Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails
Let X⊂ℙr be an integral and non-degenerate complex variety. For any q∈ℙr let rX(q) be its X-rank and S(X,q) the set of all finite subsets of X such that |S|=rX(q) and q ∈ 〈S〉, where 〈〉 denotes the linear span. We consider the case |S(X,q)|>1 (i.e. when q is not X -identifiable) and study the s...
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| Format: | Article |
| Language: | English |
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Emerald Publishing
2021-04-01
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| Series: | Arab Journal of Mathematical Sciences |
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| Online Access: | https://www.emerald.com/insight/content/doi/10.1016/j.ajmsc.2019.09.001/full/pdf |
| _version_ | 1827912860820307968 |
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| author | E. Ballico |
| author_facet | E. Ballico |
| author_sort | E. Ballico |
| collection | DOAJ |
| description | Let X⊂ℙr be an integral and non-degenerate complex variety. For any q∈ℙr let rX(q) be its X-rank and S(X,q) the set of all finite subsets of X such that |S|=rX(q) and q ∈ 〈S〉, where 〈〉 denotes the linear span. We consider the case |S(X,q)|>1 (i.e. when q is not X -identifiable) and study the set W(X)q:=∩ S∈S(X,q)〈S〉, which we call the non-uniqueness set of q. We study the case dimX=1 and the case X a Veronese embedding of ℙn. We conclude the paper with a few remarks concerning this problem over the reals. |
| first_indexed | 2024-03-13T02:22:20Z |
| format | Article |
| id | doaj.art-bfeddef92c5a41b589ac4fa4462d3cd2 |
| institution | Directory Open Access Journal |
| issn | 1319-5166 2588-9214 |
| language | English |
| last_indexed | 2024-03-13T02:22:20Z |
| publishDate | 2021-04-01 |
| publisher | Emerald Publishing |
| record_format | Article |
| series | Arab Journal of Mathematical Sciences |
| spelling | doaj.art-bfeddef92c5a41b589ac4fa4462d3cd22023-06-30T09:18:58ZengEmerald PublishingArab Journal of Mathematical Sciences1319-51662588-92142021-04-01271415210.1016/j.ajmsc.2019.09.001Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability failsE. Ballico0Department of Mathematics, University of Trento, Povo, ItalyLet X⊂ℙr be an integral and non-degenerate complex variety. For any q∈ℙr let rX(q) be its X-rank and S(X,q) the set of all finite subsets of X such that |S|=rX(q) and q ∈ 〈S〉, where 〈〉 denotes the linear span. We consider the case |S(X,q)|>1 (i.e. when q is not X -identifiable) and study the set W(X)q:=∩ S∈S(X,q)〈S〉, which we call the non-uniqueness set of q. We study the case dimX=1 and the case X a Veronese embedding of ℙn. We conclude the paper with a few remarks concerning this problem over the reals.https://www.emerald.com/insight/content/doi/10.1016/j.ajmsc.2019.09.001/full/pdfX-rankVeronese embeddingSymmetric tensor rankAdditive decompositionReal X-rank |
| spellingShingle | E. Ballico Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails Arab Journal of Mathematical Sciences X-rank Veronese embedding Symmetric tensor rank Additive decomposition Real X-rank |
| title | Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails |
| title_full | Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails |
| title_fullStr | Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails |
| title_full_unstemmed | Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails |
| title_short | Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails |
| title_sort | reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails |
| topic | X-rank Veronese embedding Symmetric tensor rank Additive decomposition Real X-rank |
| url | https://www.emerald.com/insight/content/doi/10.1016/j.ajmsc.2019.09.001/full/pdf |
| work_keys_str_mv | AT eballico reconstructionofahomogeneouspolynomialfromitsadditivedecompositionswhenidentifiabilityfails |