Algebraic Boundaries Among Typical Ranks for Real Binary Forms of Arbitrary Degree
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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.
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Maria Chiara Brambilla [email protected] Giovanni Staglianò [email protected]
1
Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy
2
Università degli Studi di Catania, Viale A. Doria 5, 95125 Catania, Italy
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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
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