Homogeneous polynomial

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a homogeneous polynomial, sometimes called quantic in older texts, is a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> whose nonzero terms all have the same <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a>. For example, 
 
 
 
 
 x
 
 5
 
 
 +
 2
 
 x
 
 3
 
 
 
 y
 
 2
 
 
 +
 9
 x
 
 y
 
 4
 
 
 
 
 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}}
 
 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial 
 
 
 
 
 x
 
 3
 
 
 +
 3
 
 x
 
 2
 
 
 y
 +
 
 z
 
 7
 
 
 
 
 {\displaystyle x^{3}+3x^{2}y+z^{7}}
 
 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 <a href="/facts/Homogeneous_function/wF9XX7s5">homogeneous function</a>. 
An algebraic form, or simply form, is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> defined by a homogeneous polynomial. A binary form is a form in two variables. A form is also a function defined on a <a href="/facts/Vector_space/M1pxMLx2">vector space</a>, which may be expressed as a homogeneous function of the coordinates over any <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a>.
A polynomial of degree 0 is always homogeneous; it is simply an element of the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> or <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> of the coefficients, usually called a constant or a scalar. A form of degree 1 is a <a href="/facts/Linear_form/oGh7Asrz">linear form</a>. A form of degree 2 is a <a href="/facts/Quadratic_form/l2n1jXPe">quadratic form</a>. In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the <a href="/facts/Euclidean_distance/9qDoQKQe">Euclidean distance</a> is the <a href="/facts/Square_root/AfrzfBdQ">square root</a> of a quadratic form.
Homogeneous polynomials are ubiquitous in mathematics and physics. They play a fundamental role in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, as a <a href="/facts/Projective_algebraic_variety/aB2XdB7B">projective algebraic variety</a> is defined as the set of the common zeros of a set of homogeneous polynomials.

Homogeneous polynomial open-in-new

Homogeneous polynomial