Abstract
In this paper we introduce the open Waring rank of a form of degree d in n variables and prove that this rank is bounded from above by
whenever n, d ≥ 3. This proves the same upper bound for the classical Waring rank of a form, improving the result of Białynicki-Birula and Schinzel (see[4]) and giving, as far as we know, the best upper bound known.
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References
J. Alexander and A. Hirschowitz Polynomial interpolation in several variables. J. Algebraic Geom. 4 (1995), 201–222.
E. Ballico and A. de Paris Generic Power Sum Decompositions and Bounds for the Waring Rank. arXiv:1312.3494, 2013.
A. Bernardi, A. Gimigliano, and M. Idà Computing symmetric rank for symmetric tensors. J. Symbolic Comput. 46 (2011), 34–53.
A. Białynicki-Birula and A. Schinzel Representations of multivariate polynomials by sums of univariate polynomials in linear forms. Colloq. Math. 112 (2008), 201–233.
G. Blekherman and Z. Teitler On maximum, Typical, and Generic Ranks. arXiv:1402.2371, 2014.
E. Carlini, M. V. Catalisano, and A. V. Geramita The solution to the Waring problem for monomials and the sum of coprime monomials. J. Algebra 370 (2012), 5–14.
Comas G., Seiguer M.: the rank of a binary form, Found. Comput. Math. 11, 65–78 (2011)
R. Hartshorne Algebraic geometry. Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52.
A. Iarrobino and V. Kanev Power sums, Gorenstein algebras, and determinantal loci, volume 1721 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman.
J. Kleppe Representing a Homogeneous Polynomial as a Sum of Powers of Linear Forms. available at http://folk.uio.no/johannkl/kleppe-master.
J. M. Landsberg Tensors: geometry and applications, volume 128 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012.
J. M. Landsberg and Z. Teitler On the ranks and border ranks of symmetric tensors. Found. Comput. Math. 10 (2010), 339–366.
J. J. Sylvester Sur une extension d’un théorème de Clebsch relatif aux courbes du quatrième degré. Math. Acad. Sci. Paris 102 (1886), 1532–1534.
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Supported by the project “Secant varieties, computational complexity, and toric degenerations” realised within the Homing Plus programme of Foundation for Polish Science, co-financed from European Union, Regional Development Fund. This paper is a part of “Computational complexity, generalised Waring type problems and tensor decompositions” project within “Canaletto”, the executive program for scientific and technological cooperation between Italy and Poland, 2013–2015.
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Jelisiejew, J. An upper bound for the Waring rank of a form. Arch. Math. 102, 329–336 (2014). https://doi.org/10.1007/s00013-014-0632-6
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DOI: https://doi.org/10.1007/s00013-014-0632-6