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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''.
A binary form (of degree ''n'') is a homogeneous polynomial ''a''<sub>''n''</sub>''x''<sup>''n''</sup> + ''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.
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