An invariant regarding Waring's problem for cubic polynomials

TL;DR

This paper computes the defining equation of the 7-secant variety to the Veronese variety for cubic polynomials in five variables, completing the Alexander-Hirschowitz classification and providing new invariant-based criteria for Waring's problem.

Contribution

It introduces the explicit equation of the 7-secant variety, derived from a 45x45 matrix determinant, as the last invariant needed in the classification.

Findings

Degree of the 7-secant variety is 15.

Provides a new invariant condition for expressing cubics as sums of seven cubes.

Connects the invariant to a determinant of a linear matrix, linking geometric and algebraic methods.

Abstract

We compute the equation of the 7-secant variety to the Veronese variety (P^4,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.