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invariant of a binary form
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 the special linear group acting on the variables ''x'' and ''y''.
==Terminology==
(詳細はrepresentation theory, given any representation ''V'' of the group ''SL''2(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 form a graded algebra, and 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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