“Catalecticant”的意思、由来-开放百科全书

词条

Catalecticant

释义

Binary forms
Catalecticants of quartic forms
References
External links

{{quotebox|quote=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.
|source={{harvtxt|Sylvester|1852}}, quoted by {{harvtxt|Miller|2010}}|align=right|width=30%}}

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 {{harvtxt|Sylvester|1852}}; see {{harvtxt|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 {{harv|Sturmfels|1993}}.

The catalecticant of a binary form can be given as the determinant of a catalecticant matrix {{harv|Eisenbud|1988}}, also called a Hankel matrix, that is a square matrix 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 2 4th powers. For a ternary quartic the catalecticant vanishes when the form is a sum of 5 4th powers. For quaternary quartics the catalecticant vanishes when the form is a sum of 9 4th powers. For quinary quartics the catalecticant vanishes when the form is a sum of 14 4th powers. {{harv|Elliot|1915|loc=p.295}}

References

{{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Linear sections of determinantal varieties | doi=10.2307/2374622 | year=1988 | journal=American Journal of Mathematics | issn=0002-9327 | volume=110 | issue=3 | pages=541–575 | mr=944327}}
{{Citation | last1=Elliott | first1=Edwin Bailey | title=An introduction to the algebra of quantics. | origyear=1895 | url=https://books.google.com/books?id=Az5tAAAAMAAJ | publisher=Oxford. Clarendon Press | edition=2nd | jfm=26.0135.01 | year=1913}}
{{Citation | last1=Sturmfels | first1=Bernd | author1-link=Bernd Sturmfels | title=Algorithms in invariant theory | publisher=Springer-Verlag | location=Berlin, New York | series=Texts and Monographs in Symbolic Computation | isbn=978-3-211-82445-0 | doi=10.1007/978-3-211-77417-5 | year=1993 | mr=1255980}}
{{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}}

External links

{{citation|first=Jeff |last=Miller|year=2010|url=http://jeff560.tripod.com/c.html |title=Earliest Known Uses of Some of the Words of Mathematics (C)}}

1 : Invariant theory

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