A binary form (of degree ''n'') is a homogeneous polynomial Σ<math>\sum_{{su|b=''i''=0|p=''}^n \binom{n''}{i} (a_{{su|p=''n''|b=''-i''}})''a''<sub>''n''−''i''</sub>''x''<sup>''^{n''−''-i''</sup>''}y''<sup>''^i''</sup> = ''a''<sub>''a_nx^n''</sub>''x''<sup>''n''</sup> + (\binom{n}{su|p=''n''|b=1}})''a''<sub>'' a_{n''−-1</sub>''}x''<sup>''^{n''−-1</sup>''}y'' + ...\cdots + ''a''<sub>0</sub>''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'' + '' by''</math> and ''<math>y''</math> to ''<math>cx'' + '' 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'' + 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 '' <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'' + 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 '' <math>n'' + 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.
