An Invariant Regarding Waring’s Problem for Cubic Polynomials

By Giorgio Ottaviani. Nagoya mathematical journal, 2007

49 citations recorded by the source corpus.

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 invariant of plane cubics as a pfaffian.

Subfields: Computational Mathematics, Electrical and Electronic Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3093904495e822aa1d6b31177134e3a8c7b4d09b

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Citations in the corpus

Classical Algebraic Geometry: A Modern View (2012, background): “The case (n, d) = (4, 3) was studied only recently in [422].”
On generic identifiability of symmetric tensors of subgeneric rank (2015, methodology): “We remark that the defective Veronese varieties according to the classification of Alexander and Hirschowitz [2] (see [30] for the equations of the defective secant varieties) yield the following dual varieties (i) σn(n+3)/2(v4(P )), for n = 2, 3, 4, contains quartic hypersurfaces which are double q”
Stratification of the fourth secant variety of Veronese varieties via the symmetric rank (2010, background): “We like to stress here that the defining ideals of σ2(X1,d) and σ3(X2,d) are known (see [21] and [25] respectively) and this allows the authors of [6] to give algorithms for the X-rank of points in σs(X) with s = 2, 3.”
Non-defectivity of Grassmannians of planes (2009, background): “A natural symmetric operator such that its determinant is a cube appears already in [ Ot ], where the coecient 2 appears at the same place.”