Real and complex rank for real symmetric tensors with low ranks

We study the case of a real homogeneous polynomial $P$ 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 $ 3\deg(P)-1$, then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.


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 [15], independent component analysis [11], complexity of matrix multiplication [20], complexity problem of P versus NP [21], quantum physics [16], [5] and phylogenetic [1]. 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 (cfr. [17] 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 L 1 , . . . , L r : (1) P = c 1 L d 1 + · · · + c d L d r . with c i = 0. Most of the papers concerning the abstract theory of the symmetric tensor rank, require that the base field is algebraically closed. In this case we may take c i = 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 reals. In the real case we may take c i = 1 for all i if d is odd, while we take c i ∈ {−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 r C (P ) (resp. r R (P )). Obviously r C (P ) ≤ r R (P ), and in many cases such an equality is strict.
In [12] P. Comon and G. 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 ( [12], [9], [2], [7]).
In this paper we want to study the relation among r C (P ) and r R (P ) in the special circumstance in which r C (P ) < r R (P ). In particular we will show that, in a certain range (say, r C (P ) + r R (P ) ≤ 3 deg(P ) − 1), all homogeneous polynomials P of that degree with r R (P ) = r C (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 ∈ S d R m+1 be a real homogeneous polynomial of degree d in m + 1 variables such that r C (P ) < r R (P ) and r C (P ) + r R (P ) ≤ 3 deg(P ) − 1; therefore its real and complex decomposition are " 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 L i 's and the N i 's are linear forms in the same two "variables". If C ′ is a conic, then L i 's and N i '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 L 1 , . . . , L k and N 1 , . . . , N h 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 bivariant forms, one for the variables of l and one for the variables of r).

Notation and statements
Before giving the precise statement of Theorem 1 we need to introduce the main algebraic geometric tools that we will use all along the paper.
Let ν d : P m → P N , N := m+d d − 1, denote the degree d Veronese embedding of P m (say, defined over C). Set X m,d := ν d (P m ). For any P ∈ P N , the symmetric rank or symmetric tensor rank or, just, the rank r C (P ) of P is the minimal cardinality of a finite set S ⊂ P m (C) such that P ∈ ν d (S) , where denote the linear span (here the linear span is with respect to complex coefficients), and we will say that S evinces r C (P ). Notice that the Veronese embedding ν d is defined over R, i.e. ν d (P m (R)) ⊂ P N (R). For each P ∈ P N (R) the real symmetric rank r R (P ) of P is the minimal cardinality of a finite set S ⊂ P m (R) such that P ∈ ν d (S) R , where R means the linear span with real coefficients, and we will say that S evinces r R (P ). The integer r R (P ) is well-defined, because ν d (P m (R)) spans P N (R). Let us fix some notation: If C ⊂ P m is either a curve or a subspace and S ⊂ P m is a finite set, we will use the following abbreviations: Theorem 1. Let P ∈ P N (R) be such that r C (P ) + r R (P ) ≤ 3d − 1 and r C (P ) = r R (P ). Fix any set S C ⊂ P m (C) and S R ⊂ P m (R) evincing r C (P ) and r R (P ) respectively. Then one of the following cases (a), (b), (c) occurs: (a) There is a line l ⊂ P m defined over R and with the following properties: (i) S C and S R coincide out of the line l in a set Sl: and S R,l evinces r R (P l ); There is a conic C ⊂ P m defined over R and with the following properties: (i) S C and S R coincide out of the conic C in a set SĈ : and there are 2 disjoint lines l, r ⊂ P m defined over R with the following properties: (i) S C and S R coincide out of the union Γ := l ∪ r in a set SΓ:

The proof
Remark 1. Let S ⊂ P N (R). It will be noteworthy in the sequel that S can be used to span both a real space S R ⊂ P N (R) and a complex space S C ⊂ P N (C) of the same dimension and S C ∩ P N (R) = S R . In the following we will always use to denote C .
Remark 2. Fix P ∈ P N and a finite set S ⊂ P N such that S evinces r C (P ). Fix any E S. Then the set {P } ∪ E ∩ S \ E is a single point (call it P 1 ) and S \ E evinces r C (P 1 ). Now assume P ∈ P N (R) and S ⊂ P N (R). Then P 1 ∈ P N (R). If S evinces r R (P ), then S \ E evinces r R (P 1 ).

Lemma 1.
Let C ⊂ P m be a reduced curve of degree t with t = 1, 2. Fix finite sets A, B ⊂ P m . Fix an integer d > t such that: Assume the existence of P ∈ ν d (A) ∩ ν d (B) and P / ∈ ν d (S ′ ) for any S ′ A and any S ′ B. Then AĈ = BĈ .
Proof. The case t = 1 is [4, 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 h 0 (I C (t)) > 0 and the linear system |I C (t)| has no base points outside C. Since A ∪ B is a finite set, there . Look at the residual exact sequence (also called the Castelnuovo's exact sequence): We can now repeat the same proof of [4, Lemma 8] but starting with (2)  We are now going to prove Theorem 1.

Proof of Theorem 1
Fix P ∈ P N (R) such that r C (P ) + r R (P ) ≤ 3d − 1 and r C (P ) = r R (P ). Fix any set S C ⊂ P m (C) evincing r C (P ) and any S R ⊂ P m (R) evincing r R (P ). By applying [3], Lemma 1, we immediately get that We are going to study separately these two cases in items (1) and (2) below.
(1) In this step we assume the existence of a line l ⊂ P m such that This hypothesis, together with r R (P ) = r C (P ), immediately implies property (aiii) of the statement of the theorem.
(1.1) Assume h 1 (I S C ∪S R (d − 1)) = 0. First of all, observe that the line l ⊂ P m is well defined over R since it contains at least 2 points of S R (Remark 1). Then, by Lemma 1, we have that S C and S R have to coincide out of the line l: S C \ S C,l = S R \ S R,l := Sl, and this proves (ai) of the statement of the theorem in this case (1.1).
Since P ∈ ν d (Sl ∪ S C ) and S C,l ⊂ l, the set ν d (Sl) ∪ {P } ∩ ν d (l) is a single point, P l ∈ P N (R).
Since P ∈ ν d (S C ) and P / ∈ ν d (S ′ C ) for any S ′ S C , the set ν d (Sl) ∪ {P } ∩ ν d (S C,l ) is a single point, P C (Remark 2).
Then obviously: P C = P l ∈ P N (R). Since S C evinces r C (P ), then S C,l evinces P l (Remark 2). In the same way we see that ν d (Sl) ∪ {P } ∩ ν d (S R,l ) = {P l } and that S R,l evinces r R (P l ). This proves (aii) of Theorem 1 in this case (1.1).
First of all, observe that there exists a line r ⊂ P m such that ♯(r ∩ ( By the same reason, if we write: ) = 0 (e.g. by [6], Lemma 34, or by [14], Theorem 3.8). Lemma 1 gives: -Assume for the moment l ∩ r = ∅. In this case, Remark 2 indicates that we can consider case (b) of the statement of the theorem. Therefore (3) proves (bi) in the case that the conic C in (b) in the statement of the theorem is reduced. Moreover, condition (biv) is satisfied, because ♯(r ∩ (S C ∪ S R )l) ≥ d + 1.
-Now assume l ∩ r = ∅. We will check that we are in case (a) with respect to the line l if ♯(S C,r ∪ S R,r ) = d + 1, while we are in case (c) with respect to the lines l and r if ♯(S C,r ∪ S R,r ) ≥ d + 2, and the case ♯(S C,l ∪ S R,l ) = ♯(S C,r ∪ S R,r ) = d + 1 cannot occur.
-Assume for the moment m ≥ 4. Hence ♯(S C ∪S R ) Γ ≤ d and h 1 (I (S C ∪S R ) Γ (d− 1)) = 0. Therefore: is a single real point: S C,Γ evinces r C (O) and S R,Γ evinces r R (O). Now, (4) implies that if we are either in case (a) or in case (c) of the theorem, we can simply study what happens at S C, Γ and at S R, Γ , which means that we can reduce our study to the case m = 3, since Γ = P 3 .
The linear system |I Γ (2)| on Γ has no base points outside Γ itself. Since S C ∪S R is finite, there is a smooth quadric surface W containing Γ such that Moreover, such a W can be found among the real smooth quadrics, since l and r are real lines.
Hence Lemma 1 applied to the point O defined in (5), gives: Now, this last equality, together with the facts that ν d (S C,W ) and ν d (S R,W ) are linearly independent and (S C ∪ S R ) W ⊂ Γ, gives: (1.2.1) Observe that (7) implies that the case ♯(S C ∪ S R ) r = ♯(S C ∪ S R ) l = d + 1 cannot happen because there is no contribution from h 1 (l, I l∩(S C ∪S R ),l (d)) + h 1 (r, I r∩(S C ∪S R ),r (d)) since both terms, in this case, are equal to 0. So, we can assume that at least ♯(S C ∪ S R ) l > d + 1. ( To prove that we are in case (a) with respect to l it is sufficient to prove S C,r = S R,r .
Hence we can consider case (a), and (3) proves property (ai) also for the case (1.2) that we are treating. The point P l that we need to get (aii) can be identified with the point O ′ defined in (6) while (aiii) comes from our hypotheses.
This gives all cases (a) of Theorem 1.
(1.2.3) Assume that both ♯(S C ∪ S R ) r ≥ d + 2 and ♯(S C ∪ S R ) l ≥ d + 2. We need to prove that we are in case (c). Recall that S C,Γ = S R,Γ and that h 1 (I S C,Γ ∪Γ (d)) = 0. The latter equality implies, as in Remark 2, that {P } ∪ ν d (S C,Γ ) ∩ ν d (Γ) is a single real point O 1 , that S C,Γ evinces r C (O 1 ) and that S R,Γ evinces O 1 . Now O 1 plays the role of O Γ of case (ciii) in Theorem 1.
Since ν d (l) ∩ ν d (r) = ∅ and O 1 ∈ ν d (Γ) , the sets {O 1 }∪ν d (l) ∩ ν d (l) (resp. {O 1 }∪ν d (r) ∩ ν d (r) ) are formed by a unique point O 2 (resp. O 3 ). Remark 2 gives that O i ∈ P N (R), i = 1, 2, S C,l evinces r C (O 2 ), S R,l evinces r R (O 2 ), S C,r evinces r C (O 3 ) and S R,r evinces r R (O 3 ). The hypotheses of the case (1.2.3) coincide with (cii) of the statement of the theorem, while (3) gives also property (ci). Moreover O 2 and O 3 defined above coincide with O l and O r in (civ) of Theorem 1, therefore we have also proved case (c) of Theorem 1.
(2) Now assume the existence of a conic C ⊂ P m such that (8) deg(S C ∪ S R ) C ≥ 2d + 2.
(2.1) Assume that C is smooth. Therefore (9) proves (bi) 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 (3)), we have concluded the proof of (bi).
Moreover the hypothesis (8) coincides with (biii) of the statement of the theorem since ♯(S C,C ) is obviously strictly smaller than ♯(S R,C ). This concludes (biii).
The fact that ♯(S C,C ) < ♯(S R,C ) also implies that ♯(S R,C ) ≥ 5. Since each point of S R is real, C is real. Remark 2 gives that ν d (S R,C ) evinces r R (P ′ ). Since S R,C ⊂ C, S R,C also evinces the real symmetric tensor rank of P ′ with respect to the degree 2d rational normal curve ν d (C). The point P ′ defined in (10) plays the role of the point P C appearing in (bii) of the statement of the theorem. Therefore, we have just proved (bii) of Theorem 1.