Homogeneous polynomial explained

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} + 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.^[2] 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.^[3] 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.^[4] 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( λ x _{1,} \ldots, λ x ^{dP(x} _{1 ,\ldots,x} _{n),} for every λ in any field containing the coefficients of P. Conversely, if the above relation is true for infinitely many λ then the polynomial is homogeneous of degree d.

In particular, if P is homogeneous then

P(x _{1,\ldots,x} _{n)=0} ⇒ P( λ x _{1,} \ldots, λ x _{n)=0,} for every λ . 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.

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} = ((d + n -1 )!)/(d!(n -1 )!) .

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} _{i =1} x _{i (\partial P)/(\partial x _{i})} , where style (\partial P)/(\partial x _{i}) denotes the formal partial derivative of 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:

{ ^{hP}(x} _{0 ,x} _{1,...,} x _{n)} = x ^{d} _{0} P \left ( (x _{1})/(x _{0}) , ... , (x _{n})/(x _{0}) \right ), where d is the degree of P. For example, ifP(x _{1,x} _{2,x} _{3)=x} ^{3} ^{3} + x _{1} x _{2+7,} then^{hP(x} ^{0 ,x} _{1,x} _{2,x} _{3)=x} ^{3} ^{3} + x _{0} x _{1x} _{2} + 7 x ^{3 .} _{0}

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

P(x _{1,...,} x ^{hP}( 1 ,x} _{1 , ... ,} 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 and References

1. Book: Cox . David A. . David A. Cox. Little . John . O'Shea . Donal . Donal O'Shea. 2005 . Using Algebraic Geometry . 2nd . Graduate Texts in Mathematics . Springer . 978-0-387-20733-9 . 185 . 2 .
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.