Homogeneous Polynomials
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 binary form is a form in two variables.
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 Polynomials: Algebraic Forms in General, Basic Properties, Homogenization, History
Famous quotes containing the word homogeneous:
“If we Americans are to survive it will have to be because we choose and elect and defend to be first of all Americans; to present to the world one homogeneous and unbroken front, whether of white Americans or black ones or purple or blue or green.... If we in America have reached that point in our desperate culture when we must murder children, no matter for what reason or what color, we don’t deserve to survive, and probably won’t.”
—William Faulkner (1897–1962)