Catalecticant
Concept in mathematical invariant theory
title: "Catalecticant" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["invariant-theory"] description: "Concept in mathematical invariant theory" topic_path: "general/invariant-theory" source: "https://en.wikipedia.org/wiki/Catalecticant" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Concept in mathematical invariant theory ::
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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.
References
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