homogeneous polynomial
Definition
Noun: A homogeneous polynomial is a polynomial in which every non-zero term has the same total degree. The total degree of a term is the sum of the exponents of the variables in that term.
Usage
A homogeneous polynomial is characterized by the uniform degree of all its constituent monomials. It is a standard concept in algebra and algebraic geometry.
Examples
Advanced Usage
- Homogenization: The process of converting a non-homogeneous polynomial into a homogeneous one by introducing an auxiliary variable. For example, (x^2 + x + 1) can be homogenized to (x^2 + xz + z^2).
- Homogeneous Equation: An equation set to zero, (P(x, y, ...) = 0), where (P) is a homogeneous polynomial. Such equations define homogeneous algebraic sets.
Variants and Related Words
- Homogeneous (adjective): Of the same kind or nature; uniform in structure or composition. In mathematics, it describes functions or polynomials with the scaling property (f(\lambda \mathbf{x}) = \lambda^k f(\mathbf{x})).
- Polynomial (noun): An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
- Form (noun): Often used synonymously with "homogeneous polynomial," especially in contexts like "quadratic form" (degree 2) or "cubic form" (degree 3).
Synonyms
- Form (in the context of algebra)
- Homogeneous form
Related Concepts
- Degree: The sum of the exponents of the variables in a term; for a homogeneous polynomial, this is constant for all terms.
- Monomial: A single term of a polynomial, a product of a constant and variables raised to non-negative integer powers. A homogeneous polynomial is a sum of monomials of identical degree.
Noun
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a polynomial consisting of terms all of the same degree
