Homogeneous Polynomial

In mathematics, a homogeneous polynomial 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. An algebraic form, or simply form, is another name for a homogeneous polynomial.

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 homogeneous polynomial of degree 1 is a linear form,. A homogeneous polynomial of degree 2 is 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.

Read more about Homogeneous Polynomial:  Algebraic Forms in General, Basic Properties, Homogenization, History

Famous quotes containing the word homogeneous:

    O my Brothers! love your Country. Our Country is our home, the home which God has given us, placing therein a numerous family which we love and are loved by, and with which we have a more intimate and quicker communion of feeling and thought than with others; a family which by its concentration upon a given spot, and by the homogeneous nature of its elements, is destined for a special kind of activity.
    Giuseppe Mazzini (1805–1872)