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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''.
In terms of [[representation theory]], given any representation ''V'' of the group ''SL''<sub>2</sub>('''C''') one can ask for the ring of invariant polynomials on ''V''. Invariants of a binary form of degree ''n'' correspond to taking ''V'' to be the (''n'' + 1)-dimensional irreducible representation, and covariants correspond to taking ''V'' to be the sum of the irreducible representations of dimensions 2 and ''n'' + 1.
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 or septimics, octics or octavics, nonics, and decics or decimics. "Quantic" is an old name for a form of arbitrary degree. Forms in 1, 2, 3, 4, ... variables are called unary, binary, ternary, quaternary, ... forms.
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