Riducibili

Riducibili is an Italian term used in mathematics and related disciplines to describe objects that can be reduced or decomposed into simpler components. The concept is closely tied to the distinction between reducible and irreducible objects, where irreducible means that no nontrivial decomposition exists.

In algebra, polynomials are called reducibile over a field if they can be factored as a product of polynomials with smaller positive degrees. If no such factorization exists, the polynomial is irreducibile. This distinction underpins factorization theory and algorithms for polynomial factoring.

In linear algebra and representation theory, a vector space or a module with a group action is called reducibile if it contains a nontrivial invariant subspace or submodule, allowing a decomposition into simpler subrepresentations or direct sums. When every representation is a direct sum of irreducible subrepresentations, the representation is said to be completely reducible.

In algebraic geometry, a polynomial equation or a variety is described as reducibile if its zero set splits into two or more proper subvarieties. This mirrors the idea that the geometric object can be viewed as a union of simpler components.

The term also appears in physics, where reducible representations often arise in the study of symmetries and their associated operators, with irreducible representations acting as the fundamental building blocks.

Etymology-wise, riducibili derives from ridurre, “to reduce.” See also reducible, irreducible, and complete reducibility.