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 {\displaystyle 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 + 3 x 2 y + z 7 {\displaystyle x^{3}+3x^{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. 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. 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. 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 , … , λ x n ) = λ d P ( x 1 , … , x n ) , {\displaystyle P(\lambda x_{1},\ldots ,\lambda x_{n})=\lambda ^{d}\,P(x_{1},\ldots ,x_{n})\,,}
for every λ {\displaystyle \lambda } in any field containing the coefficients of P. Conversely, if the above relation is true for infinitely many λ {\displaystyle \lambda } then the polynomial is homogeneous of degree d. In particular, if P is homogeneous then
P ( x 1 , … , x n ) = 0 ⇒ P ( λ x 1 , … , λ x n ) = 0 , {\displaystyle P(x_{1},\ldots ,x_{n})=0\quad \Rightarrow \quad P(\lambda x_{1},\ldots ,\lambda x_{n})=0,}
for every λ . {\displaystyle \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 [ x 1 , … , x n ] {\displaystyle 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 . {\displaystyle R_{d}.} The above unique decomposition means that R {\displaystyle R} is the direct sum of the R d {\displaystyle R_{d}} (sum over all nonnegative integers). The dimension of the vector space (or free module) R d {\displaystyle 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
( d + n − 1 n − 1 ) = ( d + n − 1 d ) = ( d + n − 1 ) ! d ! ( n − 1 ) ! . {\displaystyle {\binom {d+n-1}{n-1}}={\binom {d+n-1}{d}}={\frac {(d+n-1)!}{d!(n-1)!}}.}
… excerpt ends here. Continue reading the full article.
