Catalecticant

But the catalecticant of the biquadratic function of x, y was first brought into notice as an invariant by Mr Boole; and the discriminant of the quadratic function of x, y is identical with its catalecticant, as also with its Hessian. Meicatalecticizant would more completely express the meaning of that which, for the sake of brevity, I denominate the catalecticant.

— Sylvester (1852), quoted by Miller (2010)

In mathematical invariant theory, the catalecticant of a form of even degree is a polynomial in its coefficients that vanishes when the form is a sum of an unusually small number of powers of linear forms. It was introduced by Sylvester (1852); see Miller (2010). The word catalectic refers to an incomplete line of verse, lacking a syllable at the end or ending with an incomplete foot.

Binary forms

The catalecticant of a binary form of degree 2n is a polynomial in its coefficients that vanishes when the binary form is a sum of at most n powers of linear forms (Sturmfels 1993).

The catalecticant of a binary form can be given as the determinant of a catalecticant matrix (Eisenbud 1988), also called a Hankel matrix, that is a square matrix with constant (positive sloping) skew-diagonals, such as

${\begin{bmatrix}a&b&c&d&e\\b&c&d&e&f\\c&d&e&f&g\\d&e&f&g&h\\e&f&g&h&i\end{bmatrix}}.$

Catalecticants of quartic forms

The catalecticant of a quartic form is the resultant of its second partial derivatives. For binary quartics the catalecticant vanishes when the form is a sum of two 4th powers. For a ternary quartic the catalecticant vanishes when the form is a sum of five 4th powers. For quaternary quartics the catalecticant vanishes when the form is a sum of nine 4th powers. For quinary quartics the catalecticant vanishes when the form is a sum of fourteen 4th powers. (Elliott 1913, p. 295)

References

• Eisenbud, David (1988), "Linear sections of determinantal varieties", American Journal of Mathematics, 110 (3): 541–575, doi:10.2307/2374622, ISSN 0002-9327, JSTOR 2374622, MR 0944327
• Elliott, Edwin Bailey (1913) [1895], An introduction to the algebra of quantics. (2nd ed.), Oxford. Clarendon Press, JFM 26.0135.01
• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-211-77417-5, ISBN 978-3-211-82445-0, MR 1255980
• Miller, Jeff (2010), Earliest Known Uses of Some of the Words of Mathematics (C)
• Sylvester, J. J. (1852), "On the principles of the calculus of forms", Cambridge and Dublin Mathematical Journal: 52–97