Homogeneous polynomial

In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree.^[1] For example, $x^{5}+2x^{3}y^{2}+9xy^{4}$ is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial $x^{3}+3x^{2}y+z^{7}$ 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 homogeneous function.

An algebraic form, or simply form, is a function defined by a homogeneous polynomial.^{[notes 1]} A binary form is a form in two variables. A form is also a function defined on a vector space, 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.^{[notes 2]} A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form.

Homogeneous polynomials are ubiquitous in mathematics and physics.^{[notes 3]} 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 P is homogeneous of degree d, then

P(\lambda x_{1},\ldots ,\lambda x_{n})=\lambda ^{d}\,P(x_{1},\ldots ,x_{n})\,,

for every $\lambda$ in any field containing the coefficients 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(x_{1},\ldots ,x_{n})=0\quad \Rightarrow \quad P(\lambda x_{1},\ldots ,\lambda x_{n})=0,

for every $\lambda .$ 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 $R=K[x_{1},\ldots ,x_{n}]$ over a field (or, more generally, a ring) K, the homogeneous polynomials of degree d form a vector space (or a module), commonly denoted $R_{d}.$ The above unique decomposition means that $R$ is the direct sum of the $R_{d}$ (sum over all nonnegative integers).

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

{\binom {d+n-1}{n-1}}={\binom {d+n-1}{d}}={\frac {(d+n-1)!}{d!(n-1)!}}.

Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if $P$ is a homogeneous polynomial of degree $d$ in the indeterminates $x_{1},\ldots ,x_{n},$ one has, whichever is the commutative ring of the coefficients,

dP=\sum _{i=1}^{n}x_{i}{\frac {\partial P}{\partial x_{i}}},

where $\textstyle {\frac {\partial P}{\partial x_{i}}}$ denotes the formal partial derivative of $P$ with respect to $x_{i}.$

Homogenization

A non-homogeneous polynomial P(x₁,...,x_n) can be homogenized by introducing an additional variable x₀ and defining the homogeneous polynomial sometimes denoted ^hP:^[2]

{^{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),

where d is the degree of P. For example, if

P(x_{1},x_{2},x_{3})=x_{3}^{3}+x_{1}x_{2}+7,

then

^{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 variable x₀ = 1. That is

P(x_{1},\dots ,x_{n})={^{h}\!P}(1,x_{1},\dots ,x_{n}).

Notes

^ However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms homogeneous polynomial and form are sometimes considered as synonymous.
^ Linear forms are defined only for finite-dimensional vector space, and have thus to be distinguished from linear functionals, which are defined for every vector space. "Linear functional" is rarely used for finite-dimensional vector spaces.
^ Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems.

References

^ Cox, David A.; Little, John; O'Shea, Donal (2005). Using Algebraic Geometry. Graduate Texts in Mathematics. Vol. 185 (2nd ed.). Springer. p. 2. ISBN 978-0-387-20733-9.
^ Cox, Little & O'Shea 2005, p. 35

External links

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

[2] However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms homogeneous polynomial and form are sometimes considered as synonymous.

[3] Linear forms are defined only for finite-dimensional vector space, and have thus to be distinguished from linear functionals, which are defined for every vector space. "Linear functional" is rarely used for finite-dimensional vector spaces.

[4] Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems.

[1] Cox, David A.; Little, John; O'Shea, Donal (2005). Using Algebraic Geometry. Graduate Texts in Mathematics. Vol. 185 (2nd ed.). Springer. p. 2. ISBN 978-0-387-20733-9.

[5] Cox, Little & O'Shea 2005, p. 35

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