An invariant regarding Waring's problem for cubic polynomials

2007/12/15 by Giorgio Ottaviani, Ottaviani, Giorgio
Engineering · Mathematics · #14L35 #14M12 #14M20 #15A72 #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics and Applications #Tensor decomposition and applications #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.0712.2527

openalex publication_date 2007/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 invariantof plane cubics as a pfaffian.

Related

• Eigenvectors of tensors and algorithms for Waring decomposition
• Waring identifiable subspaces over finite fields
• 16,051 formulas for Ottaviani's invariant of cubic threefolds
• The Waring Problem of Complex Binary Forms
• VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on A⁶