homogeneous polynomial

{{short description|Polynomial whose nonzero terms all have the same degree}}

In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree.{{cite book |last1=Cox |first1=David A. |author1link = David A. Cox|last2=Little |first2=John |last3=O'Shea |first3=Donal |author3link = Donal O'Shea|date=2005 |title=Using Algebraic Geometry |edition=2nd |series=Graduate Texts in Mathematics |publisher=Springer |isbn=978-0-387-20733-9 |volume=185 |page=2 |url=https://books.google.com/books?id=QFFpepgQgT0C&pg=PP1}} 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.

An algebraic form, or simply form, is a function defined by a homogeneous polynomial.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. 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.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. 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.Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems. 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 {{math|P}} is a homogeneous polynomial of degree {{math|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 {{math|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:{{harvnb|Cox|Little|O'Shea|2005|p=35}}

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

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

: $^h\!P(x_0,x_1,x_2,x_3)=x_3^3 + x_0 x_1x_2 + 7 x_0^3.$

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

References

External links

{{Commonscatinline|Homogeneous polynomials}}
{{mathworld|urlname=HomogeneousPolynomial|title = Homogeneous Polynomial}}

Category:Multilinear algebra

Category:Algebraic geometry