Catalecticant

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 ; see. 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.

The catalecticant of a binary form can be given as the determinant of a catalecticant matrix, 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.