On Comon's and Strassen's conjectures

Comon's conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon's conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.


Introduction
Let X Ă P N be an irreducible and reduced non-degenerate variety. The rank rank X ppq with respect to X of a point p P P N is the minimal integer h such that p lies in the linear span of h distinct points of X. In particular, if Y Ď X we have that rank X ppq ď rank Y ppq.
Since the h-secant variety Sec h pXq of X is the subvariety of P N obtained as the closure of the union of all ph´1q-planes spanned by h general points of X, for a general point p P Sec h pXq we have rank X ppq " h.
When the ambient projective space is a space parametrizing tensors we enter the area of tensor decomposition. A tensor rank decomposition expresses a tensor as a linear combination of simpler tensors. More precisely, given a tensor T , lying in a given tensor space over a field k, a tensor rank-1 decomposition of T is an expression of the form where the U i 's are linearly independent rank one tensors, and λ i P k˚. The rank of T is the minimal positive integer h such that T admits such a decomposition.
Comon's conjecture [CGLM08], which states the equality of the rank and symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors [Str73] are two of the most important and guiding problems in the area of tensor decomposition.
More precisely, Comon's conjecture predicts that the rank of a homogeneous polynomial F P krx 0 , . . . , x n s d with respect to the Veronese variety V n d is equal to its rank with respect to the Segre variety S n -pP n q d into which V n d is diagonally embedded, that is rank V n d pF q " rank S n pF q. Strassen's conjecture was originally stated for triple tensors and then generalized to a number of different contexts. For instance, for homogeneous polynomials it says that if F P krx 0 , . . . , x n s d and G P kry 0 , . . . , y m s d are homogeneous polynomials in distinct sets of variables then rank V n`m`1 d pF`Gq " rank V n d pF q`rank V m d pGq. In Sections 2 and 3, while surveying the state of the art on Comon's and Strassen's conjectures, we push a bit forward some standard techniques, based on catalecticant matrices and more generally on flattenings, to extend some results on these conjectures, known in the setting of Veronese and Segre varieties, for Segre-Veronese and Segre-Grassmann varieties that is to the context of mixed tensors.
In Section 4 we introduce a method to improve a classical result on Comon's conjecture. By standard arguments involving catalecticant matrices it is not hard to prove that Comon's conjecture holds for the general polynomial in krx 0 , . . . , x n s d of symmetric rank h as soon as h ă`n`t d 2 u n˘, see Proposition 2.2. We manage to improve this bound looking for equations for the ph´1q-secant variety Sec h´1 pV n d q, not coming from catalecticant matrices, that are restrictions to the space of symmetric tensors of equations of the ph´1q-secant variety Sec h´1 pS n q. We will do so by embedding the space of degree d polynomials into the space of degree d`1 polynomials by mapping F to x 0 F and then considering suitable catalecticant matrices of x 0 F rather than those of F itself.
Implementing this method in Macaulay2 we are able to prove for instance that Comon's conjecture holds for the general cubic polynomial in n`1 variables of rank h " n`1 as long as n ď 30. Note that for cubics the usual flattenings work for h ď n.

Notation
Let n " pn 1 , . . . , n p q and d " pd 1 , . . . , d p q be two p-uples of positive integers. Set d " d 1`¨¨¨`dp , n " n 1`¨¨¨`np , and N pn, dq " . . , V p be vector spaces of dimensions n 1`1 ď n 2`1 ď¨¨¨ď n p`1 , and consider the product P n " PpV1 qˆ¨¨¨ˆPpVp q. The line bundle O P n pd 1 , . . . , d p q " O PpV1 q pd 1 q b¨¨¨b O PpV1 q pd p q induces an embedding σν n d : PpV1 qˆ¨¨¨ˆPpVp q ÝÑ PpSym d 1 V1 b¨¨¨b Sym dp Vp q " P N pn,dq´1 , prv 1 s, . . . , rv p sq a Segre-Veronese variety. It is a smooth variety of dimension n and degree When p " 1, SV n d is a Veronese variety. In this case we write V n d for SV n d , and ν n d for the Veronese embedding. When d 1 "¨¨¨" d p " 1, SV n 1,...,1 is a Segre variety. In this case we write S n for SV n 1,...,1 , and σ n for the Segre embedding. Note that where n 1 " pN pn 1 , d 1 q´1, . . . , N pn p , d p q´1q.
1.1. Flattenings. Let V 1 , ..., V p be k-vector spaces of finite dimension, and consider the tensor product as a linear map r T : VÅ Ñ V A c . Clearly, if the rank of T is at most r then the rank of r T is at most r as well. Indeed, a decomposition of T as a linear combination of r rank one tensors yields a linear subspace of V A c , generated by the corresponding rank one tensors, containing r T pVÅ q Ď V A c . The matrix associated to the linear map r T is called an pA, Bq-flattening of T . In the case of mixed tensors we can consider the embedding In particular, if n " 1 we may interpret a tensor F P Sym d 1 V 1 as a degree d 1 homogeneous polynomial on PpV1 q. In this case the matrix associated to the linear map r F : VÅ Ñ V B is nothing but the a 1 -th catalecticant matrix of F , that is the matrix whose rows are the coefficient of the partial derivatives of order a 1 of F .
Similarly, by considering the inclusion bi for any i " 1, ..., p, we get the so called skew-flattenings. We refer to [Lan12] for details on the subject.
Remark 1.2. The partial derivatives of an homogeneous polynomials are particular flattenings. The partial derivatives of a polynomial F P krx 0 , ..., x n s d are`n`s n˘h omogeneous polynomials of degree d´s spanning a linear space H B s F Ď Ppkrx 0 , ..., x n s d´s q. If F P krx 0 , ..., x n s d admits a decomposition as in (0.1) then F P Sec h pV n d q, and conversely a general F P Sec h pV n d q can be written as in (0.1). If F " λ 1 L d 1`. ..`λ h L d h is a decomposition then the partial derivatives of order s of F can be decomposed as linear combinations of L d´s 1 , ..., L d´s h as well.
Therefore, the linear space 1.3. Rank and border rank. Let X Ă P N be an irreducible and reduced non-degenerate variety. We define the rank rank X ppq with respect to X of a point p P P N as the minimal integer h such that there exist h points in linear general position The border rank rank X ppq of p P P N with respect to X is the smallest integer r ą 0 such that p is in the Zariski closure of the set of points q P P N such that rank X pqq " r. In particular rank X ppq ď rank X ppq.
Recall that given an irreducible and reduced non-degenerate variety X Ă P N , and a positive integer h ď N the h-secant variety Sec h pXq of X is the subvariety of P N obtained as the Zariski closure of the union of all ph´1q-planes spanned by h general points of X.
In other words rank X ppq is computed by the smallest secant variety Sec h pXq containing p P P N . Now, let Y, Z be subvarieties of an irreducible projective variety X Ă P N , spanning two linear subspaces P N Y :" xY y , P N Z :" xZy Ď P N . Fix two points p Y P P N Y , p Z P P N Z , and consider a point p P xp Y , p Z y. Clearly It is natural to ask under which assumptions (1.4) is indeed an equality. Consider the Segre-Veronese embedding σν n d : PpV1 qˆ¨¨¨ˆPpVp q Ñ PpSym d 1 V1 b¨¨¨b Sym dp Vp q " P N pn,dq´1 with V 1 -¨¨¨-V p -V k-vector spaces of dimension n`1. Its composition with the diagonal embedding i : PpV˚q Ñ PpV1 qˆ¨¨¨ˆPpVp q is the Veronese embedding of ν n d of degree d " d 1`¨¨¨`dp . Let V n d Ď SV n d be the corresponding Veronese variety. We will denote by Π n,d the linear span of V n d in P N pn,dq´1 .
In the notations of Section 1.3 set X " SV n d and Y " V n d . For any symmetric tensor T P Π n,d we may consider its symmetric rank srkpT q :" rank V n d pT q and its rank rankpT q :" rank SV n d pT q as a mixed tensor. Comon's conjecture predicts that in this particular setting the inequality (1.4) is indeed an equality [CGLM08].
Conjecture 1 has been generalized in a number of directions for complex border rank, real rank and real border rank, see [Lan12, Section 5.7.2] for a full overview.
Note that when d " 2 Comon's conjecture is true. Indeed, Sec h pS n q is cut out by the size ph`1qp h`1q minors of a general square matrix and Sec h pV n 2 q is cut out by the size ph`1qˆph`1q minors of a general symmetric matrix, that is Sec h pV n 2 q " Sec h pS n q X Π n,2 . Conjecture 1 has been proved in several special cases. For instance, when the symmetric rank is at most two [CGLM08], when the rank is less than or equal to the order [ZHQ16], for tensors belonging to tangential varieties to Veronese varieties [BB13], for tensors in C 2 b C n b C n [BL13], when the rank is at most the flattening rank plus one [Fri16], for the so called Coppersmith-Winograd tensors [LM17], for symmetric tensors in C 4 b C 4 b C 4 and also for symmetric tensors of symmetric rank at most seven On the other hand, a counter-example to Comon's conjecture has recently been found by Y. Shitov [Shi17a]. The counter-example consists of a symmetric tensor T in C 800ˆC800ˆC800 which can be written as a sum of 903 rank one tensors but not as a sum of 903 symmetric rank one tensors. It is important to stress that for this tensor T rank and border rank are quite different. Comon's conjecture for border ranks is still completely open [Shi17a, Problem 25].
Even though it has been recently proven false in full generality, we believe that Comon's conjecture is true for a general symmetric tensor, perhaps it is even true for those tensor for which rank T " rank T .
In what follows we use simple arguments based on flattenings to give sufficient conditions for Comon's conjecture, recovering a known result, and its skew-symmetric analogue. In the Segre-Grassmann setting we argue in the same way by using skew-flattenings.
Proposition 2.2. [IK99] For any integer h ă`n`t d 2 u n˘t here exists an open subset U h Ď SecpV d n q such that for any T P U h the rank and the symmetric rank of T coincide, that is rankpT q " srkpT q Proof. First of all, note that we always have rankpT q ď srkpT q. Furthermore, Section 1.1 yields that for any pA, Bq-flattening r T : VÅ Ñ V B the inequality rankpT q ě dimp r T pVÅ qq holds. Since T is symmetric and its catalecticant matrices are particular flattenings we get that rankpT q ě dimpH B s T q for any s ě 0.
Now, for a general T P Sec h pV n d q we have srkpT q " h, and if h ă`n`s n˘, where s " t d 2 u, then Lemma 2.1 yields dimpH B s T q " h. Therefore, under these conditions we have the following chain of inequalities dimpH B s T q ď rankpT q ď srkpT q " dimpH B s T q and hence rankpT q " srkpT q. Now, consider the Segre-Plücker embedding PpV 1 qˆ. . .ˆPpV p q Ñ Pp For any skew-symmetric tensor T P Π n,d we may consider its skew rank skrkpT q that is its rank with respect to the Grassmannian Grpd, nq Ď Π n,d , and its rank rankpT q as a mixed tensor. Playing the same game as in Proposition 2.2 we have the following.
Proposition 2.3. For any integer h ă`n t d 2 u˘t here exists an open subset U h Ď Sec h pGrpd, nqq such that for any T P U h the rank and the skew rank of T coincide, that is rankpT q " skrkpT q Proof. As before for any tensor T we have rankpT q ď skrkpT q. For any pA, Bq-skew-flattening r T : VÅ Ñ V B we have skrkpT q ě dimp r T pVÅ qq. Furthermore, since r T is in particular a flattening also the inequality rankpT q ě dimp r T pVÅ qq holds. Now, for a general T P Sec h pGrpd, nqq we have skrkpT q " h, and if h ă`n s˘, where s " t d 2 u, Lemma 2.1 yields skrkpT q " dimp r T s pVÅ qq, where r T s is the skew-flattening corresponding to the partition ps, d´sq of d. Therefore, we deduce that dimp r T s pVÅ qq ď rankpT q ď skrkpT q " dimp r T s pVÅ qq and hence rankpT q " skrkpT q.
Remark 2.4. Propositions 2.2, 2.3 suggest that whenever we are able to write determinantal equations for secant varieties we are able to verify Comon's conjecture. We conclude this section suggesting a possible way to improve the range where the general Comon's conjecture holds giving a conjectural way to produce determinantal equations for some secant varieties. Set n " pn, . . . , nq, pd`1q-times, n 1 " pn, . . . , nq, d-times, and consider the corresponding Segre varieties X :" S n , X 1 :" S n 1 and Veronese varieties Y " V n d`1 , Y 1 :" V n d . Fix the polynomial x d`1 0 P Y and let Π be the linear space spanned by the polynomials of the form x 0 F , where F is a polynomial of degree d. This allow us to see Y 1 Ď Π. Note that polynomials of the form x 0 L d 1 lie in the tangent space of Y at L d`1 1 , and therefore rank Y px 0 L bd q " 2. Hence for a polynomial F of degree d we have rank Y px 0 F q ď 2 rank Y 1 pF q. Our aim is to understand when the equality holds.
We may mimic the same construction for the Segre varieties X and X 1 , and use determinantal equations for the secant varieties of X 1 to give determinantal equations of the secant varieties of X and henceforth conclude Comon's conjecture. In particular, as soon as d is odd and d ă n, this produces new determinantal equations for Sec h pX 1 q and Sec h pY 1 q with 2h ă`n`d`1 2 n˘. Therefore, this would give new cases in which the general Comon's conjecture holds. Unfortunately, we are only able to successfully implement this procedure in very special cases, see Section 4.

Strassen's conjecture
Another natural problem consists in giving hypotheses under which in (1.5) equality holds. Consider the triple Segre embedding σ n : PpV1 qˆPpV2 qˆPpV3 q " P aˆPbˆPc Ñ PpV1 bV2 bV1 q " P N pn,dq´1 , and let S n be the corresponding Segre variety. Now, take complementary subspaces P a 1 , P a 2 Ă P a , P b 1 , P b 2 Ă P b , P c 1 , P c 2 Ă P c , and let S pa 1 ,b 1 ,c 1 q , S pa 2 ,b 2 ,c 2 q be the Segre varieties associated respectively to P a 1ˆP b 1ˆP c 1 and P a 2ˆP b 2ˆP c 2 .
In the notations of Section 1.3 set X " S n , Y " S pa 1 ,b 1 ,c 1 q and Z " S pa 1 ,b 1 ,c 1 q . Strassen's conjecture states that the additivity of the rank holds for triple tensors, or in onther words that in this setting the inequality (1.5) is indeed an equality [Str73].
Even though Conjecture 2 was originally stated in the context of triple tensors that is bilinear forms, with particular attention to the complexity of matrix multiplication, a number of generalizations are immediate. For instance, we could ask the same question for higher order tensors, symmetric tensors, mixed tensors and skew-symmetric tensors. It is also natural to ask for the analogue of Conjecture 2 for border rank. This has been answered negatively [Sch81].
Conjecture 2 and its analogues have been proven when either T 1 or T 2 has dimension at most two, when rankpT 1 q can be determined by the so called substitution method [LM17], when dimpV 1 q " 2 both for the rank and the border rank [BGL13], when T 1 , T 2 are symmetric that is homogeneous polynomials in disjoint sets of variables, either T 1 , T 2 is a power, or both T 1 and T 2 have two variables, or either T 1 or T 2 has small rank [CCC15], and also for other classes of homogeneous polynomials [CCO17], [Tei15].
As for Comon's conjecture a counterexample to Strassen's conjecture has recentely been given by Y. Shitov [Shi17b]. In this case Y. Shitov proved that over any infinite field there exist tensors T 1 , T 2 such that the inequality in Conjecture 2 is strict.
In what follows we give sufficient conditions for Strassen's conjecture, recovering a known result, and for its mixed and skew-symmetric analogues.
Proposition 3.1. [IK99] Let V 1 , V 2 be k-vector spaces of dimensions n`1, m`1, and consider V " V 1 ' V 2 . Let F P Sym d pV 1 q Ă Sym d pV q and G P Sym d pV 2 q Ă Sym d pV q be two homogeneous polynomials. If there exists an integer s ą 0 such that dimpH B s F q " srkpF q, dimpH B s G q " srkpGq then srkpF`Gq " srkpF q`srkpGq.
Proof. Clearly, srkpF`Gq ď srkpF q`srkpGq holds in general. On the other hand, our hypothesis yields srkpF q`srkpGq " dimpH B s F q`dimpH B s F q " dimpH B s F`G q ď srkpF`Gq where the last inequality follows from Remark 1.2.
Remark 3.2. The argument used in the proof of Proposition 3.1 works for F P P N pn,dq general only if for the generic rank we have t`n`d dn`1 u ď`n`t d 2 u n˘. For instance, when n " 3, d " 6 the generic rank is 21 while the maximal dimension of the spaces spanned by partial derivatives is 20. Proposition 3.3. Let V 1 , . . . , V p and W 1 , . . . , W p be k-vector spaces of dimension n 1`1 , . . . , n p`1 and m 1`1 , . . . , m p`1 respectively. Consider U i " V i ' W i for every 1 ď i ď p. Let T 1 P Sym d 1 V 1 b¨¨¨b Sym dp V p Ă Sym d 1 U 1 b¨¨¨bSym dp U p and T 2 P Sym d 1 W 1 b¨¨¨bSym dp W p Ă Sym d 1 U 1 b¨¨¨bSym dp U p be two mixed tensors.
If for any i P t1, .., pu there exists a pair pa i , b i q with a i`bi " d i and pA, Bq-flattenings r T 1 : VÅ Ñ V B , r T 2 : VÅ Ñ V B as in (1.1) such that dimp r T 1 pVÅ qq " rankpT 1 q, dimp r T 2 pVÅ qq " rankpT 2 q then rankpT 1`T2 q " rankpT 1 q`rankpT 2 q.
Arguing as in the proof of Proposition 3.3 with skew-symmetric flattenings we have an analogous statement in the Segre-Grassmann setting.
Proposition 3.4. Let V 1 , . . . , V p and W 1 , . . . , W p be k-vector spaces of dimension n 1`1 , . . . , n p`1 and m 1`1 , . . . , m p`1 respectively. Consider U i " V i ' W i for every 1 ď i ď p, and let T 1 P Ź d 1 V 1 b¨¨¨b Ź dp V p Ă Ź d 1 U 1 b¨¨¨b Ź dp U p and T 2 P Ź d 1 W 1 b¨¨¨b Ź dp W p Ă Ź dp U 1 b¨¨¨b Ź dp U p be two skew-symmetric tensors with d i ď mintn i`1 , m i`1 u. If for any i P t1, . . . , pu there exists a pair pa i , b i q with a i`bi " d i and pA, Bq-skew-flattenings r T 1 : VÅ Ñ V B , r T 2 : VÅ Ñ V B as in (1.1) such that dimp r T 1 pVÅ qq " rankpT 1 q, dimp r T 2 pVÅ qq " rankpT 2 q then rankpT 1`T2 q " rankpT 1 q`rankpT 2 q.

On the rank of x 0 F
In this section, building on Remark 2.4, we present new cases in which Comon's conjecture holds. Recall, that for a smooth point x P X, the a-osculating space T a x X of X at x is roughly the smaller linear subspace locally approximating X up to order a at x, and the a-osculating variety T a X of X is defined as the closure of the union of all the osculating spaces For any 1 ď a ď d´1 the osculating space T a rL d s V n d of order a at the point rL d s P V d can be written as Equivalently, T a rL d s V n d is the space of homogeneous polynomials whose derivatives of order less than or equal to a in the direction given by the linear form L vanish. Note that dimpT a rL d s V n d q "`n`a n˘´1 and T b rL d s V n d Ď T a rL d s V n d for any b ď a. Moreover, for any 1 ď a ď d and rL d s P V n d we can embed a copy of V n a into the osculating space T a rL d s V n d by considering and Remark 2.4 yields that This embedding extends to an embedding at the level of Segre varieties, and, in the notation of Remark 2.4, we have that Sec h pS n 1 q Ď Sec 2h pS n q. Assume that for a polynomial F P Sec h pV n d q we have F P Sec h´1 pS n 1 q. Then x 0 F P Sec 2h´2 pS n q. Now, if we find a determinantal equation of Sec 2h´2 pV n d`1 q coming as the restriction to Π, the space of symmetric tensors, of a determinantal equation of Sec 2h´2 pS n q, and not vanishing at x 0 F then x 0 F R Sec 2h´2 pS n q and hence F R Sec h´1 pS n 1 q proving Comon's conjecture for F . This will be the leading idea to keep in mind in what follows. The determinantal equations involved will always come from minors of suitable catalecticant matrices, that can be therefore seen as the restriction to Π of determinantal equations for the secants of the Segre coming from non symmetric flattenings.
It is easy to give examples where the inequality (4.2) is strict. When n " 1 the generic rank is g d " r d`1 2 s. Then for d odd we have g d " g d´1 while for d even we have g d " g d´1`1 . Hence where V d :" V 1 d is the rational normal curve. It is natural to ask if the inequality is indeed an equality as long as the rank is subgeneric. In the case n " 1 we have the following result.
Proof. Clearly, it is enough to prove the statement for k h " 1. Let p P Sec h pV d q be a general point. Then p P @ rx 0 L d 1 s, . . . , rx 0 L d h s D with L i general linear forms. In particular p P H :" Note that dimpHq " 2h´1. Now, assume that p is contained also in Sec 2h´1 pV d`1 q. Then there exists a linear subspace H 1 Ă P d`1 of dimension 2h´2 passing through p intersecting V d`1 at 2h´1 points q 1 , . . . , q r counted with multiplicity. Let q i 1 , . . . , q ir be the points among the q i coinciding with some of the rL d`1 i s and such that the intersection multiplicity of H 1 and V d`1 at q i j is one, and q j 1 , . . . , q jr be the points among the q i coinciding with some of the rL d`1 i s and such that the intersection multiplicity of H 1 and V d at q j k is greater that or equal to two.
Set Π :" xH, H 1 y, then dimpΠq " 2h´1`2h´2´i r´2 j r and Π intersects V d`1 at 2h`p2h´1´i r´2 j r q points counted with multiplicity. Consider general points b 1 , . . . , b s P V d`1 with s " i r`2 j r , and the linear space Π 1 " xΠ, b 1 , . . . , b s y. Therefore, dimpΠ 1 q " 4h´3 and Π 1 intersects V d`1 at 4h´1 points counted with multiplicity. Since 2h ď d`3 2 adding enough general points to Π 1 we may construct a hyperplane in P d`1 intersecting V d`1 at d`2 points counted with multiplicity, a contradiction. Proposition 4.3 can be applied to get results on the rank of a special class of matrices called Hankel matrices.
Let In particular all the matrices of the form M d and N d considered above are Hankel matrices. Let M pa, bq be the vector space of aˆb matrices with coefficients in the base field k. For any h ď minta, bu let Rank r pM pa, bqq Ď M pa, bq be the subvariety consisting of all matrices of rank at most h. Now, consider the map β : N ÝÑ NˆN given by βp2nq " pn`1, n`1q and βp2n`1q " pn`2, n`1q. For any d ě 1 we can view the subspace H d Ď M pβpdqq formed by matrices of the form M d as the subspace of Hankel matrices. Now, given any linear morphism f : M pa, bq Ñ M pc, dq we can ask if for some s ď mintc, du we have f pRank h pM pa, bqqq Ď Rank s pM pc, dqq.
-if d is odd and`n`k n˘ă 2`n`k´1 n˘, where k " t d`1 2 u, then again it returns that the method does not apply; -if d is odd,`n`k n˘ě 2`n`k´1 n˘a nd 2h´1 ą`n`k n˘t hen it returns that the method does not apply since 2h´2 must be smaller than the number of order k partial derivatives; -if d is odd,`n`k n˘ě 2`n`k´1 n˘a nd 2h´1 ď`n`k n˘t hen Comon, in the spirit of Remark 4.1, produces a polynomial of the form F " h ÿ i"1 pa i,0 x 0`¨¨¨`ai,n x n q d then substitutes random rational values to the a i,j , computes the polynomial G " x 0 F , the catalecticant matrix D of order k partial derivatives of G, extracts the most up left 2h´1ˆ2h´1 minor P of D, and compute the determinant detpP q of P ; -if detpP q " 0 then Comon returns that the method does not apply, otherwise it returns that Comon's conjecture holds for the general degree d polynomial in n`1 variables of rank h.
Note that since the function random is involved Comon may return that the method does not apply even though it does. Clearly, this event is extremely unlikely. Thanks to this function we are able to prove that Comon's conjecture holds in some new cases that are not covered by Proposition 2.2. Since the case n " 1 is covered by Proposition 4.3 in the following we assume that n ě 2.
Theorem 4.8. Assume n ě 2 and set h "`n`t d 2 u n˘. Then Comon's conjecture holds for the general degree d homogeneous polynomial in n`1 variables of rank h in the following cases: -d " 3 and 2 ď n ď 30; -d " 5 and 3 ď n ď 8; -d " 7 and n " 4.
Proof. The proof is based on Macaualy2 computations using the function Comon exactly as shown in Example 4.9 below.