In
mathematics, a homogeneous polynomial, sometimes called
quantic in older texts, is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, which may be expressed as a homogeneous function of the coordinates over any
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
.
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
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the
Euclidean distance is the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
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.
Properties
A homogeneous polynomial defines a
homogeneous function. This means that, if a
multivariate polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
''P'' is homogeneous of degree ''d'', then
:
for every
in any
field containing the
coefficients of ''P''. Conversely, if the above relation is true for infinitely many
then the polynomial is homogeneous of degree ''d''.
In particular, if ''P'' is homogeneous then
:
for every
This property is fundamental in the definition of a
projective variety.
Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the homogeneous components of the polynomial.
Given a
polynomial ring