# Real and Complex Rank for Real Symmetric Tensors with Low Ranks

Authors: Edoardo Ballico; Alessandra Bernardi
Journal: Algebra
 (2013)

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## Abstract

We study the case of a real homogeneous polynomialP whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that if the sum of the complex and the real ranks of P is at most 3deg(P)-1, then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.


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## Body

## 1. Introduction

The problem of decomposing a tensor into a minimal sum of rank-1 terms is raising interest and attention from many applied areas as signal processing for telecommunications [1], independent component analysis [2], complexity of matrix multiplication [3], complexity problem of P versus NP [4], quantum physics [5, 6], and phylogenetics [7]. The particular instance in which the tensor is symmetric and hence representable by a homogeneous polynomial is one of the most studied and developed ones (cf. [8] and references therein). In this last case, we say that the rank of a homogeneous polynomial P of degree d is the minimum integer r needed to write it as a linear combination of pure powers of linear forms L1,…,Lr:
(1)P=c1L1d+⋯+cdLrd,
with ci≠0. Most of the papers concerning the abstract theory of the symmetric tensor rank require the base field to be algebraically closed. In this case, we may take ci=1 for all i without loss of generality. However, for the applications, it is very important to consider the case of real polynomials and look at their real decomposition. Namely, one can study separately the case in which the linear forms appearing in (1) are complex or real. In the real case we may take ci=1 for all i if d is odd, while we take ci∈{-1,1} if d is even. When we look for a minimal complex (resp., real) decomposition as in (1), we say that we are computing the complex symmetric rank (resp., real symmetric rank) of P and we will indicate it rℂ(P) (resp., rℝ(P)). Obviously
(2)rℂ(P)≤rℝ(P),
and in many cases such an equality is strict.In [9] Comon and Ottaviani studied the real case for bivariate symmetric tensors. Even in this case, there are many open conjectures, and, up to now, few cases are completely settled [9–12].In this paper, we want to study the relation betweenrℂ(P) and rℝ(P) in the special circumstance in which rℂ(P)<rℝ(P). In particular, we will show that in a certain range (say, rℂ(P)+rℝ(P)≤3deg(P)-1), all homogeneous polynomials P of that degree with rℝ(P)≠rℂ(P) are characterized by the existence of a curve with the property that the sets evincing the real and the complex ranks coincide out of it (see Theorem 1 for the precise statement). More precisely, let P∈Sdℝm+1 be a real homogeneous polynomial of degree d in m+1 variables such that rℂ(P)<rℝ(P) and rℂ(P)+rℝ(P)≤3deg(P)-1; therefore, its real and complex decomposition are
(3)P=a1L1d+⋯+akLkd+c1M1d+⋯+csMsd,P=N1d+⋯+Nhd+c1M1d+⋯+csMsd,
respectively, with k>h, ai∈{-1,1}, cj∈{-1,1}, M1,…,Ms∈S1ℝm+1, h+k>d+2. Moreover, there exists a curve C′⊂ℙm such that [L1],…,[Lk],[N1],…,[Nh] “depends only from the variables of C′” and C′ is either a line or a reduced conic or a disjoint union of two lines. If C′ is a line (item (a) in Theorem 1) then both the Li’s and the Ni’s are linear forms in the same two “variables.” If C′ is a conic, then Li’s and Ni’s depend on 3 “variables” and their projectivizations lie on C′. See item (c) of Theorem 1 for the geometric interpretation of the reduction of L1,…,Lk and N1,…,Nh to bivariate forms involved with C′ when C′=l⊔r is a disjoint union of two lines l and r (we have two sets of bivariate forms, one for the variables of l and one for the variables of r).


## 2. Notation and Statements

Before giving the precise statement of Theorem1 we need to introduce the main algebraic geometric tools that we will use all along the paper.Letνd:ℙm→ℙN, N∶=(m+dd)-1, denote the degree d Veronese embedding of ℙm (say, defined over ℂ). Set Xm,d:=νd(ℙm). For any P∈ℙN, the symmetric rank or symmetric tensor rank or, just, the rankrℂ(P) of P is the minimal cardinality of a finite set S⊂ℙm(ℂ) such that P∈〈νd(S)〉, where 〈〉 denote the linear span (here the linear span is with respect to complex coefficients), and we will say that Sevincesrℂ(P). Notice that the Veronese embedding νd is defined over ℝ, that is, νd(ℙm(ℝ))⊂ℙN(ℝ). For each P∈ℙN(ℝ) the real symmetric rankrℝ(P) of P is the minimal cardinality of a finite set S⊂ℙm(ℝ) such that P∈〈νd(S)〉ℝ, where 〈〉ℝ means the linear span with real coefficients, and we will say that S evinces rℝ(P). The integer rℝ(P) is well defined because νd(ℙm(ℝ)) spans ℙN(ℝ).Let us fix some notation: ifC⊂ℙm is either a curve or a subspace and S⊂ℙm is a finite set, we will use the following abbreviations:
(4)SC∶=S∩C,SC^∶=S∖(S∩C).Theorem 1.
LetP∈ℙN(ℝ) be such that rℂ(P)+rℝ(P)≤3d-1 and rℂ(P)≠rℝ(P). Fix any set Sℂ⊂ℙm(ℂ) and Sℝ⊂ℙm(ℝ) evincing rℂ(P) and rℝ(P), respectively. Then one of the following cases (a), (b), and (c) occurs.(a)
There is a linel⊂ℙm defined over ℝ and with the following properties:
(i)
Sℂ and Sℝ coincide out of the line l in a set Sl^:
(5)Sℂ∖Sℂ∩l=Sℝ∖Sℝ∩l=:Sl^;(ii)
there is a pointPl∈〈νd(Sℂ,l)〉∩〈νd(Sℝ,l)〉 such that Sℂ,l evinces rℂ(Pl) and Sℝ,l evinces rℝ(Pl);(iii)
♯(Sℂ,l∪Sℝ,l)≥d+2 and ♯(Sℂ,l)<♯(Sℝ,l).(b)
There is a conicC⊂ℙm defined over ℝ and with the following properties:
(i)
Sℂ and Sℝ coincide out of the conic C in a set SC^:
(6)Sℂ∖Sℂ,C=Sℝ∖Sℝ∩C=:SC^;(ii)
there is a pointPC∈〈νd(Sℂ,C)〉∩〈νd(Sℝ,C)〉 such that Sℂ,C evinces rℂ(PC) and Sℝ,C evinces rℝ(PC);(iii)
♯(Sℂ,C∪Sℝ,C)≥2d+2 and ♯(Sℂ,C)<♯(Sℝ,C);(iv)
ifC is reducible, say C=l1∪l2 with Q=l1∩l2, then ♯((Sℂ∪Sℝ)∩(li∖Q))≥d+1 for i∈{1,2}.(c)
m≥3 and there are 2 disjoint lines l,r⊂ℙm defined over ℝ with the following properties:
(i)
Sℂ and Sℝ coincide out of the union Γ:=l∪r in a set SΓ^:
(7)Sℂ∖Sℂ∩(l∪r)=Sℝ∖Sℝ∩(l∪r)∶=SΓ^;(ii)
♯(Sℂ,l∪Sℝ,l)≥d+2 and ♯(Sℂ,r∪Sℝ,r)≥d+2;(iii)
the set〈νd(SΓ^)〉∩〈νd(Γ)〉 is a single point, OΓ∈ℙN(ℝ):Sℂ,Γ evinces rℂ(OΓ) and Sℝ,Γ evinces OΓ;(iv)
the set〈{OΓ}∪νd(l)〉∩〈νd(l)〉 (resp., 〈{OΓ}∪νd(r)〉∩〈νd(r)〉) is formed by a unique point Ol∈ℙN(ℝ) (resp. Or∈ℙN(ℝ)):Sℂ,l (resp., Sℂ,r) evinces rℂ(Ol) (resp., rℂ(Or));Sℝ,l (resp., Sℝ,r) evinces rℝ(Ol) (resp., rℝ(Or)).


## 3. The Proof

Remark 2.
LetS⊂ℙN(ℝ). It will be noteworthy in the sequel that S can be used to span both a real space 〈S〉ℝ⊂ℙN(ℝ) and a complex space 〈S〉ℂ⊂ℙN(ℂ) of the same dimension and 〈S〉ℂ∩ℙN(ℝ)=〈S〉ℝ. In the following, we will always use 〈〉 to denote 〈〉ℂ.Remark 3.
FixP∈ℙN and a finite set S⊂ℙN such that S evinces rℂ(P). Fix any E⊊S. Then the set 〈{P}∪E〉∩〈S∖E〉 is a single point (call it P1) and S∖E evinces rℂ(P1). Now assume P∈ℙN(ℝ) and S⊂ℙN(ℝ). Then P1∈ℙN(ℝ). If S evinces rℝ(P), then S∖E evinces rℝ(P1).Lemma 4.
LetC⊂ℙm be a reduced curve of degree t with t=1,2. Fix finite sets A,B⊂ℙm. Fix an integer d>t such that
(8)h1(ℐ(A∪B)C^(d-t))=0.
Assume the existence of P∈〈νd(A)〉∩〈νd(B)〉 and P∉〈νd(S′)〉 for any S′⊊A and any S′⊊B. Then
(9)AC^=BC^.Proof.
The caset=1 is [13, Lemma 8]. If t=2, then either C is a conic or m≥3 and C is a disjoint union of 2 lines. In both cases, we have h0(ℐC(t))>0, and the linear system |ℐC(t)| has no base points outside C. Since A∪B is a finite set, there is M∈|ℐC(t)| such that M∩(A∪B)=C∩(A∪B). Look at the following residual exact sequence (also called the Castelnuovos exact sequence):
(10)0→ℐ(A∪B)C^(d-t)→ℐA∪B(d)→ℐ(A∪B)∩M,M(d)→0.
We can now repeat the same proof of [13, Lemma 8] but starting with (10) instead of the exact sequence used there (cf. first displayed formula in the proof of [13, Lemma 8]).
We will therefore getAM^=BM^. Now, since M∩(A∪B)=C∩(A∪B), we are done.We are now going to prove Theorem1.Proof of Theorem1.
FixP∈ℙN(ℝ) such that rℂ(P)+rℝ(P)≤3d-1 and rℂ(P)≠rℝ(P).
Fix any setSℂ⊂ℙm(ℂ) evincing rℂ(P) and any Sℝ⊂ℙm(ℝ) evincing rℝ(P).
By applying [14], Lemma 4, we immediately get that
(11)h1(ℐSℂ∪Sℝ(d))>0.
Since ♯(Sℂ)+♯(Sℝ)≤3d-1, either there is a line l⊂ℙm such that ♯(Sℂ,l∪Sℝ,l)≥d+2 or there is a conic C such that ♯(Sℂ,C∪Sℝ,C)≥2d+2 (Theorem 3.8, [15]). We are going to study separately these two cases in items (1) and (10) below.
(1) In this step, we assume the existence of a linel⊂ℙm such that
(12)♯(Sℂ,l∪Sℝ,l)≥d+2.
This hypothesis, together withrℝ(P)≠rℂ(P), immediately implies property (a)-(iii) of the statement of the theorem.
We are now going to distinguish the caseh1(ℐSℂ∪Sℝ(d-1))=0 (item (1.1) below) from the case h1(ℐSℂ∪Sℝ(d-1))>0 (item (1.2) below).
(1.1) Assumeh1(ℐSℂ∪Sℝ(d-1))=0.
First of all, observe that the linel⊂ℙm is well defined over ℝ since it contains at least 2 points of Sℝ (Remark 2). Then, by Lemma 4, we have that Sℂ and Sℝ have to coincide out of the line l:
(13)Sℂ∖Sℂ,l=Sℝ∖Sℝ,l:=Sl^,
and this proves (a)-(i) of the statement of the theorem in this case (1.1).
The fact that♯(Sℂ)<♯(Sℝ) implies that ♯(Sℝ,l)>(d+2)/2 and ♯(Sl^)≤d; hence, h1(ℐSl^∪l(d))=0.
Therefore, we have thatdim(〈νd(Sl^∪l)〉)=♯(Sl^)+d+1, dim(〈νd(Sl^)〉)=♯(Sl^)-1 and dim(〈νd(l)〉)=d+1, and Grassmann’s formula gives 〈νd(Sl^)〉∩〈νd(l)〉=∅.
SinceP∈〈νd(Sl^∪Sℂ)〉 and Sℂ,l⊂l, the set 〈νd(Sl^)∪{P}〉∩〈νd(l)〉 is a single point, Pl∈ℙN(ℝ).
SinceP∈〈νd(Sℂ)〉 and P∉〈νd(Sℂ′)〉 for any S′⊊Sℂ, the set 〈νd(Sl^)∪{P}〉∩〈νd(Sℂ,l)〉 is a single point, PC (Remark 3).
Then obviously(14)PC=Pl∈ℙN(ℝ).
Since Sℂ evinces rℂ(P), then Sℂ,l evinces Pl (Remark 3). In the same way, we see that 〈νd(Sl^)∪{P}〉∩〈νd(Sℝ,l)〉={Pl} and that Sℝ,l evinces rℝ(Pl). This proves (a)-(ii) of Theorem 1 in this case (1.1).
(1.2) Assumeh1(ℐSℂ∪Sℝ(d-1))>0.
First of all, observe that there exists a liner⊂ℙm such that ♯(r∩(Sℂ∪Sℝ)l^)≥d+1, because ♯(Sℂ∪Sℝ)l^≤3d-1-d-2≤2(d-1)+1.
By the same reason, if we writeC:=l∪r, we get that ♯(Sℂ∪Sℝ)C^≤3d-1-d-2-d-1≤d-2 and hence h1(ℐ(Sℂ∪Sℝ)C^(d-2))=0 (e.g., by [16], Lemma 34, or by [15], Theorem 3.8). Lemma 4 gives
(15)Sℂ,C^=Sℝ,C^.
Assume for the momentl∩r≠∅. In this case, Remark 3 indicates that we can consider case (b) of the statement of the theorem. Therefore, (15) proves (b)-(i) in the case that the conic C in (b) in the statement of the theorem is reduced. Moreover, condition (b)-(iv) is satisfied because ♯(r∩(Sℂ∪Sℝ)l^)≥d+1.
Now assumel∩r=∅. We will check that we are in case (a) with respect to the line l if ♯(Sℂ,r∪Sℝ,r)=d+1, while we are in case (c) with respect to the lines l and r if ♯(Sℂ,r∪Sℝ,r)≥d+2, and the case ♯(Sℂ,l∪Sℝ,l)=♯(Sℂ,r∪Sℝ,r)=d+1 cannot occur.
SetΓ:=l∪r. Since r∩l=∅, we have dim〈Γ〉=3 and hence m≥3.
Assume for the momentm≥4. Hence, ♯(Sℂ∪Sℝ)〈Γ〉^≤d and h1(ℐ(Sℂ∪Sℝ)〈Γ〉^(d-1))=0. Therefore
(16)Sℂ,〈Γ〉^=Sℝ,〈Γ〉^,
and the set 〈{P}∪νd(Sℂ,〈Γ〉^)〉∩〈νd(〈Γ〉)〉 is a single real point:
(17)O:=〈{P}∪νd(Sℂ,〈Γ〉^)〉∩〈νd(〈Γ〉)〉∈ℙN(ℝ),Sℂ,Γ evinces rℂ(O) and Sℝ,Γ evinces rℝ(O). Now, (16) implies that if we are either in case (a) or in case (c) of the theorem, we can simply study what happens at Sℂ,〈Γ〉 and at Sℝ,〈Γ〉, which means that we can reduce our study to the case m=3, since 〈Γ〉=ℙ3.
Until step (2) below, we will assumem=3.
The linear system|ℐΓ(2)| on 〈Γ〉 has no base points outside Γ itself. Since Sℂ∪Sℝ is finite, there is a smooth quadric surface W containing Γ such that
(18)Sℂ,W∪Sℝ,W=Sℂ,Γ∪Sℝ,Γ.
Moreover, such a W can be found among the real smooth quadrics, since l and r are real lines.
Since♯(Sℂ,〈Γ〉∪Sℝ,〈Γ〉)W^≤d-1, we have h1(ℐ(Sℂ∪Sℝ)W^(d-2))=0. Hence, Lemma 4 applied to the point O defined in (17) gives
(19)(Sℂ,〈Γ〉)W^=(Sℝ,〈Γ〉)W^,〈{O}∪νd(Sℂ,〈Γ,〉)W^〉∩〈νd(W)〉 is a single real point,
(20)O′=〈{O}∪νd(Sℂ,〈Γ,〉)W^〉∩〈νd(W)〉∈ℙN(ℝ),
and Sℂ,W evinces rℂ(O′). If (O′,Sℂ,W,Sℝ,W) is either as in case (a) or in case (c) of the statement of the theorem, then (O,Sℂ,〈Γ〉,Sℝ,〈Γ〉) is in the same case. Consider the system |(1,0)| of lines on the smooth quadric surface W containing Γ. We have that h1(W,𝒪W(d-2,d))=0, and hence the restriction map H0(W,𝒪W(d))→H0(Γ,𝒪Γ(d)) is surjective. Therefore,
(21)h1(W,ℐW∩(Sℂ∪Sℝ)(d))=h1(l,ℐl∩(Sℂ∪Sℝ),l(d))+h1(r,ℐr∩(Sℂ∪Sℝ),r(d)).
Now, this last equality, together with the facts that νd(Sℂ,W) and νd(Sℝ,W) are linearly independent and (Sℂ∪Sℝ)W⊂Γ, gives
(22)dim(〈νd(Sℂ,W)〉∩〈νd(Sℝ,W)〉)=♯(Sℂ∩Sℝ)l+♯(Sℂ∩Sℝ)r+h1(l,ℐl∩(Sℂ∪Sℝ),l(d))+h1(r,ℐr∩(Sℂ∪Sℝ),r(d)).
(1.2.1) Observe that (22) implies that the case ♯(Sℂ∪Sℝ)r=♯(Sℂ∪Sℝ)l=d+1 cannot happen because there is no contribution from h1(l,ℐl∩(Sℂ∪Sℝ),l(d))+h1(r,ℐr∩(Sℂ∪Sℝ),r(d)) since both terms, in this case, are equal to 0. So, we can assume that at least ♯(Sℂ∪Sℝ)l>d+1.
(1.2.2) Assume♯(Sℂ∪Sℝ)r=d+1 and ♯(Sℂ∪Sℝ)l>d+1.
To prove that we are in case (a) with respect tol it is sufficient to prove Sℂ,r=Sℝ,r.
We havedim〈νd(Sℂ∪Sℝ)〉=♯(Sℂ∪Sℝ)-1-h1(ℐSℂ∪Sℝ(d)). Since νd(Sℂ) and νd(Sℝ) are linearly independent, ♯(Sℂ∪Sℝ)=♯(Sℂ)+♯(Sℝ)-♯(Sℂ∩Sℝ), ♯(Sℂ,l∪Sℝ,l)=♯(Sℂ,l)+♯(Sℝ,l)-♯(Sℂ,l∩Sℝ,l) and h1(ℐSℂ∪Sℝ(d))=h1(l,ℐSℂ,l∪Sℝ,l), Grassmann’s formula gives that 〈νd(Sℂ)〉∩〈νd(Sℝ)〉 is generated by 〈νd(Sℂ,l)〉∩〈νd(Sℝ,l)〉. Since P∉〈νd(S′)〉 for any S′⊊Sℂ, we get Sℂ=Sℂ,l∪(Sℂ∩Sℝ)l^, that is,  Sl^=Sℝ∖Sℂ,l. Hence, we can consider case (a), and (15) proves property (a)-(i) also for the case (1.2) that we are treating. The point Pl that we need to get (a)-(ii) can be identified with the point O′ defined in (20) while (a)-(iii) comes from our hypotheses.
This gives all cases (a) of Theorem1.
(1.2.3) Assume that both♯(Sℂ∪Sℝ)r≥d+2 and ♯(Sℂ∪Sℝ)l≥d+2.
We need to prove that we are in case (c). Recall thatSℂ,Γ^=Sℝ,Γ^ and that h1(ℐSℂ,Γ^∪Γ(d))=0. The latter equality implies, as in Remark 3, that 〈{P}∪νd(Sℂ,Γ)〉∩〈νd(Γ)〉 is a single real point O1, that Sℂ,Γ evinces rℂ(O1), and that Sℝ,Γ evinces O1. Now O1 plays the role of OΓ of case (c)-(iii) in Theorem 1.
Since〈νd(l)〉∩〈νd(r)〉=∅ and O1∈〈νd(Γ)〉, the sets 〈{O1}∪νd(l)〉∩〈νd(l)〉 (resp., 〈{O1}∪νd(r)〉∩〈νd(r)〉) are formed by a unique point O2 (resp., O3). Remark 3 gives that Oi∈ℙN(ℝ), i=1,2, Sℂ,l evinces rℂ(O2), Sℝ,l evinces rℝ(O2), Sℂ,r evinces rℂ(O3), and Sℝ,r evinces rℝ(O3). The hypotheses of the case (1.2.3) coincide with (c)-(ii) of the statement of the theorem, while (15) gives also property (c)-(i). Moreover, O2 and O3 defined above coincide with Ol and Or in (c)-(iv) of Theorem 1; therefore, we have also proved case (c) of Theorem 1.
(2) Now assume the existence of a conicC⊂ℙm such that
(23)deg(Sℂ∪Sℝ)C≥2d+2.
Since♯(Sℂ∪Sℝ)C≤3d-1-2d-2≤d-1, we have h1(ℐ(Sℂ∪Sℝ)C^(d-2))=0. By Lemma 4 we have
(24)Sℂ,C^=Sℝ,C^,
the set 〈{P}∪νd(Sℂ,C^)〉∩〈νd(〈C〉)〉 is a single point:
(25)P′:=〈{P}∪νd(Sℂ,C^)〉∩〈νd(〈C〉)〉,
and Sℂ,C evinces rℂ(P′). Moreover, if C is defined over ℝ, then P′∈ℙN(ℝ) and Sℝ,C evinces rℝ(P′). Hence, ♯(Sℂ,C)<♯(Sℂ∩Sℝ).
(2.1) Assume thatC is smooth. Therefore, (24) proves (b)-(i) of the statement of the theorem in the case where C is smooth. Since the reduced case is proved above (immediately after the displayed formula (15)), we have concluded the proof of (b)-(i).
Moreover, the hypothesis (23) coincides with (b)-(iii) of the statement of the theorem since ♯(Sℂ,C) is obviously strictly smaller than ♯(Sℝ,C). This concludes (b)-(iii).
The fact that♯(Sℂ,C)<♯(Sℝ,C) also implies that ♯(Sℝ,C)≥5. Since each point of Sℝ is real, C is real. Remark 3 gives that νd(Sℝ,C) evinces rℝ(P′). Since Sℝ,C⊂C, Sℝ,C also evinces the real symmetric tensor rank of P′ with respect to the degree 2d rational normal curve νd(C). The point P′ defined in (25) plays the role of the point PC appearing in (b)-(ii) of the statement of the theorem. Therefore, we have just proved (b)-(ii) of Theorem 1.
We treat the case (2.2) below for the sake of completeness, but we can observe that this concludes the proof of Theorem1.
(2.2) Assume thatC is reducible, say C=L1∪L2 with L1 and L2 lines and ♯((Sℂ∪Sℝ)L1)≥♯((Sℂ∩Sℝ)L2). If ♯((Sℂ∪Sℝ)∩(L2∖L2∩L1))≤d, then we proved in step (1) that we are in case (a) with respect to the line L1. Hence, we may assume ♯((Sℂ∪Sℝ)∩(L2∖L2∩L1))≥d+1. Thus, even condition (b)-(iv) is satisfied as already remarked above after the displayed formula (15).



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