Algebraic Boundaries Among Typical Ranks for Real Binary Forms of Arbitrary Degree

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-020-09474-9 Algebraic Boundaries Among Typical Ranks for Real Binary Forms of Arbitrary Degree Maria Chiara Brambilla1 · Giovanni Staglianò2 Received: 8 November 2019 / Revised: 3 July 2020 / Accepted: 7 September 2020 © The Author(s) 2020 Abstract We show that the algebraic boundaries of the regions of real binary forms with fixed typical rank are always unions of dual varieties to suitable coincident root loci. Keywords Typical rank · Real rank boundary · Algebraic boundary · Binary form · Multiple root locus · Coincident root locus · Waring problem Mathematics Subject Classification Primary 15A69; Secondary 14P10 · 14N05 Introduction Let f ∈ Rd = K[x, y]d be a binary form of degree d, where K = R or C. By definition, see e.g., [27],  the K-rank of f is the minimum integer r such that f admits a decomposition f = ri=1 αi (i )d , where αi ∈ K and i ∈ K[x, y]1 for i = 1, . . . , r . The C-rank of a form, also called complex Waring rank, has been widely studied by many authors. The case of binary forms was considered and completely solved by Sylvester [36], who proved that the generic rank, i.e., the complex rank of a general complex binary form of degree d, is  d+1 2  (see also [9]). The generic complex rank of forms in more variables is described by the celebrated Alexander–Hirschowitz theorem [1] (see also [3]). Communicated by Peter Bürgisser. The first named author is partially supported by MIUR and INDAM. B Maria Chiara Brambilla Giovanni Staglianò 1 Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy 2 Università degli Studi di Catania, Viale A. Doria 5, 95125 Catania, Italy 123 Foundations of Computational Mathematics On the other hand, the real Waring rank has been studied only in recent years and most of the questions are still open. Clearly the real case is particularly relevant for the applications. In fact, the notion of tensor rank, which generalizes the Waring rank, has recently attracted great interest in applied mathematics, chemometrics, complexity theory, signal processing, quantum information theory, machine learning, and other current fields of research; see e.g., [10,11,18,25,27,28,32,33]. When we work on the real field, the notion of generic rank is replaced by the notion of typical ranks. A rank is called typical for real binary forms of degree d if it occurs in an open subset of Rd , with respect to the Euclidean topology. More precisely, denoting by Rd,r the interior of the semi-algebraic set { f ∈ Rd : rk R ( f ) = r } in the real vector space Rd , a rank r is typical exactly when Rd,r is not empty. By [2] it is known that a rank r is typical for forms of degree d if and only if d+1 2 ≤ r ≤ d. Let us now assume d+1 ≤ r ≤ d. Following [5,29], we define the topological 2 boundary ∂(Rd,r ) as the set-theoretic difference of the closure of Rd,r and the interior of the closure of Rd,r . Thus, if f ∈ ∂(Rd,r ) then every neighborhood of f contains a generic form of real rank equal to r and also a generic form of real rank different from r . We have that ∂(Rd,r ) is a semi-algebraic subset of Rd of pure codimension one. We define the algebraic boundary ∂alg (Rd,r ), also called real rank boundary, as the Zariski closure of the topological boundary ∂(Rd,r ) in the complex projective space P(C[x, y]d ). The algebraic boundary for maximum rank r = d coincides with the discriminant hypersurface. Indeed by [6,12], we know that the open set Rd,d corresponds to the locus of real-rooted forms, that is forms with all distinct and real roots. In the opposite case, the algebraic boundary for minimum rank r =  d+1 2  has been described in [29]. It is irreducible when d is odd, and it has two irreducible components when d is even. From these general results, it follows a complete description of all the algebraic boundaries with low degree d ≤ 6. In [5], we completely described the algebraic boundaries for the next two cases, d = 7 and d = 8. More precisely, we show in [5] that all the boundaries between two typical ranks are unions of dual varieties to suitable coincident root loci. Coincident root loci are well-studied varieties which parametrize binary forms with multiple roots, see Sect. 1.1 for the precise definitions. In this paper, we study the algebraic boundaries for forms of arbitrary degree, and our main result is the following: Theorem 0.1 (Theorems 3.2 and 3.3) For any degree d and any typical rank d+1 2 ≤ r ≤ d, the algebraic boundary ∂alg (Rd,r ) is a union of dual varieties to coincident root loci. Finally, we remark that the study of algebraic boundaries for forms with more than two variables is a challenging and quite open problem, see [30,37]. The paper is organized as follows: In the preliminary Sect. 1, we recall some basic notions and results about coincident root loci, higher associated subvarieties and apolar maps. Section 2 is devoted to the detailed analysis of the pullbacks, via apolar maps, of higher associated varieties to coincident root loci. The main result of this section is Theorem 2.1, whose two corollaries (Corollaries 2.3 and 2.4) are key tools in the 123 Foundations of Computational Mathematics proof of Theorem 0.1. In Sect. 3, we prove Theorem 0.1: More precisely, we consider the case of odd degree in Theorem 3.2 and the case of even degree in Theorem 3.3. 1 Preliminary 1.1 Coincident Root Loci Let r be a positive integer. A partition of r is an equivalence class, under reordering, n of lists of positive integers λ = [λ1 , . . . , λn ] such that i=1 λi = r . We denote by |λ| the length n of the partition. Alternatively, the partition λ can be represented by the list  of integers m 1 , . . . , m k defined as m j = |{i : λi = j}|, and clearly kj=1 jm j = r . Given a partition λ as above, the coincident root locus λ ⊂ Pr = P(C[x, y]r ) associated with λ is the set of binary forms f of degree r which admit a factorization n f = i=1 iλi for some linear forms 1 , . . . , n ∈ C[x, y]1 . These varieties have been extensively studied, see e.g., [7,8,23,26,38]. We have a unirational parameterization of degree m 1 !m 2 ! · · · m k !: 1 · · × P1 −→ λ ⊆ Pr , (1 , . . . , n ) → P × · n times n  iλi . i=0 In particular, the dimension of λ is n. The degree of λ was determined by Hilbert [20]. He showed that deg(λ ) = n! λ1 λ2 · · · λn . m 1 !m 2 ! · · · m k ! If λ and μ are two partitions of r , we have μ ⊆ λ if and only if λ is a refinement of μ (equiv., μ is a coarsening of λ). In [7] and subsequently in [26], it has been shown that the singular locus Sing(λ ) is given by the union of μ for some suitable coarsenings μ of λ (see [7, Definition 5.2] and [26, Proposition 2.1] for the precise description). In particular, one has that λ is smooth if and only if λ1 = · · · = λn . Otherwise, the singular locus is of codimension 1. The dual variety (λ )∨ of λ ⊂ P(C[x, y]r ) is a subvariety of th (...truncated)