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. Glossary of invariant theory A binary form (of degree n) is a homogeneous polynomial Σ ()an−ixn−iyi = anxn + ()an−1xn−1y + ... + a0yn. The group SL2(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 a0, ..., an and on the polynomials in these variables. An invariant is a polynomial in these n + 1 variables a0, ..., an that is invariant under this action. More generally a covariant is a polynomial in a0, ..., an, 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 SL2(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. A form f is itself a covariant of degree 1 and order n. The discriminant of a form is an invariant. The resultant of two forms is a simultaneous invariant of them. The Hessian covariant of a form is the determinant of the Hessian matrix It is a covariant of order 2n− 4 and degree 2.