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, is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial 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 edit

A homogeneous polynomial defines a homogeneous function. This means that, if a multivariate polynomial P is homogeneous of degree d, then


for every   in any field containing the coefficients of P. Conversely, if the above relation is true for infinitely many   then the polynomial is homogeneous of degree d.

In particular, if P is homogeneous then


for every   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   over a field (or, more generally, a ring) K, the homogeneous polynomials of degree d form a vector space (or a module), commonly denoted   The above unique decomposition means that   is the direct sum of the   (sum over all nonnegative integers).

The dimension of the vector space (or free module)   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


Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if P is a homogeneous polynomial of degree d in the indeterminates   one has, whichever is the commutative ring of the coefficients,


where   denotes the formal partial derivative of P with respect to  

Homogenization edit

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


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




A homogenized polynomial can be dehomogenized by setting the additional variable x0 = 1. That is


See also edit

Notes edit

  1. ^ 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.
  2. ^ 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.
  3. ^ Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems.

References edit

  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.
  2. ^ Cox, Little & O'Shea 2005, p. 35

External links edit