Lowerdegree

Lowerdegree is a descriptive term used in mathematics to refer to parts of a polynomial, series, or function that have lower degree than a specified reference degree. It is not a fixed, standalone object with a single formal definition, but rather a way to talk about a decomposition or comparison of degrees.

In one-variable polynomials, the lower-degree portion of a polynomial p(x) relative to its leading term is the sum of all terms with degree less than the highest degree. For example, if p(x) = 3x^4 + 2x^3 + x^2 + 7x + 5, the lower-degree part (relative to degree 4) is 2x^3 + x^2 + 7x + 5. The constant term 5 is a degree-0 component, often grouped with the lower-degree terms.

In multivariate polynomials, the total degree of a monomial is the sum of its exponents. Lower-degree terms are those whose total degree is less than a chosen threshold or, more generally, less than the leading degree of a given polynomial or subset under consideration. This concept is useful in asymptotic analysis, polynomial approximation, and polynomial decomposition.

People sometimes study lower-degree components to simplify computations, perform Taylor or Maclaurin expansions, or separate leading behavior from subordinate contributions. The notion extends to power series, where lower-degree terms are the initial terms of increasing degree, and to algorithmic contexts where lower-degree polynomials are processed before higher-degree ones.

See also: Polynomial, Degree, Taylor series, Asymptotic notation, Polynomial decomposition.