Invariant of a binary form: Difference between revisions

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In mathematical [[invariant theory]], an '''invariant of a binary form''' is a polynomial in the coefficients of a [[Homogeneous polynomial|binary form]] in two variables ''x'' and ''y'' that remains invariant under the [[special linear group]] acting on the variables ''x'' and ''y''.

{{TOC limit|2}}

==Terminology==

{{main|Glossary of invariant theory}}

A binary form (of degree ''n'') is a homogeneous polynomial &Sigma;<math>\sum_{{su|b=''i''=0|p=''}^n \binom{n''}{i} (a_{{su|p=''n''|b=''-i''}})''a''_{''n''&minus;''i''}''x''^{''^{n''&minus;''-i''}''}y''^{''^i''} = ''a''_{''a_nx^n''}''x''^''n'' + (\binom{n}{su|p=''n''|b=1}})''a''_{'' a_{n''&minus;-1}''}x''^{''^{n''&minus;-1}''}y'' + ...\cdots + ''a''₀''y''<sup>''a_0y^n''</supmath>. The group ''SL''<submath>2SL_2(\mathbb{C})</submath>('''C''') acts on these forms by taking ''<math>x''</math> to ''<math>ax''&nbsp; +&nbsp;'' by''</math> and ''<math>y''</math> to ''<math>cx''&nbsp; +&nbsp;'' dy''</math>. This induces an action on the space spanned by ''a''<sub>0</submath>a_0, ...\ldots, ''a''<sub>''n''a_n</submath> and on the polynomials in these variables. An '''invariant''' is a polynomial in these ''<math>n''&nbsp; +&nbsp; 1</math> variables ''a''<sub>0</submath>a_0, ...\ldots, ''a''<sub>''n''a_n</submath> that is invariant under this action. More generally a '''covariant''' is a polynomial in ''a''<submath>0a_0, \ldots, a_n</submath>, ..., ''a''<submath>''n''x</submath>, ''x'', ''<math>y''</math> that is invariant, so an invariant is a special case of a covariant where the variables ''<math>x''</math> and ''<math>y''</math> do not occur. More generally still, a '''simultaneous invariant''' is a polynomial in the coefficients of several different forms in ''<math>x''</math> and&nbsp;'' <math>y''</math>.

In terms of [[representation theory]], given any representation ''<math>V''</math> of the group ''SL''<sub>2</submath>SL_2('''\mathbb{C'''})</math> one can ask for the ring of invariant polynomials on ''<math>V''</math>. Invariants of a binary form of degree ''<math>n''</math> correspond to taking ''<math>V''</math> to be the <math>(''n''&nbsp; +&nbsp; 1)</math>-dimensional irreducible representation, and covariants correspond to taking ''<math>V''</math> to be the sum of the irreducible representations of dimensions 2 and&nbsp;'' <math>n''&nbsp; +&nbsp; 1</math>.

The invariants of a binary form form a [[graded algebra]], and {{harvtxt|Gordan|1868}} proved that this algebra is finitely generated if the base field is the complex numbers.

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\frac{\partial g}{\partial x} & \frac{\partial g}{\partial y}

\end{bmatrix}</math>

is a simultaneous invariantcovariant of two forms ''f'', ''g''.

==The ring of invariants==

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===Covariants of a binary linear form===

For linear forms ''ax''<math>F_1(x,y) = Ax + ''by''By</math> the only invariants are constants. The algebra of covariants is generated by the form itself of degree 1 and order 1.

===Covariants of a binary quadric===

The algebra of invariants of the quadratic form ''ax''<supmath>F_2(x,y) = Ax^2</sup> + 2''bxy''2Bxy + ''cy''<sup>Cy^2</supmath> is a polynomial algebra in 1 variable generated by the discriminant ''b''<supmath>B^2 - AC</supmath> &minus; ''ac'' of degree 2. The algebra of covariants is a polynomial algebra in 2 variables generated by the discriminant together with the form ''f'' itself (of degree 1 and order 2). {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVI, XX}}

===Covariants of a binary cubic===

The algebra of invariants of the cubic form ''ax''<supmath>F_3(x,y) = Ax^3</sup> + 3''bx''²''y''3Bx^2y + 3''cxy''^{3Cxy^2} + ''dy''<sup>Dy^3</supmath> is a polynomial algebra in 1 variable generated by the discriminant ''D''<math>\Delta = 3''b''^{3B^2C^2}''c''² + 6''abcd''6ABCD- &minus;4B^3D 4''b''³''d''- &minus;4C^3A 4''c''³''a''- &minus; ''a''²''d''^{A^2D^2</supmath> of degree 4. The algebra of covariants is generated by the discriminant, the form itself (degree 1, order 3), the Hessian ''<math>H''</math> (degree 2, order 2) and a covariant ''<math>T''</math> of degree 3 and order 3. They are related by the [[Syzygy (mathematics)|syzygy]] 4''h''<supmath>4H^3}=''Df''^{^2}-''T''<sup>^2</supmath> of degree 6 and order 6. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVII, XX}}

===Covariants of a binary quartic===

The algebra of invariants of a quartic form is generated by invariants ''<math>i''</math>, ''<math>j''</math> of degrees 2, 3.:
<math display="block">
\begin{aligned}
F_4(x, y)&=A x^4+4 B x^3 y+6 C x^2 y^2+4 D x y^3+E y^4\\
i_{F_4}&=A E-4 B D+3 C^2 \\
j_{F_4}&=A C E+2 B C D-C^3-B^2 E-A D^2
\end{aligned}
</math>

This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series ''E''<submath>4E_4</submath> and ''E''<submath>6E_6</submath>. The algebra of covariants is generated by these two invariants together with the form ''<math>f''</math> of degree 1 and order 4, the Hessian ''<math>H''</math> of degree 2 and order 4, and a covariant ''<math>T''</math> of degree 3 and order 6. They are related by a syzygy ''jf''<supmath>jf^3</sup>&minus;'' - Hf''²''i''^2i +4''H''^{4H^3} +'' T''<sup>^2 = 0</supmath>=0 of degree 6 and order 12. {{harv|Schur|1968|loc=II.8}} {{harv|Hilbert|1993|loc=XVIII, XXII}}

===Covariants of a binary quintic===

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===Covariants of a binary septic===

The ring of invariants of binary septics is anomalous and has caused several published errors. Cayley claimed incorrectly that the ring of invariants is not finitely generated. {{harvtxt|Sylvester|Franklin|1879}} gave lower bounds of 26 and 124 for the number of generators of the ring of invariants and the ring of covariants and observed that an unproved "fundamental postulate" would imply that equality holds. However {{harvtxt|von Gall|1888}} showed that Sylvester's numbers are not equal to the numbers of generators, which are 30 for the ring of invariants and at least 130 for the ring of covariants, so Sylvester's fundamental postulate is wrong. {{harvtxt|von Gall|1888}} and {{harvtxt|Dixmier|Lazard|19861988}} showed that the algebra of invariants of a degree 7 form is generated by a set with 1 invariant of degree 4, 3 of degree 8, 6 of degree 12, 4 of degree 14, 2 of degree 16, 9 of degree 18, and one of each of the degrees 20, 22, 26, 30. {{harvtxt|Cröni|2002}} gives 147 generators for the ring of covariants.

===Covariants of a binary octavic===

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==Invariants of several binary forms==

The covariants of a binary form are essentially the same as joint invariants of a binary form and a binary linear form. More generally, on can ask for the joint invariants (and covariants) of any collection of binary forms. Some cases that have been studied are listed below.

{| class="wikitable"

|+ (basic invariant, basic covariant)

! Degree of forms !! 1 !! 2 !! 3 !! 4 !! 5

|-

| 1

| (1, 3)

|| (2, 5)

|| (4, 13)

|| (5, 20)

|| (23, 94)

|-

| 2

|

|| (3, 6)

|| (5, 15)

|| (6, 18)

|| (29, 92)

|-

| 3

|

||

||

|| (20, 63)

||(8, 28)

|-

| 4

|

||

||

||

||

|}

Notes:

===Covariants of two linear forms===

There are 1 basic invariant and 3 basic covariants.

===Covariants of a linear form and a quadratic===

There are 2 basic invariants and 5 basic covariants.

===Covariants of a linear form and a cubic===

There are 4 basic invariants (essentially the covariants of a cubic) and 13 basic covariants.

===Covariants of a linear form and a quartic===

There are 5 basic invariants (essentially the basic covariants of a quartic) and 20 basic covariants.

===Covariants of a linear form and a quintic===

There are 23 basic invariants (essentially the basic covariants of a quintic) and 94 basic covariants.

===Covariants of a linear form and a quantic===

===Covariants of several linear forms===

The ring of invariants of ''n'' linear forms is generated by ''n''(''n''–1)/2 invariants of degree 2.

The ring of covariants of ''n'' linear forms is essentially the same as the ring of invariants of ''n''+1 linear forms.

===Covariants of two quadratics===

There are 3 basic invariants and 6 basic covariants.

===Covariants of two quadratics and a linear form===

===Covariants of several linear and quadratic forms===

The ring of invariants of a sum of ''m'' linear forms and ''n'' quadratic forms

is generated by ''m''(''m''–1)/2 + ''n''(''n''+1)/2 generators in degree 2, ''nm

''(''m''+1)/2 + ''n''(''n''–1)(''n''–2)/6 in degree 3, and m''(''m''+1)''n''(''n

''–1)/4 in degree 4.

For the number of generators of the ring of covariants, change ''m'' to ''m''+1.

===Covariants of a quadratic and a cubic===

There are 5 basic invariants and 15 basic covariants

===Covariants of a quadratic and a quartic===

There are 6 basic invariants and 18 basic covariants

===Covariants of a quadratic and a quintic===

There are 29 basic invariants and 92 basic covariants

===Covariants of a cubic and a quartic===

There are 20 basic invariants and 63 basic covariants

===Covariants of two quartics===

There are 8 basic invariants (3 of degree 2, 4 of degree 3, and 1 of degree 4) and 28 basic covariants. (Gordan gave 30 covariants, but Sylvester showed that two of these are reducible.)

* The basic invariants of a linear form are essentially the same as its basic covariants.

===Covariants of many cubics or quartics===

* For two quartics, there are 8 basic invariants (3 of degree 2, 4 of degree 3, and 1 of degree 4) and 28 basic covariants. (Gordan gave 30 covariants, but Sylvester showed that two of these are reducible.)

Multiple forms:

The numbers of generators of invariants or covariants were given by {{harvtxt|Young|1899}}.

* Covariants of several linear forms: The ring of invariants of <math>n</math> linear forms is generated by <math>n(n-1)/2</math> invariants of degree 2. The ring of covariants of <math>n</math> linear forms is essentially the same as the ring of invariants of <math>n+1</math> linear forms.

* Covariants of several linear and quadratic forms:

** The ring of invariants of a sum of <math>m</math> linear forms and <math>n</math> quadratic forms is generated by <math>m(m-1)/2 + n(n+1)/2</math> generators in degree 2, <math>nm(m+1)/2 + n(n-1)(n-2)/6</math> in degree 3, and <math>m(m+1)n(n-1)/4</math> in degree 4.

** For the number of generators of the ring of covariants, change <math>m</math> to <math>m+1</math>.

* Covariants of many cubics or quartics: See {{harvtxt|Young|1898}}.

==See also==

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{{Reflist}}

*{{Citation | last1=Brouwer | first1=Andries E. | last2=Popoviciu | first2=Mihaela | title=The invariants of the binary nonic | doi=10.1016/j.jsc.2010.03.003 | year=2010a | journal=Journal of Symbolic Computation | issn=0747-7171 | volume=45 | issue=6 | pages=709–720 | mr=2639312| arxiv=1002.0761 | s2cid=30297 }}

*{{Citation | last1=Brouwer | first1=Andries E. | last2=Popoviciu | first2=Mihaela | title=The invariants of the binary decimic | doi=10.1016/j.jsc.2010.03.002 | year=2010b | journal=Journal of Symbolic Computation | issn=0747-7171 | volume=45 | issue=8 | pages=837–843 | mr=2657667| arxiv=1002.1008 | s2cid=12702092 }}

*{{Citation | first=Holger |last=Cröni |title=Zur Berechnung von Kovarianten von Quantiken |type=Dissertation |publisher=Univ. des Saarlandes |___location=Saarbrücken |year=2002}}
*{{Citation | last1=Dixmier | first1=Jacques | last2=Lazard | first2=D. | title=Minimum number of fundamental invariants for the binary form of degree 7 | doi=10.1016/S0747-7171(88)80026-9 | year=1988 | journal=Journal of Symbolic Computation | issn=0747-7171 | volume=6 | issue=1 | pages=113–115 | mr=961375| doi-access=free }}

*{{Citation | last1=von Gall | first1=August Freiherr | title=Das vollständige Formensystem einer binären Form achter Ordnung | doi=10.1007/BF01444117 | year=1880 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=17 | issue=1 | pages=31–51 | mr=1510048| s2cid=120828980 | url=https://zenodo.org/record/2373565 }}

*{{Citation | last1=von Gall | first1=August Freiherr | title=Das vollständige Formensystem der binären Form 7^terOrdnung | doi=10.1007/BF01206218 | year=1888 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=31 | issue=3 | pages=318–336 | mr=1510486| s2cid=121051862 | url=https://zenodo.org/record/2481603 }}

*{{Citation | last1=Gordan | first1=Paul | title=Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist | doi=10.1515/crll.1868.69.323 | year=1868 | journal= J.Journal F.für Mathdie reine und angewandte Mathematik | volume=691868 | pages= 323–354 | issue=69| s2cid=120689164 | url=https://zenodo.org/record/1448894 }}

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*{{Citation | last1=Schur | first1=Issai | editor1-last=Grunsky | editor1-first=Helmut | title=Vorlesungen über Invariantentheorie | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Die Grundlehren der mathematischen Wissenschaften | year=1968 | volume=143 |isbn = 978-3-540-04139-9 | mr=0229674}}

*{{Citation | last1=Shioda | first1=Tetsuji | title=On the graded ring of invariants of binary octavics | year=1967 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=89 | pages=1022–1046 | mr=0220738 | jstor=2373415 | doi=10.2307/2373415 | issue=4}}

*{{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 | last2=Franklin | first2=F. | title=Tables of the Generating Functions and Groundforms for the Binary Quantics of the First Ten Orders | doi=10.2307/2369240 | year=1879 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=2 | issue=3 | pages=223–251 | mr=1505222| jstor=2369240 }}

*{{Citation | last1=Sylvester | first1=James Joseph | title=Tables of the Generating Functions and Groundforms of the Binary Duodecimic, with Some General Remarks, and Tables of the Irreducible Syzygies of Certain Quantics | url=http://www.jstor.org/stable/=2369149 | publisher=The Johns Hopkins University Press | year=1881 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=4 | issue=1 | pages= 41–61 | doi=10.2307/2369149}}

* {{cite journal |last1=Young |first1=A. |title=The Irreducible Concomitants of any Number of Binary Quartics |journal=Proceedings of the London Mathematical Society |date=November 1898 |volume=s1-30 |issue=1 |pages=290–307 |doi=10.1112/plms/s1-30.1.290}}

==External links==