Illinois Journal of Mathematics

A note on the simultaneous Waring rank of monomials

Enrico Carlini and Emanuele Ventura

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Abstract

In this paper, we study the complex simultaneous Waring rank for collections of monomials. For general collections, we provide a lower bound, whereas for special collections we provide a formula for the simultaneous Waring rank. Our approach is algebraic and combinatorial. We give an application to ranks of binomials and maximal simultaneous ranks. Moreover, we include an appendix of scripts written in the algebra software Macaulay2 to experiment with simultaneous ranks.

Article information

Source
Illinois J. Math., Volume 61, Number 3-4 (2017), 517-530.

Dates
Received: 27 November 2017
Revised: 7 May 2018
First available in Project Euclid: 22 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1534924838

Digital Object Identifier
doi:10.1215/ijm/1534924838

Mathematical Reviews number (MathSciNet)
MR3845732

Zentralblatt MATH identifier
06932515

Subjects
Primary: 51N35: Questions of classical algebraic geometry [See also 14Nxx] 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80] 05E40: Combinatorial aspects of commutative algebra

Citation

Carlini, Enrico; Ventura, Emanuele. A note on the simultaneous Waring rank of monomials. Illinois J. Math. 61 (2017), no. 3-4, 517--530. doi:10.1215/ijm/1534924838. https://projecteuclid.org/euclid.ijm/1534924838


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References

  • A. Anandkumar, D. Hsu, R. Ge, S. M. Kakade and M. Telgarsky, Tensor decompositions for learning latent variable models, J. Mach. Learn. Res. 15 (2014), 2773–2832.
  • E. Angelini, F. Galuppi, M. Mella and G. Ottaviani, On the number of Waring decompositions for a generic polynomial vector, J. Pure Appl. Algebra 222 (2018), 680–965.
  • E. Ballico, A. Bernardi and M. V. Catalisano, Higher secant varieties of $\mathbb{P}^{n}\times \mathbb{P}^{1}$ embedded in bidegree $(a,b)$, Comm. Algebra 40 (2012), no. 10, 3822–3840.
  • K. Baur and J. Draisma, Secant dimensions of low-dimensional homogeneous varieties, Adv. Geom. 10 (2015), no. 1, 1–29.
  • G. Blekherman and Z. Teitler, On maximum, typical and generic ranks, Math. Ann. 362 (2015), no. 3–4, 1021–1031.
  • J. Buczyński, K. Han, M. Mella and Z. Teitler, On the locus of points of high rank, Eur. J. Math. 4 (2018), 113–136.
  • J. Buczyński and Z. Teitler, Some examples of forms of high rank, Collect. Math. 67 (2016), no. 3, 431–441.
  • E. Carlini, M. V. Catalisano and A. V. Geramita, The solution to Waring's problem for monomials and the sum of coprime monomials, J. Algebra 370 (2012), 5–14.
  • E. Carlini, M. V. Catalisano and A. Oneto, Waring loci and Strassen conjecture, Adv. Math. 314 (2017), 630–662.
  • E. Carlini, L. Chiantini, M. V. Catalisano, A. V. Geramita and J. Woo, Symmetric tensors: Rank, strassen's conjecture and $e$-computability, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XVIII (2018), 363–390.
  • E. Carlini and J. Chipalkatti, On Waring's problem for several algebraic forms, Comment. Math. Helv. 78 (2003), no. 3, 494–517.
  • C. Dionisi and C. Fontanari, Grassmann defectivity à la Terracini, Matematiche (Catania) 56 (2001), 245–255.
  • D. R. Grayson and M. E. Stillman, Macaulay2–-A software system for research in algebraic geometry; available at http://www.math.uiuc.edu/Macaulay2/.
  • A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, vol. 1721, Springer-Verlag, Berlin, 1999.
  • R. P. Stanley, Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge Univ. Press, Cambridge, 1997.
  • V. Strassen, Vermeidung von Divisionen, J. Reine Angew. Math. 264 (1973), 184–202.
  • Z. Teitler, Sufficient conditions for Strassen's additivity conjecture, Illinois J. Math. 59 (2015), no. 4, 1071–1085.
  • A. Terracini, Sulla rappresentazione delle coppie di forme ternarie mediante somme di potenze di forme lineari, Ann. Mat. Pura Appl. XXIV (1915), 91–100.