Homogeneous polynomials | Multilinear algebra | Algebraic geometry

Homogeneous polynomial

In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial. A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form. A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form. Homogeneous polynomials are ubiquitous in mathematics and physics. They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials. (Wikipedia).

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Related pages

Multilinear map | Vector space | Coefficient | Multilinear form | Polarization of an algebraic form | Polynomial | Projective variety | Polynomial ring | Degree of a polynomial | Quasi-homogeneous polynomial | Free module | Homogeneous function | Binomial coefficient | Mathematics | Function (mathematics) | Field (mathematics) | Formal derivative | Ring (mathematics) | Basis (linear algebra) | Direct sum | Quadratic form | Symbol of a differential operator | Euclidean distance | Square root | Diagonal form | Geometry | Commutative ring | Schur polynomial | Module (mathematics) | Hilbert series and Hilbert polynomial