## Nagoya Mathematical Journal

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

### An invariant regarding Waring's problem for cubic polynomials

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 3 March 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.nmj/1236089982

**Mathematical Reviews number (MathSciNet)**

MR2502909

**Zentralblatt MATH identifier**

1205.14064

**Subjects**

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]

#### Citation

Ottaviani, Giorgio. An invariant regarding Waring's problem for cubic polynomials. Nagoya Math. J. 193 (2009), 95--110. https://projecteuclid.org/euclid.nmj/1236089982