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. 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 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 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 [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 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