ⟶^{M}  F_{0}  ⟶ I_{A} ⟶ 0

where ϕ : S(−d_{1}) ⊕ S(−d_{2}) → F_{0} is the map induced by the right-most vertical arrow which corresponds to the inclusion I_{Z} ⊂ I_{A} and M is a Hilbert-Burch matrix of A. Then, by the mapping cone construction, a resolution of I_{B} is given by

0 → F_{0}(−d_{1} − d_{2})^{∨} ⟶^{(ϕ|M)^{t}}  \begin{matrix}S(−d_{1}) ⊕ S(−d_{2}) \\ ⊕ \\ F_{1}(−d_{1} − d_{2})^{∨} \end{matrix} → I_{B} → 0.

The Hilbert functions between the two linked sets of points are linked by the following formula on their first differences:

(2.1)  Dh_{B}(i) + Dh_{A}(d_{1} + d_{2} − i − 2) = Dh_{Z}(i),

see [20, Theorem 3] or [29, Corollary 5.2.19].

2.3. Cayley-Bacharach properties. We recall here also some basic fact about Cayley-Bacharach properties for sets of points in projective space. They are used to understand the possible Hilbert functions of sets of points A ∪ B where A and B are apolar to the same homogeneous form. These properties have been crucial to provide certificates for identifiability of forms in [5, 3, 4].

Definition 2.5. Let Z ⊂ ℙ^{n} be a set of points. We say that it satisfies the Cayley-Bacharach property in degree d (CB(d)) if, for all p ∈ Z, it holds that every form of degree d vanishing at Z ∖ {p} also vanishes at p.

Proposition 2.6. [3, Proposition 2.25] Let f ∈ S_{d} and let A be a non-redundant set of points apolar to f. Let B a second non-redundant set of points apolar to f such that A ∩ B = ∅. Then, the set Z = A ∪ B satisfies CB(d).

Theorem 2.7. [5, Theorem 4.9] Let Z ⊂ ℙ^{n} be a set of points satisfying CB(d). Then, for any i ∈ {0, …, d + 1},

∑_{j=0}^{i} Dh_{Z}(j) ≤ ∑_{j=0}^{i} Dh_{Z}(d + 1 − j).

2.4. Secant varieties and Terracini loci. The geometric framework to deal with Waring decompositions is the one of secant varieties of Veronese varieties, see [11, 7]. Given an irreducible algebraic variety X ⊂ ℙ^{N}, the r-th secant variety is an irreducible variety defined as the Zariski closure of the union of linear spaces spanned by r-tuples of points on X, i.e.,

σ_{r}(X) := \overline{⋃_{p_{1},...,p_{r}∈X}⟨p_{1}, …, p_{r}⟩} ⊂ ℙ^{N}.

If X is the d-th Veronese embedding of ℙS_{1}, then, by Definition 1.1,

σ_{r}(ν_{d}(ℙ^{n})) = \overline{\{[f] ∈ ℙS_{d} : rk(f) ≤ r\}} ⊂ ℙS_{d}.

Secant varieties are clearly nested into each other by definition, but, if the variety X is non-degenerate, then we have that its secant varieties eventually fill the ambient space, see [11, Exercise 2.6]. For example, the minimal r such that σ_{r}(ν_{d}(ℙ^{n})) = ℙS_{d} is the Waring rank of the general degree-d form in n + 1 variables.

Understanding which secant variety fills the ambient space reduces to the computation of dimensions of secant varieties. By a simple parameter count, the expected dimension of the r-th secant variety of an algebraic variety X ⊂ ℙ^{N} is

exp. dim σ_{r}(X) := min{r dim(X) + r − 1, N}.

The expected dimension is always an upper bound for the actual dimension and if dim σ_{r}(X) < exp. dim σ_{r}(X) then we say that X is r-defective. The classification of defective varieties is a very classical problem with roots in the late XIX century. By Palatini [33], if σ_{r}(X) is not an hypersurface and does not fill the ambient space, then it is at least of codimension 2 inside σ_{r+1}(X). As an immediate corollary, projective curves are never defective. A classification of defective surfaces is due to Severi and Terracini, see [13, Theorem 14 and Theorem 15], while the classification of defective threefolds is due to Scorza [37]. The case of fourfolds has been treated first by Scorza in [38] (see also [39]) and then completed in the recent preprint [15].

If we denote by X^{(r)} the r-th symmetric product of the variety X, the r-th abstract secant variety of X is

abσ_{r}(X) := \overline{\{((p_{1}, …, p_{r}), q) : q ∈ ⟨p_{1}, …, p_{r}⟩\}} ⊂ X^{(r)} × ℙ^{N}.

Throughout the paper, we always denote by π_{i} the projection onto the i-th factor from a cartesian product. The r-th abstract secant variety is irreducible of dimension r dim(X) + r − 1. Hence, if σ_{r}(X) ⊊ ℙ^{N}, non-defectiveness is equivalent to say that the projection π_{2} : abσ_{r}(X) → ℙ^{N} has generically finite fibers. The r-th Terracini locus of X is

𝕋_{r}(X) := \overline{\{(p_{1}, …, p_{r}) : \begin{matrix}p_{i}∈X^{smooth},\{p_{1},...,p_{r}\} are linearly independent, \\ T_{p_{1}} (X),...,T_{pr} (X) are linearly dependent \end{matrix}\}} ⊂ X^{(r)}.

The main tool to study dimensions of secant varieties is the Terracini’s Lemma which describes the general tangent space to secant varieties, see [7, Lemma 1]. In particular, it claims that, given general points p_{1}, …, p_{r} ∈ X and a general point p ∈ ⟨p_{1}, …, p_{r}⟩, then

T_{p}σ_{r}(X) = ⟨T_{p_{1}}(X), …, T_{p_{r}}(X)⟩.

If X is not r-defective and σ_{r}(X) does not fill the space, then 𝕋_{r}(X) is a proper subvariety of X^{(r)}. In other words, the preimage of the Terracini’s locus π_{1}^{−1}(𝕋_{r}(X)) is contained in the locus where the differential of the projection π_{2} : abσ_{r}(X) → ℙ^{N} drops rank.

The complete classification of defective Veronese varieties is known as the Alexander-Hirschowitz Theorem [1]. In the 1980s, Alexander and Hirschowitz defined a very powerful machinery to deal with problems on polynomial interpolation such as the computation of the dimension of the linear system of degree-d hypersurfaces of ℙ^{n} having r singularities at general points. The latter problem is equivalent to compute dimensions of secant varieties of the Veronese varieties and, a posteriori, to compute the general Waring rank in S_{d}.

Theorem 2.8 (Alexander-Hirschowitz Theorem, [1]). The r-th secant variety of the Veronese variety ν_{d}(ℙ^{n}) is non-defective except for • d = 2, n ≥ 2 and 2 ≤ r ≤ n;

• d = 3, n = 4, r = 7;

• d = 4, (n, r) ∈ {(2, 5), (3, 9), (4, 14)}.

3. The examples

In this section, we give a negative answer to Question 1.3 by showing that there exist ternary forms admitting apolar sets of points computing their Waring rank with different Hilbert function as well as different regularity.

The idea is to start from a special set of points Z_{1} and, by means of liaison theory, to construct a second set of points of the same cardinality Z_{2}, but different Hilbert function, such that ⟨ν_{d}(Z_{1})⟩∩⟨ν_{d}(Z_{2})⟩ is non-empty, i.e., there is a degree-d form f whose Z_{1} and Z_{2} are both apolar to. After the construction, we prove that indeed Z_{1} and Z_{2} compute the rank of f.

3.1. First example. Degree-10 ternary forms of rank 22.

Theorem 3.1. There exists a form f of degree 10 and of rank 22 having two sets of points computing its rank and presenting different Hilbert functions.

In order to help the reader to follow the several steps of the proof, before getting into the proof, we summarize the idea of how we will proceed.

(1) Starting from a set Z_{1} of 22 general points in projective plane lying on a quintic curve, we construct a new set of points Z_{2} of 22 points by means of liaison theory such that ⟨ν_{10}(Z_{1})⟩ ∩ ⟨ν_{10}(Z_{2})⟩ ≠ ∅, i.e., there exists a form f which both Z_{1} and Z_{2} are apolar to, see Claim 3.2. Then, we have to show that the form f has indeed rank 22.

(2) We denote by X the variety of degree-10 forms which admit an apolar set of 22 points lying on a quintic curve and observe that the general element of X has rank 22, see Claim 3.3 and Claim 3.4. Our construction produces forms that are clearly in X: we need to show that these are general enough to have rank 22.

(3) We consider the variety Y of degree-10 forms obtained from our construction. In Claim 3.5 and Claim 3.6, we prove that Y is dense in X through a dimension count. This is enough to conclude that the general element of Y has rank 22.

Proof of Theorem 3.1. Let Z_{1} ⊂ ℙ^{2} be a set of 22 general points on a quintic curve. Then,

(3.1)  \begin{matrix}h_{Z_{1}} : & 1 & 3 & 6 & 10 & 15 & 20 & 22 & 22 & ⋯ \\ Dh_{Z_{1}} : & 1 & 2 & 3 & 4 & 5 & 5 & 2 & − \end{matrix}

Let L ⊂ ℙ^{2} be a general line and let Y be a set of five general points on L. Consider A = Z_{1} ∪ Y. Clearly there are no quintics through A since there is a unique quintic through Z_{1} and the line L is chosen generically. Moreover, Y imposes independent conditions on the space of 6-ics through Z_{1}: indeed, it is enough to observe that dim I(Z_{1})_{6} − dim I(Z_{1})_{5} = 5 ≥ ℓ(Y), see e.g. [12, Lemma 1.9-(i)]. Hence,

(3.2)  \begin{matrix}h_{A} : & 1 & 3 & 6 & 10 & 15 & 21 & 27 & 27 & ⋯ \\ Dh_{A} : & 1 & 2 & 3 & 4 & 5 & 6 & 6 & − \end{matrix}

Let X be a complete intersection of two general 7-ics containing A. Then, we define Z_{2} to be the residue of A through X. Hence, by (2.1), we have

(3.3)  \begin{aligned}& & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ Dh_{X}(i) & : & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\ Dh_{Z_{2}}(i) & : & 1 & 2 & 3 & 4 & 5 & 6 & 1 & − & − & − \\ Dh_{A}(12 − i) & : & & & & & & & 6 & 6 & 5 & 4 & 3 & 2 & 1 \end{aligned}

Note that, since A is regular in degree 6, it has generators of degree at most 7. Then, by the Peskine-Szpiro Theorem, see [29, Theorem 6.4.4], Z_{2} is smooth and Z_{1} ∩ Z_{2} = ∅. Hence, Z_{2} is a set of 22 distinct points disjoint from Z_{1} and with Hilbert function different than Z_{1}.

Claim 3.2. Z_{1} and Z_{2} are apolar to a common degree-10 ternary form.

Proof. By Proposition 2.2, it is enough to compute the Hilbert function of the union Z_{1} ∪ Z_{2}. Note that, by construction, Z_{1} ∪ Z_{2} is linked to the set Y of five collinear points. Hence, by (2.1)

0 1 2 3 4 5 6 7 8 9 10 11 12
Dh_{X}(i)  : 1 2 3 4 5 6 7 6 5 4 3 2 1
Dh_{Z_{1}∪Z_{2}}(i) : 1 2 3 4 5 6 7 6 4 3 2 1
Dh_{Y} (12 − i) :  1 1 1 1 1

Hence, by Proposition 2.2, dim⟨ν_{10}(Z_{1})⟩ ∩ ⟨ν_{10}(Z_{2})⟩ = 0. I.e., there exists f ∈ S_{10} such that [f] ∈ ⟨ν_{10}(Z_{1})⟩ ∩ ⟨ν_{10}(Z_{2})⟩. □

Now, we need to show that the general degree-10 form f constructed as in Claim 3.2 is indeed of rank 22. Let X be the variety of degree-10 forms admitting an apolar set of 22 reduced points lying on a quintic.

Claim 3.3. The variety X is irreducible and has codimension 2.

Proof. Note that the variety of degree-10 ternary forms f ∈ S_{10} admitting a quintic in the apolar ideal is a hypersurface defined by the vanishing of the determinant of the central catalecticant matrix associated to the map

cat_{5}(f) : R_{5} → S_{5}, g ↦ g ∘ f.

The fact that the variety {det(cat_{5}(f)) = 0} is irreducible follows from [21]. Fixed an irreducible plane quintic C ⊂ ℙ^{2}, by Riemann-Roch Theorem, we have that ⟨ν_{10}(C)⟩ = ℙ^{44}. Since curves are non-defective, dim σ_{22}ν_{10}(C) = 43, see Section 2.4. Therefore the general form f ∈ ⟨ν_{10}(C)⟩ has rank 23 with respect to ν_{10}(C) and X has codimension one in the hypersurface of forms admitting a quintic in the apolar ideal. In other words, if we consider the incidence variety of forms lying on the 22-nd secant variety of the degree-10 Veronese embedding of a plane quintic

I = {(C, [f]) : [f] ∈ σ_{22}(ν_{10}(C))} ⊂ ℙS_{5} × ℙS_{10},

then X = π_{2}(I). The projection π_{2} is generically one-to-one because the kernel of cat_{5}(f) for a general form [f] ∈ X is one-dimensional and, generically, it is also generated by an irreducible quintic (since it is a open, this can be checked by a direct computation). Given a general irreducible quintic C, we have that π_{1}^{−1}(C) ≅ σ_{22}(ν_{10}(C)). Hence, dim X = dim I = dim ℙS_{5} + dim σ_{22}(ν_{10}(C)) = 20 + 43 = 63 inside ℙS_{10} = ℙ^{65}. □

Claim 3.4. The general form [f] ∈ X has rank 22.

Proof. By Alexander-Hirschowitz Theorem (Theorem 2.8),

dim σ_{21}(ν_{10}(ℙ^{2})) = 21 · 2 + 20 = 62,

i.e., σ_{21}(ν_{10}(ℙ^{2})) has codimension 3. Hence, by Claim 3.3, X ∩ σ_{21}(ν_{10}(ℙ^{2})) is a proper subvariety of X. □

Consider Y ⊂ X the set of degree-10 forms that are constructed as in Claim 3.2. Let Hilb_{22}^{∘}(ℙ^{2}) be the open subset of the Hilbert scheme of 22 points in ℙ^{2} and let Hilb_{22,5}^{∘}(ℙ^{2}) denote the set of all sets of 22 general points lying on a quintic of ℙ^{2}. Then, consider the incidence variety

J = {(Z_{1}, Z_{2}, [f]) : ⟨ν_{10}(Z_{1})⟩∩⟨ν_{10}(Z_{2})⟩ = [f]} ⊂ Hilb_{22,5}^{∘}(ℙ^{2})×Hilb_{22}^{∘}(ℙ^{2})×ℙS_{10}.

By construction, Y = π_{3}(J). Moreover, from the construction described at the beginning of the proof, the projection π_{1} is dominant, i.e., for a general choice of a set Z_{1} of 22 points on a quintic in ℙ^{2}, there exists Z_{2} ∈ Hilb_{22}^{∘}(ℙ^{2}) and f ∈ S_{10} such that (Z_{1}, Z_{2}, [f]) ∈ J. With a similar construction, we can also show that the projection π_{2} is dominant.

Claim 3.5. The projection π_{2} : J → Hilb_{22}^{∘}(ℙ^{2}) is dominant.

Proof. Let Z_{2} be a general set of 22 points in ℙ^{2}. Take a set Y of three general points collinear on a general line L and let A = Z_{2} ∪ Y. Since there are no quintics through Z_{2}, then the three points impose independent conditions on 6-ics through Z_{2}, see [12, Lemma 1.9]. Hence,

\begin{matrix}h_{A} : & 1 & 3 & 6 & 10 & 15 & 21 & 25 & 25 & ⋯ \\ Dh_{A} : & 1 & 2 & 3 & 4 & 5 & 6 & 4 & − \end{matrix}

Let {G_{1}, G_{2}, G_{3}} be a basis for the space of 6-ics passing through A, i.e., [I_{A}]_{6} = ⟨G_{1}, G_{2}, G_{3}⟩. The restriction of [I_{A}]_{6} on L defines a non-complete linear series g_{3}^{2} away from the set of points Y. The linear series on L is generically base-point free, then it defines a regular map on ℙ^{2} which maps L into a singular cubic curve C. Indeed, since being base-point free is an open condition, this can be checked directly on a specific example with the support of the algebra software Macaulay2 [23], see Section 4.1. For example, without loss of genericity, let L = {x_{0} = 0} and Y = {(0 : 1 : 0), (0 : 0 : 1), (0 : 1 : 1)}. Hence, for i = 1, 2, 3, the restriction of G_{i} on L is given by G_{i}(0, x_{1}, x_{2}) = x_{1}x_{2}(x_{1} − x_{2})H_{i}(x_{1}, x_{2}) and we consider the map φ : L → ℙ^{2}, p ↦ (H_{1}(p) : H_{2}(p) : H_{3}(p)). Let C = φ(L). The general cubic C constructed as above is nodal. Indeed, since being nodal is an open condition on the space of rational plane cubics, it is enough to exhibit one example in which the above construction provides a nodal curve. This is checked with the support of the algebra software Macaulay2 [23], see Section 4.1.

Now, let X be the set of two distinct points on L which are pre-images of the singular point of C and consider the set of five points X ∪ Y ⊂ L. By construction, X ∪Y does not impose independent conditions on the space of 6-ics passing through A. This implies that

\begin{aligned}h_{X∪A} & : & 1 & 3 & 6 & 10 & 15 & 21 & 26 & 27 & 27 & ⋯ \\ Dh_{X∪A} & : & 1 & 2 & 3 & 4 & 5 & 6 & 5 & 1 & − \end{aligned}

Now, we consider the linkage to X ∪ A via a general complete intersection of type (7, 7), i.e., U = (X ∪ A) ∪ Z_{1}. Then, Z_{1} is a set of 22 points with the following Hilbert function

(3.4) \begin{aligned}& & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ Dh_{U}(i) & : & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\ Dh_{Z_{1}}(i) & : & 1 & 2 & 3 & 4 & 5 & 5 & 2 & − & − & − \\ Dh_{X∪A}(12 − i) & : & & & & & & 1 & 5 & 6 & 5 & 4 & 3 & 2 & 1 \end{aligned}

Note that X ∪ A has regularity 7, but, by our explicit computations in Section 4.1, we also know that it is generated in degree 7. Therefore, again by Peskine-Szpiro Theorem as before, see [29, Theorem 6.4.4], we deduce that Z_{1} is smooth and Z_{1} ∩Z_{2} = ∅. Hence, Z_{1} ∈ Hilb_{22,5}^{∘}(ℙ^{2}). Moreover, by construction, U = (Z_{1} ∪Z_{2})∪ (X ∪ Y). I.e., the union Z_{1} ∪ Z_{2} is linked through U to the set X ∪ Y of 5 collinear points. Hence, by (2.1),

0 1 2 3 4 5 6 7 8 9 10 11 12
Dh_{U}(i)  : 1 2 3 4 5 6 7 6 5 4 3 2 1
Dh_{X∪Y} (12 − i) :  1 1 1 1 1
Dh_{Z_{1}∪Z_{2}}(i) : 1 2 3 4 5 6 7 6 4 3 2 1

Hence, Dh_{Z_{1}∪Z_{2}}(11) = 1 and, by Proposition 2.2, there exists f ∈ S_{10} such that (Z_{1}, Z_{2}, [f]) ∈ J. □

Claim 3.6. Y is dense in X. In particular, the general [f] ∈ Y has rank 22.

Proof. Since the 22-nd secant variety of ν_{10}(ℙ^{2}) and of ν_{10}(C), for any quintic C ⊂ ℙ^{2}, are non-defective, we have that the Terracini loci in Hilb_{22,5}^{∘}(ℙ^{2}) and Hilb_{22}^{∘}(ℙ^{2}) are proper sub-varieties. In particular, we can restrict the projection π_{3} to the dense subspace of J given by the triplets (Z_{1}, Z_{2}, [f]) such that Z_{1} and Z_{2} do not belong to the corresponding Terracini locus. This implies that, for the general [f] ∈ Y, the fiber π_{3}^{−1}([f]) is finite, see [6, Proposition 6.2]. In particular, dim Y = dim J.

Since π_{2} is dominant, for a general Z_{2} ∈ Hilb_{22}^{∘}(ℙ^{2}), we have that dim J = dim Hilb_{22}^{∘}(ℙ^{2}) + dim π_{2}^{−1}(Z_{2}). Now, dim Hilb_{22}^{∘}(ℙ^{2}) = 44. Following the construction in Claim 3.5, we see that dim π_{2}^{−1}(Z_{2}) = 19. Indeed, it depends on the choice of three collinear points, that is five parameters (2 to fix a line and 3 to fix three points on it), and the choice of the general complete intersection of two 7-ics, that is the choice of a pencil in the 8-dimensional projective space of 7-ics through the set of 27 points A, i.e., other 14 = 2(9 − 2) parameters. In particular, dim J = 63 = dim X. Since X is irreducible, we conclude the first part of the statement. The second part of the statement follows directly from Claim 3.4. □

Therefore, the general element of Y evinces the statement of the theorem. □

Remark 3.7. In the proof of Claim 3.3 we observed that, given a general quintic curve C ⊂ ℙ^{2}, the general degree-10 form in ⟨ν_{10}(C)⟩ = ℙ^{44} has rank 23 with respect to the curve. However, since the general rank of degree-10 ternary forms is equal to 22 by Alexander-Hirschowitz Theorem (Theorem 2.8), we also deduce that the general form in ⟨ν_{10}(C)⟩ has rank 22 with respect to the Veronese variety ν_{10}(ℙ^{2}). In other words, there exist degree-10 ternary forms whose apolar ideal has minimal generators in degree five but minimal Waring decompositions are given by sets of points whose ideal does not involve such minimal generators.

Remark 3.8. For sake of completeness, we observe that, in the example constructed in Theorem 3.1, the space of decompositions is discrete. This has been checked employing Bertini’s Theorem and numerical computations, see [2] for a reference about the methodology used. In particular, a degeneration argument could not be used to prove an analogous statement.

3.2. Second example. Degree-13 ternary forms of rank 30.

Theorem 3.9. There exists a form f of degree 13 and of rank 30 having two sets of points computing its rank and admitting different Hilbert functions and regularities.

Proof. Let A be a set of 12 general points lying on a cubic. I.e.,

\begin{matrix}h_{A} : & 1 & 3 & 6 & 9 & 12 & 12 & ⋯ \\ Dh_{A} : & 1 & 2 & 3 & 3 & 3 & − \end{matrix}

Then, we consider the set Z_{1} of 30 points linked to A through a complete intersection X = A ∪ Z_{1} of 42 points defined by a general 6-ic and a 7-ic. Namely, by (2.1),

\begin{matrix}Dh_{X} : & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 5 & 4 & 3 & 2 & 1 & − \\ Dh_{Z_{1}} : & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 2 & 1 & − \end{matrix}

We link Z_{1} to another set Z_{2} of 30 points through a complete intersection Y = Z_{1} ∪ Z_{2} of 60 points defined by a 6-ic and degree-10 form. Namely, by (2.1),

(3.5) \begin{aligned}Dh_{Y} (i) & : & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 6 & 6 & 6 & 5 & 4 & 3 & 2 & 1 & − \\ Dh_{Z_{1}}(14 − i) & : & & & & & & & 1 & 2 & 6 & 6 & 5 & 4 & 3 & 2 & 1 \\ Dh_{Z_{2}}(i) & : & 1 & 2 & 3 & 4 & 5 & 6 & 5 & 4 & − \end{aligned}

We immediately observe that Z_{1} and Z_{2} are both apolar to a form of degree 13. Indeed, from the first difference of Y = Z_{1} ∪ Z_{2} in (3.5), we have h_{14}^{1}(Z_{1} ∪ Z_{2}) = 1. By Proposition 2.2, ⟨ν_{13}(Z_{1})⟩ ∩ ⟨ν_{13}(Z_{2})⟩ = [f] for some degree 13 form f.

Claim 3.10. f has rank 30.

Proof. Let Z_{1} be as above and Z′ ⊂ ℙ^{2} be another set of points apolar to f. Our goal is to show that ℓ(Z′) ≥ 30.

First, assume that Z_{1} ∩ Z′ = ∅ and let \overline{Z} = Z_{1} ∪ Z′. By Lemma 2.1-(9) since ∑_{i=0}^{5} Dh_{Z_{1}}(i) = 21 then also ∑_{i=0}^{5} Dh\overline{Z}(i) = 21, i.e., Dh\overline{Z}(i) = i + 1 for i ∈ \{0, … 5\}. By Lemma 2.1-(10), the function Dh\overline{Z} cannot increase after the first degree i for which Dh\overline{Z}(i) ≤ i. Moreover, by Proposition 2.3, we cannot have consecutive values of the first difference Dh\overline{Z} equal to some value strictly smaller than 6. Finally, recall that by Proposition 2.6, \overline{Z} satisfies CB(13) and then, by Theorem 2.7, Dh\overline{Z}(14) ≥ 1. Therefore, we deduce

\begin{aligned}& & ⋯ & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ Dh\overline{Z} & : & ⋯ & 5 & 6 & ≥ 6 & ≥ 6 & ≥ 6 & ≥ 6 & ≥ 5 & ≥ 4 & ≥ 3 & ≥ 2 & ≥ 1 \end{aligned}

In particular, ℓ(\overline{Z}) ≥ ∑_{i=0}^{∞} Dh\overline{Z}(i) ≥ 60. Hence, ℓ(Z′) ≥ 30.

Assume now that Z_{1} ∩ Z′ ≠ ∅. As observed in [3, Remark 3.3], if Z′′ = Z′ ∖ (Z′ ∩ Z_{1}), then there exists a form f_{0} ∈ S_{13} such that [f_{0}] ∈ ⟨ν_{13}(Z_{1})⟩ ∩ ⟨ν_{13}(Z′′)⟩. Indeed, if Z_{1} = {ℓ_{1}, …, ℓ_{30}} and Z′ = {ℓ_{1}, …, ℓ_{k}, m_{k+1}, …, m_{r}}, then,

f = ∑_{i=1}^{30} a_{i}ℓ_{i}^{13} = ∑_{i=1}^{k} b_{i}ℓ_{i}^{13} + ∑_{i=k+1}^{r} b_{i}m_{i}^{13}

and

f_{0} = ∑_{i=1}^{k}(a_{i} − b_{i})ℓ_{i}^{13} + ∑_{i=k+1}^{30} a_{i}ℓ_{i}^{13} = ∑_{i=k+1}^{r} b_{i}m_{i}^{13}.

Therefore, we can proceed with the same argument as before by considering f_{0} instead of f and \overline{Z} = Z_{1} ∪ Z′′ to deduce that ℓ(Z_{1}) ≥ ℓ(Z′′) ≥ 30. □

Therefore, we have that Z_{1} and Z_{2} have different Hilbert function and regularity by (3.5) and both compute the rank of f by Claim 3.10. This concludes the proof.

□

4. Explicit Macaulay2 computations

In this last section, we provide a code in the language of the algebra software Macaulay2 [23] to construct the examples presented in the previous section.

4.1. First example. Degree-10 ternary forms of rank 22. We present here the second construction, as in the proof of Claim 3.5.

i2 : S = k[x,y,z];
i3 : -- consider a set of 22  general  points
      IZ2 = intersect  for i from 1 to 22 list
                                minors(2,vars(S)||random(QQ^1,QQ^3));
i4 : degree IZ2, for i to 10 list  hilbertFunction(i,IZ2)
o4 = (22, {1, 3, 6, 10, 15, 21, 22, 22, 22, 22, 22})
i5 : -- consider  the  set of  points Y = {(0:1:0),(0:0:1),(0:1:1)} on
     the  line x = 0 and  the  union A = Z1 u Y
      IY = intersect(ideal(x,z), ideal(x,y), ideal(x,y-z));
i6 : IA = intersect(IZ2,IY);
i7 : -- take  the  degree-6 part of the  ideal  of A
      G = ideal  super  basis(6,IA);
i8 : -- take  the  ideal  generated  by the  restriction  of G over  the  line
      x = 0 away  from  the  set of  points Y
      R = k[y,z];
i9 : G0 = sub(G,R);
i10 : G' = G0 : sub(IY,R);
i11 : -- G' is then  defined  by  three  cubics: let C be the  cubic  curve
     as  image  of the  line L via  the  map  defined  by G'
       phi = map(R,S,first  entries  mingens G');
i12 : C = ker  phi;
i13 : -- We  compute  the  singular  locus  and we  check  that it is  nodal
       VC = Proj (S/C);
i14 : singVC = singularLocus  VC;
i15 : codim  singVC, degree  singVC
o15 = (2, 1)
i16 : IO = ideal  singVC;
i17 : -- We  compute  the pre-image X of the  singular  point:
       -- that is a union  of two  simple  points
       T = QQ[a,b,c,y,z, MonomialOrder => Eliminate  3];

i18 : psi = map(T,S,{a,b,c});
i19 : J = psi(IO) + ideal(a-sub(G'_0,T),b-sub(G'_1,T),c-sub(G'_2,T));
i20 : use S;
i21 : IX=saturate(ideal(x)+sub(ideal  selectInSubring(1,gens gb J),S));
i22 : dim IX, degree IX, IX ==  radical  IX
o22 = (1, 2, true)
i23 : -- Now, the  union A u X is a set of 27  points
       IA' = intersect(IA,IX);
i24 : degree IA', for i to 10 list  hilbertFunction(i,IA')
o24 = (27, {1, 3, 6, 10, 15, 21, 26, 27, 27, 27, 27})
i25 : -- We can  also  note  that IA' is  generated  in  degree 7
       betti IA'
                0 1
o25 = total: 1 5
            0: 1 .
            1: . .
            2: . .
            3: . .
            4: . .
            5: . 2
            6: . 3
o25 : BettiTally
i26 : -- Construct a general c.i. of type  (7,7)
       B7 = super  basis(7,IA');
i27 : C1 = B7 * random(QQ^(numcols  B7), QQ^1);
i28 : C2 = B7 * random(QQ^(numcols  B7), QQ^1);
i29 : IU = ideal(C1,C2);
i30 : degree IU, for i to 12 list  hilbertFunction(i,IU)
o30 = (49, {1, 3, 6, 10, 15, 21, 28, 34, 39, 43, 46, 48, 49})
i31 : -- Consider  the  linkage  to A' through U
       IZ1 = IU : IA';
i32 : degree IZ1, for i to 10 list  hilbertFunction(i,IZ1)
o32 = (22, {1, 3, 6, 10, 15, 20, 22, 22, 22, 22, 22})
i33 : -- It can be  checked  that  IZ1 is  smooth  and  disjoint  from  IZ2
       loadPackage "Divisor";
i34 : isSmooth(IZ1, IsGraded => true), dim (IZ1 + IZ2)
o34 : (true, 0)
i35 : -- we  compute  now  the  intersection  of  their  linear  spans
       perpSpace = ideal  super  basis(10,IZ1)+ideal  super  basis(10,IZ2);
i36 : B10 = basis(10,S);

i37 : f = (B10 * gens  ker  diff(B10, transpose  mingens  perpSpace))_0_0;
i38 : -- Z1 and Z2 are  apolar  to f and  have  different  HFs
       fperp = inverseSystem f; -- computes  the  apolar  ideal  of f
i39 : isSubset(IZ1,fperp), isSubset(IZ2,fperp),
            netList {for i to 14 list  hilbertFunction(i,IZ1), for i to
                 14 list  hilbertFunction(i,IZ2)}

                       +-+-+-+--+--+--+--+--+--+--+--+--+--+--+--+

o39 = (true, true, |1|3|6|10|15|21|22|22|22|22|22|22|22|22|22|)

                       +-+-+-+--+--+--+--+--+--+--+--+--+--+--+--+
                       |1|3|6|10|15|20|22|22|22|22|22|22|22|22|22|
                       +-+-+-+--+--+--+--+--+--+--+--+--+--+--+--+

4.2. Second example. Degree-13 ternary forms of rank 30.

i2 : S = k[x,y,z];
i3 : -- consider  12  random  points  on the  rational  cubic  xz^2 - y^3 = 0
      a = entries  random(k^12,k^2);
i4 : A = (p -> {p_0^3, p_0*p_1^2, p_1^3}) \ a;
i5 : IA = intersect  ((p -> minors(2,matrix{{x,y,z},p})) \ A);
i6 : degree IA, for i to 5 list  hilbertFunction(i,IA)
o6 = (12, {1, 3, 6, 9, 12, 12})
i7 : -- consider a c.i. X of type  (6,7)  containing A
      B6 = super  basis(6,IA);     B7 = super  basis(7,IA);
i9 : F1 = B6 * random(QQ^(numcols  B6), QQ^1);
i10 : F2 = B7 * random(QQ^(numcols  B7), QQ^1);
i11 : IX = ideal(F1,F2);
i12 : degree IX, for i to 13 list  hilbertFunction(i,IX)
o12 = (42, {1, 3, 6, 10, 15, 21, 27, 32, 36, 39, 41, 42, 42, 42})
i13 : -- consider  the  link to A through X
       IZ1 = IX : IA;
i14 : degree IZ1, for i to 13 list  hilbertFunction(i,IZ1)
o14 = (30, {1, 3, 6, 10, 15, 21, 27, 29, 30, 30, 30, 30, 30, 30})
i15 : -- consider a c.i. Y of type  (6,10)  containing  Z1
       B6 = super  basis(6,IZ1);     B10 = super  basis(10,IZ1);
i17 : G1 = B6 * random(QQ^(numcols  B6), QQ^1);
i18 : G2 = B10 * random(QQ^(numcols  B10), QQ^1);
i19 : IY = ideal(G1,G2);
i20 : degree IY, for i to 14 list  hilbertFunction(i,IY)
o20 = (60, {1, 3, 6, 10, 15, 21, 27, 33, 39, 45, 50, 54, 57, 59, 60})
i21 : -- consider  the  link to Z1  through Y
       IZ2 = IY : IZ1;
i22 : degree IZ2, for i to 13 list  hilbertFunction(i,IZ2)
o22 = (30, {1, 3, 6, 10, 15, 21, 26, 30, 30, 30, 30, 30, 30, 30})

i23 : -- we  compute  now  the  intersection  of  their  linear  spans
    perpSpace = ideal  super  basis(13,IZ1)+ideal  super  basis(13,IZ2);
i24 : B13 = basis(13,S);
i25 : f = (B13 * gens  ker  diff(B13, transpose  mingens  perpSpace))_0_0;
i26 : -- note  that Z1 and Z2 are  apolar  to f
    -- and  have  different  Hilbert  functions  and  regularity
    fperp = inverseSystem f; -- computes  the  apolar  ideal  of f
i27 : isSubset(IZ1,fperp), isSubset(IZ2,fperp),
         netList {for i to 14 list  hilbertFunction(i,IZ1),
                    for i to 14 list  hilbertFunction(i,IZ2)}

                    +-+-+-+--+--+--+--+--+--+--+--+--+--+--+--+

o27 = (true, true, |1|3|6|10|15|21|27|29|30|30|30|30|30|30|30|)

                    +-+-+-+--+--+--+--+--+--+--+--+--+--+--+--+
                    |1|3|6|10|15|21|26|30|30|30|30|30|30|30|30|
                    +-+-+-+--+--+--+--+--+--+--+--+--+--+--+--+

References

1. James Alexander and André Hirschowitz, Polynomial interpolation in several variables, Journal of Algebraic Geometry 4 (1995), no. 2, 201–222.

2. Elena Angelini, Waring decompositions and identifiability via bertini and macaulay2 software, Journal of Symbolic Computation 91 (2019), 200–212.

3. Elena Angelini and Luca Chiantini, On the identifiability of ternary forms, Linear Algebra and its Applications 599 (2020), 36–65.

4. , Minimality and uniqueness for decompositions of specific ternary forms, Mathematics of Computation 91 (2022), no. 334, 973–1006.

5. Elena Angelini, Luca Chiantini, and Nick Vannieuwenhoven, Identifiability beyond kruskal’s bound for symmetric tensors of degree 4, Rendiconti Lincei 29 (2018), no. 3, 465–485.

6. Edoardo Ballico and Luca Chiantini, On the terracini locus of projective varieties, Milan J. Math. 89 (2021), 1–17.

7. Alessandra Bernardi, Enrico Carlini, Maria Virginia Catalisano, Alessandro Gimigliano, and Alessandro Oneto, The hitchhiker guide to: Secant varieties and tensor decomposition, Mathematics 6 (2018), no. 12, 314.

8. Weronika Buczyńska, Jarosław Buczyński, and Zach Teitler, Waring decompositions of monomials, Journal of Algebra 378 (2013), 45–57.

9. Enrico Carlini, Maria Virginia Catalisano, Luca Chiantini, Anthony V Geramita, and Youngho Woo, Symmetric tensors: rank, strassen’s conjecture and e-computability, Annali della Scuola Normale Superiore di Pisa. Classe di scienze 18 (2018), no. 1, 363–390.

10. Enrico Carlini, Maria Virginia Catalisano, and Anthony V Geramita, The solution to the waring problem for monomials and the sum of coprime monomials, Journal of algebra 370 (2012), 5–14.

11. Enrico Carlini, Nathan Grieve, and Luke Oeding, Four lectures on secant varieties, Connections between algebra, combinatorics, and geometry, Springer, 2014, pp. 101–146.

12. Maria Virginia Catalisano, Anthony V Geramita, and Alessandro Gimigliano, Segre-veronese embeddings of ℙ^{1} × ℙ^{1} × ℙ^{1} and their secant varieties, Collectanea Mathematica (2007), 1–24.

13. Luca Chiantini, Lectures on the structure of projective embeddings, Rend. Sem. Mat. Univ. Politec. Torino 62 (2004), no. 4, 335–388.

14. , Hilbert functions and tensor analysis, Quantum physics and geometry (2019), 125– 151.

15. Luca Chiantini, Ciro Ciliberto, and Francesco Russo, On secant defective varieties, in particular of dimension 4, arXiv preprint arXiv:2011.01105 (2020).

16. Luca Chiantini and Giorgio Ottaviani, A footnote to a footnote to a paper of B. Segre, Trends in Mathematics, to appear. arXiv preprint arXiv:2103.04659 (2021).

17. Luca Chiantini, Giorgio Ottaviani, and Nick Vannieuwenhoven, On generic identifiability of symmetric tensors of subgeneric rank, Transactions of the American Mathematical Society 369 (2017), no. 6, 4021–4042.

18. Alfred Clebsch, Ueber curven vierter ordnung., (1861).

19. Edward D Davis, Complete intersections of codimension 2 in pr: the bezout-jacobi-segre theorem revisited, Rend. Sem. Mat. Univers. Politecn. Torino 43 (1985), no. 4, 333–353.

20. Edward D Davis, Anthony V Geramita, and Ferruccio Orecchia, Gorenstein algebras and the cayley-bacharach theorem, Proceedings of the American Mathematical Society 93 (1985), no. 4, 593–597.

21. David Eisenbud, Linear sections of determinantal varieties, American Journal of Mathematics 110 (1988), no. 3, 541–575.

22. , The geometry of syzygies: A second course in commutative algebra and algebraic geometry, vol. 229, Springer, 2005.

23. Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.

24. Christopher J Hillar and Lek-Heng Lim, Most tensor problems are np-hard, Journal of the ACM (JACM) 60 (2013), no. 6, 1–39.

25. Anthony Iarrobino and Vassil Kanev, Power sums, gorenstein algebras, and determinantal loci, Springer Science & Business Media, 1999.

26. Tamara G Kolda and Brett W Bader, Tensor decompositions and applications, SIAM review 51 (2009), no. 3, 455–500.

27. Joseph M Landsberg, Tensors: geometry and applications, Representation theory 381 (2012), no. 402, 3.

28. Francis S Macaulay, The algebraic theory of modular systems, University press,, 1916.

29. Juan C Migliore, Introduction to liaison theory and deficiency modules, vol. 165, Springer Science & Business Media, 1998.

30. Bernard Mourrain and Alessandro Oneto, On minimal decompositions of low rank symmetric tensors, Linear Algebra and its Applications 607 (2020), 347–377.

31. Francesco Palatini, Sulla rappresentazione delle forme ed in particolare della cubica quinaria con la somma di potenze di forme lineari, vol. 38, Atti Acad. Torino, 1902.

32. , Sulla rappresentazione delle forme ternarie mediante la somma di potenze di forme lineari, vol. V, Tip. della R. Accademia dei Lincei, 1903.

33. , Sulle varieta algebriche per le quali sono di dimensione minore dell’ordinario senza riempire lo spazio ambiente una o alcune delle varieta formate da spazi seganti, 1909.

34. Christian Peskine and Lucien Szpiro, Liaison des variétés algébriques. i., Inventiones mathematicae 26 (1974), 271–302.

35. Kristian Ranestad and Frank-Olaf Schreyer, Varieties of sums of power, J. Reine Angew. Math. (2000), no. 525, 147–181.

36. Herbert W Richmond, On canonical forms, Quart. J. Pure Appl. Math 33 (1904), 967–984.

37. Gaetano Scorza, Determinazione delle varietà tre dimensioni di s_{r},(r ≥ 7) i cui s_{3} tangenti si tangliano a due a due, Rendiconti del Circolo Matematico di Palermo (1884-1940) 25 (1908), 193–204.

38. , Sulle varietà a quattro dimensioni di s_{r} (r > 9) i cui s_{4} tangenti si tagliano a due a due, Rendiconti del Circolo Matematico di Palermo 27 (1909), 148–178.

39. , Opere scelte, 1 (1960), 252–292.

40. James Joseph Sylvester, An essay on canonical forms, supplement to a sketch of a memoir on elimination, Collected Mathematical Paper I 34 (1851), 1851.

41. , On a remarkable discovery in the theory of canonical forms and of hyperdeterminants,, originally in Philosophical Magazine 1 (1851), 265–283.

42. Alessandro Terracini, Sulla rappresentazione delle coppie di forme ternarie mediante somme di potenze di forme lineari, Annali di Matematica Pura ed Applicata (1898-1922) 24 (1915), no. 1, 1–10.

Dip. di Ingegneria dell’Informazione e Scienze Matematiche, U. di Siena, Italy

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Dip. di Mathematics, U. of Trento, Via Sommarive 14, 38123 Povo (Trento), Italy

   Email address: alessandro.oneto@unitn.it