homogeneous polynomial

Thân thiện

A student writes a homogeneous polynomial on the chalkboard.

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.

Học thuật

Noun

1. a polynomial consisting of terms all of the same degree