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==Terminology==
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A binary form (of degree ''n'') is a homogeneous polynomial Σ{{su|b=''i''=0|p=''n''}} ({{su|p=''n''|b=''i''}})''a''<sub>''n''−''i''</sub>''x''<sup>''n''−''i''</sup>''y''<sup>''i''</sup> = ''a''<sub>''n''</sub>''x''<sup>''n''</sup> + ({{su|p=''n''|b=1}})''a''<sub>''n''−1</sub>''x''<sup>''n''−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'' + ''by'' and ''y'' to ''cx'' + ''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'' + 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 ''y''.
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*{{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}}
*{{Citation | last1=von Gall | first1=August Freiherr | title=Das vollständige Formensystem der binären Form 7<sup>ter</sup>Ordnung | doi=10.1007/BF01206218 | year=1888 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=31 | issue=3 | pages=318–336 | mr=1510486}}
*{{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=
*{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | url=https://books.google.com/books?isbn=0521449030|title=Theory of algebraic invariants | origyear=1897 | publisher=[[Cambridge University Press]] | isbn=978-0-521-44457-6 | year=1993 | mr=1266168}}
*{{Citation | last1=Kung | first1=Joseph P. S. | last2=Rota | first2=Gian-Carlo | author2-link=Gian-Carlo Rota | title=The invariant theory of binary forms | url=http://www.ams.org/journals/bull/1984-10-01/S0273-0979-1984-15188-7 | doi=10.1090/S0273-0979-1984-15188-7 | year=1984 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=10 | issue=1 | pages=27–85 | mr=722856}}
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