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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 examp ...
whose nonzero terms all have the same degree. For example, $x^5 + 2 x^3 y^2 + 9 x y^4$ is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial $x^3 + 3 x^2 y + z^7$ 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 In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
. 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-orien ...
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 (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
, 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 mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
. 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 c ...
, the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...
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 . E ...
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 In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
. 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 examp ...
''P'' is homogeneous of degree ''d'', then :$P\left(\lambda x_1, \ldots, \lambda x_n\right)=\lambda^d\,P\left(x_1,\ldots,x_n\right)\,,$ for every $\lambda$ in any field containing the
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves va ...
s of ''P''. Conversely, if the above relation is true for infinitely many $\lambda$ then the polynomial is homogeneous of degree ''d''. In particular, if ''P'' is homogeneous then :$P\left(x_1,\ldots,x_n\right)=0 \quad\Rightarrow\quad P\left(\lambda x_1, \ldots, \lambda x_n\right)=0,$ for every $\lambda.$ This property is fundamental in the definition of a
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
. 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 In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variab ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
(or a module), commonly denoted $R_d.$ The above unique decomposition means that $R$ is the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of the $R_d$ (sum over all
nonnegative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s). The dimension of the vector space (or free module) $R_d$ is the number of different monomials of degree ''d'' in ''n'' variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degree ''d'' in ''n'' variables). It is equal to the
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
:$\binom=\binom=\frac.$ Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if is a homogeneous polynomial of degree in the indeterminates $x_1, \ldots, x_n,$ one has, whichever is the
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
of the coefficients, :$dP=\sum_^n x_i\frac,$ where $\textstyle \frac$ denotes the formal partial derivative of with respect to $x_i.$

Homogenization

A non-homogeneous polynomial ''P''(''x''''1'',...,''x''''n'') can be homogenized by introducing an additional variable ''x''0 and defining the homogeneous polynomial sometimes denoted ''h''''P'': :$\left(x_0,x_1,\dots, x_n\right) = x_0^d P \left \left(\frac,\dots, \frac \right \right),$ where ''d'' is the degree of ''P''. For example, if :$P=x_3^3 + x_1 x_2+7,$ then :$^h\!P=x_3^3 + x_0 x_1x_2 + 7 x_0^3.$ A homogenized polynomial can be dehomogenized by setting the additional variable ''x''0 = 1. That is :$P\left(x_1,\dots, x_n\right)=\left(1,x_1,\dots, x_n\right).$

* Multi-homogeneous polynomial *
Quasi-homogeneous polynomial In algebra, a multivariate polynomial : f(x)=\sum_\alpha a_\alpha x^\alpha\text\alpha=(i_1,\dots,i_r)\in \mathbb^r \text x^\alpha=x_1^ \cdots x_r^, is quasi-homogeneous or weighted homogeneous, if there exist ''r'' integers w_1, \ldots, w_r, called ...
* Diagonal form *
Graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
* Hilbert series and Hilbert polynomial *
Multilinear form In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map :f\colon V^k \to K that is separately ''K''- linear in each of its ''k'' arguments. More generally, one can define multilinear forms on ...
*
Multilinear map In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W ar ...
* Polarization of an algebraic form * Schur polynomial *
Symbol of a differential operator In mathematics, the symbol of a linear differential operator is a polynomial representing a differential operator, which is obtained, roughly speaking, by replacing each partial derivative by a new variable. The symbol of a differential operator ...