Nagoya Mathematical Journal

An invariant regarding Waring's problem for cubic polynomials

Giorgio Ottaviani

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We compute the equation of the 7-secant variety to the Veronese variety $({\bf P}^{4}, \mathcal{O}(3))$, 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.

Article information

Nagoya Math. J., Volume 193 (2009), 95-110.

First available in Project Euclid: 3 March 2009

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Zentralblatt MATH identifier

Primary: 15A72: Vector and tensor algebra, theory of invariants [See also 13A50, 14L24] 14L35: Classical groups (geometric aspects) [See also 20Gxx, 51N30] 14M12: Determinantal varieties [See also 13C40] 14M20: Rational and unirational varieties [See also 14E08]


Ottaviani, Giorgio. An invariant regarding Waring's problem for cubic polynomials. Nagoya Math. J. 193 (2009), 95--110.

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