In mathematical
invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
, the catalecticant of a
form
Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens.
Form also refers to:
*Form (document), a document (printed or electronic) with spaces in which to write or enter data
...
of even degree is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
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
A catalectic line is a metrically incomplete line of verse, lacking a syllable at the end or ending with an incomplete foot. One form of catalexis is headlessness, where the unstressed syllable is dropped from the beginning of the line.
A line ...
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
Binary form is a musical form in 2 related sections, both of which are usually repeated. Binary is also a structure used to choreograph dance. In music this is usually performed as A-A-B-B.
Binary form was popular during the Baroque period, of ...
of degree 2''n'' 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 In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.:
\qquad\begin
a & b & c & d & e \\
b & c & d & e & f \\
c & d & ...
, that is a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
with constant (positive sloping) skew-diagonals, such as
:
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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*{{Citation , last1=Sylvester , first1=J. J. , author1-link=J. J. Sylvester , title=On the principles of the calculus of forms , year=1852 , journal=Cambridge and Dublin Mathematical Journal , pages=52–97
Invariant theory