Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have the same total degree.[1] 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. An algebraic form, or simply form, is another name for a homogeneous polynomial.

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 homogeneous polynomial of degree 1 is a linear form,[2]. A homogeneous polynomial of degree 2 is a quadratic form.

Homogeneous polynomials are ubiquitous in mathematics and physics. 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.

Algebraic forms in general

Algebraic form, or simply form, is another term for homogeneous polynomial. These then generalise from quadratic forms to degrees 3 and more, and have in the past also been known as quantics (a term that originated with Cayley). To specify a type of form, one has to give its degree of a form, and number of variables n. A form is over some given field K, if it maps from Kn to K, where n is the number of variables of the form.

A form f over some field K in n variables represents 0 if there exists an element

(x₁,...,x_n)

in Kⁿ with f(x₁,...,x_n) =0 such that at least one of the x_i is not equal to zero.

A quadratic form over the field of the real numbers represents 0 if and only if it is not definite.
Basic properties

The number of different homogeneous monomials of degree M in N variables is \frac{(M+N-1)!}{M!(N-1)!}

The Taylor series for a homogeneous polynomial P expanded at point x may be written as

\( \begin{matrix} P(x+y)= \sum_{j=0}^n {n \choose j} P ( &\underbrace{x,x,\dots ,x}, & \underbrace{y,y,\dots ,y} ). \\ & j & n-j\\ \end{matrix} \)

Another useful identity is

\( \begin{matrix} P(x)-P(y)= \sum_{j=0}^{n-1} {n \choose j} P ( &\underbrace{y,y,\dots ,y}, & \underbrace{(x-y),(x-y),\dots ,(x-y)} ). \\ & j & n-j\\ \end{matrix} \)

Homogenization

A non-homogeneous polynomial P(x_1,x_2,\cdots x_n) can be homogenized by introducing an additional variable x_0 and defining[3]

\( h(P(x_1,x_2,\cdots x_n)) = x_0^d P(\frac{x_1}{x_0},\frac{x_2}{x_0},\cdots \frac{x_n}{x_0}), \)

where d is the degree of P. For example, \( h(x_3^3 + x_1 x_2+7)=x_3^3 + x_0 x_1x_2 + 7 x_0^3. \)

A homogenized polynomial can be dehomogenized by setting the additional variable \( x_0=1. \)
History

Algebraic forms played an important role in nineteenth century mathematics.

The two obvious areas where these would be applied were projective geometry, and number theory (then less in fashion). The geometric use was connected with invariant theory. There is a general linear group acting on any given space of quantics, and this group action is potentially a fruitful way to classify certain algebraic varieties (for example cubic hypersurfaces in a given number of variables).

In more modern language the spaces of quantics are identified with the symmetric tensors of a given degree constructed from the tensor powers of a vector space V of dimension m. (This is straightforward provided we work over a field of characteristic zero). That is, we take the n-fold tensor product of V with itself and take the subspace invariant under the symmetric group as it permutes factors. This definition specifies how GL(V) will act.

It would be a possible direct method in algebraic geometry, to study the orbits of this action. More precisely the orbits for the action on the projective space formed from the vector space of symmetric tensors. The construction of invariants would be the theory of the co-ordinate ring of the 'space' of orbits, assuming that 'space' exists. No direct answer to that was given, until the geometric invariant theory of David Mumford; so the invariants of quantics were studied directly. Heroic calculations were performed, in an era leading up to the work of David Hilbert on the qualitative theory.

For algebraic forms with integer coefficients, generalisations of the classical results on quadratic forms to forms of higher degree motivated much investigation.
See also

diagonal form
graded algebra
Homogeneous function
multilinear form
multilinear map
polarization of an algebraic form
Schur polynomial
Symbol of a differential operator

Notes
Note notes
General notes

^ D. Cox, J. Little, D. O'Shea: Using Algebraic Geometry, 2nd ed., page 2. Springer-Verlag, 2005.
^ Linear form has to be distinguished from linear functional, which is the function defined by a linear form.
^ D. Cox, J. Little, D. O'Shea: Using Algebraic Geometry, 2nd ed., page 35. Springer-Verlag, 2005.

External links

Weisstein, Eric W., "Homogeneous Polynomial" from MathWorld.

Mathematics Encyclopedia