Real and Complex Rank for Real Symmetric Tensors with Low Ranks
We study the case of a real homogeneous polynomial 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 is at most , then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.
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 , independent component analysis , complexity of matrix multiplication , complexity problem of versus NP , quantum physics [5, 6], and phylogenetics . 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.  and references therein). In this last case, we say that the rank of a homogeneous polynomial of degree is the minimum integer needed to write it as a linear combination of pure powers of linear forms : with . 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 for all 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 for all if is odd, while we take if 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 and we will indicate it (resp., ). Obviously and in many cases such an equality is strict.
In this paper, we want to study the relation between and in the special circumstance in which . In particular, we will show that in a certain range (say, ), all homogeneous polynomials of that degree with 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 be a real homogeneous polynomial of degree in variables such that and ; therefore, its real and complex decomposition are respectively, with , , , , . Moreover, there exists a curve such that “depends only from the variables of ” and is either a line or a reduced conic or a disjoint union of two lines. If is a line (item (a) in Theorem 1) then both the ’s and the ’s are linear forms in the same two “variables.” If is a conic, then ’s and ’s depend on 3 “variables” and their projectivizations lie on . See item (c) of Theorem 1 for the geometric interpretation of the reduction of and to bivariate forms involved with when is a disjoint union of two lines and (we have two sets of bivariate forms, one for the variables of and one for the variables of ).
2. 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 , , denote the degree Veronese embedding of (say, defined over . Set . For any , the symmetric rank or symmetric tensor rank or, just, the rank of is the minimal cardinality of a finite set such that , where denote the linear span (here the linear span is with respect to complex coefficients), and we will say that evinces . Notice that the Veronese embedding is defined over , that is, . For each the real symmetric rank of is the minimal cardinality of a finite set such that , where means the linear span with real coefficients, and we will say that evinces . The integer is well defined because spans .
Let us fix some notation: if is either a curve or a subspace and is a finite set, we will use the following abbreviations:
Theorem 1. Let be such that and . Fix any set and evincing and , respectively. Then one of the following cases (a), (b), and (c) occurs.(a) There is a line defined over and with the following properties:(i) and coincide out of the line in a set : (ii) there is a point such that evinces and evinces ;(iii) and .(b) There is a conic defined over and with the following properties:(i) and coincide out of the conic in a set : (ii) there is a point such that evinces and evinces ;(iii) and ;(iv) if is reducible, say with , then for .(c) and there are disjoint lines defined over with the following properties:(i) and coincide out of the union in a set : (ii) and ;(iii) the set is a single point, : evinces and evinces ; (iv) the set (resp., ) is formed by a unique point (resp. ): (resp., ) evinces (resp., ); (resp., ) evinces (resp., ).
3. The Proof
Remark 2. Let . It will be noteworthy in the sequel that can be used to span both a real space and a complex space of the same dimension and . In the following, we will always use to denote .
Remark 3. Fix and a finite set such that evinces . Fix any . Then the set is a single point (call it ) and evinces . Now assume and . Then . If evinces , then evinces .
Lemma 4. Let be a reduced curve of degree with . Fix finite sets . Fix an integer such that Assume the existence of and for any and any . Then
Proof. The case is [13, Lemma 8]. If , then either is a conic or and is a disjoint union of lines. In both cases, we have , and the linear system has no base points outside . Since is a finite set, there is such that . Look at the following residual exact sequence (also called the Castelnuovos exact sequence):
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 get . Now, since , we are done.
We are now going to prove Theorem 1.
Proof of Theorem 1. Fix such that and .
Fix any set evincing and any evincing .
By applying , Lemma 4, we immediately get that Since , either there is a line such that or there is a conic such that (Theorem 3.8, ). 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 line such that
This hypothesis, together with , immediately implies property (a)-(iii) of the statement of the theorem.
We are now going to distinguish the case (item (1.1) below) from the case (item (1.2) below).
(1.1) Assume .
First of all, observe that the line is well defined over since it contains at least 2 points of (Remark 2). Then, by Lemma 4, we have that and have to coincide out of the line : and this proves (a)-(i) of the statement of the theorem in this case (1.1).
The fact that implies that and ; hence, .
Therefore, we have that , and , and Grassmann’s formula gives .
Since and , the set is a single point, .
Since and for any , the set is a single point, (Remark 3).
Then obviously Since evinces , then evinces (Remark 3). In the same way, we see that and that evinces . This proves (a)-(ii) of Theorem 1 in this case (1.1).
(1.2) Assume .
First of all, observe that there exists a line such that , because .
By the same reason, if we write , we get that and hence (e.g., by , Lemma 34, or by , Theorem 3.8). Lemma 4 gives
Assume for the moment . 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 in (b) in the statement of the theorem is reduced. Moreover, condition (b)-(iv) is satisfied because .
Now assume . We will check that we are in case (a) with respect to the line if , while we are in case (c) with respect to the lines and if , and the case cannot occur.
Set . Since , we have and hence .
Assume for the moment . Hence, and . Therefore and the set is a single real point: evinces and evinces . 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 and at , which means that we can reduce our study to the case , since .
Until step (2) below, we will assume .
The linear system on has no base points outside itself. Since is finite, there is a smooth quadric surface containing such that Moreover, such a can be found among the real smooth quadrics, since and are real lines.
Since , we have . Hence, Lemma 4 applied to the point defined in (17) gives is a single real point, and evinces . If is either as in case (a) or in case (c) of the statement of the theorem, then is in the same case. Consider the system of lines on the smooth quadric surface containing . We have that , and hence the restriction map is surjective. Therefore, Now, this last equality, together with the facts that and are linearly independent and , gives
(1.2.1) Observe that (22) implies that the case cannot happen because there is no contribution from since both terms, in this case, are equal to 0. So, we can assume that at least .
(1.2.2) Assume and .
To prove that we are in case (a) with respect to it is sufficient to prove .
We have . Since and are linearly independent, , and , Grassmann’s formula gives that is generated by . Since for any , we get , that is, . Hence, we can consider case (a), and (15) proves property (a)-(i) also for the case (1.2) that we are treating. The point that we need to get (a)-(ii) can be identified with the point defined in (20) while (a)-(iii) comes from our hypotheses.
This gives all cases (a) of Theorem 1.
(1.2.3) Assume that both and .
We need to prove that we are in case (c). Recall that and that . The latter equality implies, as in Remark 3, that is a single real point , that evinces , and that evinces . Now plays the role of of case (c)-(iii) in Theorem 1.
Since and , the sets (resp., ) are formed by a unique point (resp., ). Remark 3 gives that , , evinces , evinces , evinces , and evinces . 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, and defined above coincide with and in (c)-(iv) of Theorem 1; therefore, we have also proved case (c) of Theorem 1.
(2) Now assume the existence of a conic such that
Since , we have . By Lemma 4 we have the set is a single point: and evinces . Moreover, if is defined over , then and evinces . Hence, .
(2.1) Assume that is smooth. Therefore, (24) proves (b)-(i) of the statement of the theorem in the case where 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 is obviously strictly smaller than . This concludes (b)-(iii).
The fact that also implies that . Since each point of is real, is real. Remark 3 gives that evinces . Since , also evinces the real symmetric tensor rank of with respect to the degree rational normal curve . The point defined in (25) plays the role of the point 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 Theorem 1.
(2.2) Assume that is reducible, say with and lines and . If , then we proved in step (1) that we are in case (a) with respect to the line . Hence, we may assume . Thus, even condition (b)-(iv) is satisfied as already remarked above after the displayed formula (15).
The authors were partially supported by CIRM of FBK Trento (Italy), Project Galaad of INRIA Sophia Antipolis Méditerranée (France), Institut Mittag-Leffler (Sweden), Marie Curie: Promoting science (FP7-PEOPLE-2009-IEF), MIUR, and GNSAGA of INdAM (Italy).
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