Homogeneous polynomials

In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial 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.

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 _1, \ldots,x_n/math> 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

:$\binom=\binom=\frac.$

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 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'':
:$(x_0,x_1,\dots, x_n) = x_0^d P \left (\frac,\dots, \frac \right ),$
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 ''x''₀ = 1. That is
:$P(x_1,\dots, x_n)=(1,x_1,\dots, x_n).$

See also

* 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

Notes

References

External links

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{{Polynomials

Multilinear algebra
Algebraic geometry