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In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree. For example, $x^{5}+2x^{3}y^{2}+9xy^{4}$ is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial $x^{3}+3x^{2}y+z^{7}$ 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 binary form is a form in two variables.

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. 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 the degree d and the 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 such that f(x₁,...,x_n) = 0 and 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 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) is the binomial coefficient

${\binom {d+n-1}{n-1}}={\binom {d+n-1}{d}}={\frac {(d+n-1)!}{d!(n-1)!}}.$

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:

${^{h}\!P}(x_{0},x_{1},\cdots x_{n})=x_{0}^{d}P\left({\frac {x_{1}}{x_{0}}},\cdots {\frac {x_{n}}{x_{0}}}\right),$

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

$P=x_{3}^{3}+x_{1}x_{2}+7,$

then

$^{h}\!P=x_{3}^{3}+x_{0}x_{1}x_{2}+7x_{0}^{3}.$

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

$P(x_{1},\cdots x_{n})={^{h}\!P}(1,x_{1},\cdots x_{n}).$

See also

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

References

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