Minimality and uniqueness for decompositions of specific ternary forms
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- by Elena Angelini and Luca Chiantini HTML | PDF
- Math. Comp. 91 (2022), 973-1006 Request permission
Abstract:
The paper deals with the computation of the rank and the identifiability of a specific ternary form. Often, one knows some short Waring decomposition of a given form, and the problem is to determine whether the decomposition is minimal and unique. We show how the analysis of the Hilbert-Burch matrix of the set of points representing the decomposition can solve this problem in the case of ternary forms. Moreover, when the decomposition is not unique, we show how the procedure of liaison can provide alternative, maybe shorter, decompositions. We give an explicit algorithm that tests our criterion of minimality for the case of ternary forms of degree $9$. This is the first numerical case in which a new phenomenon appears: the span of $18$ general powers of linear forms contains points of (subgeneric) rank $18$, but it also contains points whose rank is $17$, due to the existence of a second shorter decomposition which is completely different from the given one.References
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Additional Information
- Elena Angelini
- Affiliation: Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Via Roma 56, 53100 Siena, Italy
- MR Author ID: 944551
- ORCID: 0000-0002-8686-199X
- Email: elena.angelini@unisi.it
- Luca Chiantini
- Affiliation: Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Via Roma 56, 53100 Siena, Italy
- MR Author ID: 194958
- ORCID: 0000-0001-5776-1335
- Email: luca.chiantini@unisi.it
- Received by editor(s): April 10, 2020
- Received by editor(s) in revised form: January 19, 2021, and June 28, 2021
- Published electronically: November 10, 2021
- Additional Notes: This work was supported by the National Group for Algebraic and Geometric Structures, and their Applications (GNSAGA – INdAM) and by the Italian PRIN 2015 - Geometry of Algebraic Varieties (B16J15002000005)
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 973-1006
- MSC (2020): Primary 14N07, 14J70, 14C20, 14N05, 15A69, 15A72
- DOI: https://doi.org/10.1090/mcom/3681
- MathSciNet review: 4379984