Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-24T16:10:42.706Z Has data issue: false hasContentIssue false
  • English
  • Français

An Invariant Regarding Waring’s Problem for Cubic Polynomials

Published online by Cambridge University Press:  11 January 2016

Giorgio Ottaviani
Giorgio Ottaviani*
Affiliation:
Dipartimento di Matematica U. Dini, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy, ottavian@math.unifi.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We compute the equation of the 7-secant variety to the Veronese variety (P4,O(3)), its degree is 15. This is the last missing invariant in the Alexander-Hirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariant of plane cubics as a pfaffian.

Keywords

15A7214L3514M1214M20
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 193 , 2009 , pp. 95 - 110
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

References

[AH] Alexander, J. and Hirschowitz, A., Polynomial interpolation in several variables, J. Alg. Geom., 4 (1995), no. 2, 201–222.Google Scholar
[Be] Beauville, A., Determinantal hypersurfaces, Michigan Math. J., 48 (2000), 39–64.Google Scholar
[BO] M. Brambilla, C. and Ottaviani, G., On the Alexander-Hirschowitz Theorem, J. of Pure and Applied Algebra, 212 (2008), 1229–1251.Google Scholar
[CH] Ciliberto, C. and Hirschowitz, A., Hypercubiques de P4 avec sept points singuliers génériques, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), no. 3, 135–137.Google Scholar
[DK] Dolgachev, I. and Kanev, V., Polar covariants of plane cubics and quartics, Adv. Math., 98 (1993), no. 2, 216–301.Google Scholar
[FH] Fulton, W. and Harris, J., Representation theory, Graduate Texts in Math. 129, Springer-Verlag, New York, 1991.Google Scholar
[IK] Iarrobino, A. and Kanev, V., Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics 1721, Springer, 1999.Google Scholar
[Mu] Mukai, S., Fano 3-folds, LMS Lecture Notes Series 179, Cambridge, 1992.Google Scholar
[RS] Ranestad, K. and Schreyer, F., Varieties of sums of powers, J. Reine Angew. Math., 525 (2000), 147–181.CrossRefGoogle Scholar
[Re] Reichstein, B., On Waring’s problem for cubic forms, Linear Algebra Appl., 160 (1992), 1–61.CrossRefGoogle Scholar
[St] Sturmfels, B., Algorithms in invariant theory, Springer, New York, 1993.CrossRefGoogle Scholar
[SS] Sturmfels, B. and Sullivant, S., Combinatorial secant varieties, Pure Appl. Math. Q., 2 (2006), no. 3, 867–891.CrossRefGoogle Scholar