In mathematics, a homogeneous polynomial 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. 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.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 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, ..., xn) in Kn such that f(x1,...,xn) = 0 and at least one of the xi is not equal to zero.
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
A non-homogeneous polynomial P(x1,...,xn) can be homogenized by introducing an additional variable x0 and defining the homogeneous polynomial sometimes denoted hP:3
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
- diagonal form
- graded algebra
- Homogeneous function
- multilinear form
- multilinear map
- polarization of an algebraic form
- Schur polynomial
- Symbol of a differential operator
- 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.
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