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In mathematical [[invariant theory]], an '''invariant of a binary form''' is a polynomial in the coefficients of a [[binary form]] in two variables ''x'' and ''y'' that remains invariant under unimodular transformations of the variables ''x'' and ''y''.
==Terminology==
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. The invariants are polynomials in these ''n''+1 variables ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub> that are invariant under this action.
The invariants of a binary form are a graded algebra, and {{harvtxt|Gordan|1868}} proved that this algebra is finitely generated if the base field is the complex numbers.
Forms of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10 are sometimes called quadrics, cubic, quartics, quintics, sextics, septics, octavics, nonics, and decimics. "Quantic" is an old name for a form of arbitrary degree. Forms in 1, 2, 3, ... variables are called unary, binary, ternary, ... forms.
==The ring of invariants==
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