• Source: 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 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






[



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



]


.


{\displaystyle {\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

Kata Kunci Pencarian:

• Catalecticant
• Hankel matrix
• James Joseph Sylvester
• Ternary quartic
• Diagonal form
• Canonizant
• Invariant of a binary form
• Glossary of classical algebraic geometry
• Glossary of invariant theory