Invariant of a binary form: Difference between revisions

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==Terminology==
 
{{main article|Glossary of invariant theory}}
A binary form (of degree ''n'') is a homogeneous polynomial &Sigma;{{su|b=''i''=0|p=''n''}} ({{su|p=''n''|b=''i''}})''a''<sub>''n''&minus;''i''</sub>''x''<sup>''n''&minus;''i''</sup>''y''<sup>''i''</sup> = ''a''<sub>''n''</sub>''x''<sup>''n''</sup> + ({{su|p=''n''|b=1}})''a''<sub>''n''&minus;1</sub>''x''<sup>''n''&minus;1</sup>''y'' + ... + ''a''<sub>0</sub>''y''<sup>''n''</sup>. The group ''SL''<sub>2</sub>('''C''') acts on these forms by taking ''x'' to ''ax''&nbsp;+&nbsp;''by'' and ''y'' to ''cx''&nbsp;+&nbsp;''dy''. This induces an action on the space spanned by ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub> and on the polynomials in these variables. An '''invariant''' is a polynomial in these ''n''&nbsp;+&nbsp;1 variables ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub> that is invariant under this action. More generally a '''covariant''' is a polynomial in ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub>, ''x'', ''y'' that is invariant, so an invariant is a special case of a covariant where the variables ''x'' and ''y'' do not occur. More generally still, a '''simultaneous invariant''' is a polynomial in the coefficients of several different forms in ''x'' and&nbsp;''y''.
 
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*{{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}}
*{{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}}
 
==External links==