wellfactorized

Wellfactorized is not a standard mathematical term; in practice it is used informally to describe a factorization that meets predefined quality criteria in a given context. The exact meaning depends on the field, but the core idea is that the object has been decomposed into factors in a way that is considered clean, canonical, or efficient.

In algebra, a well-factorized element is typically one expressed as a product of irreducible (atomic) factors, with minimal redundancy and, in domains with unique factorization, uniqueness up to units and order. A square-free factorization—no repeated irreducible factors—is often described as well factorized. In other rings or modules, the term may refer to a factorization into atoms stable under the operations of interest.

In computational contexts, well-factorized representations emphasize practical properties such as canonical ordering of factors, minimal representation size, and numerical stability. For polynomials, a well-factorized form commonly means a product of irreducible factors over a specified field.

Examples: over the rational field, the polynomial x^4 - 5x^2 + 6 factors as (x^2-2)(x^2-3); if further factorization is possible (e.g., into linear factors), a well-factorized version would present those irreducible factors with a standard order.

Limitations: as a term, wellfactorized lacks a universal definition; users should consult the context or documentation to determine the intended criteria.

See also: factorization, irreducible, square-free factorization, unique factorization domain, polynomial factorization.