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Inmathematics, ahomogeneous polynomial, sometimes calledquantic in older texts, is apolynomial whose nonzero terms all have the samedegree.^[1] For example,${\displaystyle x^5+2x^3{y}^2+9x{y}^4}$ is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial${\displaystyle x^3+3x^{2}y+z^7}$ is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always ahomogeneous function.

Analgebraic form, or simplyform, is afunction defined by a homogeneous polynomial.^{[notes 1]} Abinary form is a form in two variables. Aform is also a function defined on avector space, which may be expressed as a homogeneous function of the coordinates over anybasis.

A polynomial of degree 0 is always homogeneous; it is simply an element of thefield orring of the coefficients, usually called a constant or a scalar. A form of degree 1 is alinear form.^{[notes 2]} A form of degree 2 is aquadratic form. Ingeometry, theEuclidean distance is thesquare root of a quadratic form.

Homogeneous polynomials are ubiquitous in mathematics and physics.^{[notes 3]} They play a fundamental role inalgebraic geometry, as aprojective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials.

A homogeneous polynomial defines ahomogeneous function. This means that, if amultivariate polynomialP is homogeneous of degreed, then

${\displaystyle P(\lambda x_1,\dots ,\lambda x_n)={\lambda }^{d}P(x_1,\dots ,x_n),}$

for every${\displaystyle \lambda }$ in anyfield containing thecoefficients ofP. Conversely, if the above relation is true for infinitely many${\displaystyle \lambda }$ then the polynomial is homogeneous of degreed.

In particular, ifP is homogeneous then

${\displaystyle P(x_1,\dots ,x_n)=0\Rightarrow P(\lambda x_1,\dots ,\lambda x_n)=0,}$

for every${\displaystyle \lambda .}$ This property is fundamental in the definition of aprojective variety.

Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called thehomogeneous components of the polynomial.

Given apolynomial ring${\displaystyle R=K[x_1,\dots ,x_n]}$ over afield (or, more generally, aring)K, the homogeneous polynomials of degreed form avector space (or amodule), commonly denoted${\displaystyle R_d.}$ The above unique decomposition means that${\displaystyle R}$ is thedirect sum of the${\displaystyle R_d}$ (sum over allnonnegative integers).

The dimension of the vector space (orfree module)${\displaystyle R_d}$ is the number of different monomials of degreed inn variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degreed inn variables). It is equal to thebinomial coefficient

${\displaystyle (\genfrac{}{}{0ex}{}{d+n-1}{n-1})=(\genfrac{}{}{0ex}{}{d+n-1}{d})=\frac{(d+n-1)!}{d!(n-1)!}.}$

Homogeneous polynomial satisfyEuler's identity for homogeneous functions. That is, ifP is a homogeneous polynomial of degreed in the indeterminates${\displaystyle x_1,\dots ,x_n,}$ one has, whichever is thecommutative ring of the coefficients,

${\displaystyle dP=\sum _{i=1}^n{x}_i\frac{\mathrm{\partial }P}{\mathrm{\partial }{x}_i},}$

where${\displaystyle \frac{\mathrm{\partial }P}{\mathrm{\partial }{x}_i}}$ denotes theformal partial derivative ofP with respect to${\displaystyle x_i.}$

A non-homogeneous polynomialP(x₁,...,x_n) can be homogenized by introducing an additional variablex₀ and defining the homogeneous polynomial sometimes denoted^hP:^[2]

${\displaystyle {}^{h}P(x_0,x_1,\dots ,x_n)=x_0^{d}P\left(\frac{x_1}{x_0},\dots ,\frac{x_n}{x_0}\right),}$

whered is thedegree ofP. For example, if

${\displaystyle P(x_1,x_2,x_3)=x_3^3+x_1{x}_2+7,}$

then

${\displaystyle {}^{h}P(x_0,x_1,x_2,x_3)=x_3^3+x_0{x}_1{x}_2+7x_0^3.}$

A homogenized polynomial can be dehomogenized by setting the additional variablex₀ = 1. That is

${\displaystyle P(x_1,\dots ,x_n)={}^{h}P(1,x_1,\dots ,x_n).}$

• Multi-homogeneous polynomial
• Quasi-homogeneous polynomial
• Diagonal form
• Graded algebra
• Hilbert series and Hilbert polynomial
• Multilinear form
• Multilinear map
• Polarization of an algebraic form
• Schur polynomial
• Symbol of a differential operator

• Media related toHomogeneous polynomials at Wikimedia Commons
• Weisstein, Eric W."Homogeneous Polynomial".MathWorld.

• Homogeneous polynomials
• Multilinear algebra
• Algebraic geometry

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