Invariante

Invariante, or invariant, is a property of an object that remains unchanged under a specified set of transformations or operations. In mathematics and related fields, invariants help classify objects, prove theorems, and constrain possible behavior by focusing on features that do not vary.

Geometric and algebraic invariants are common. Distances and angles are invariant under rigid motions in the plane or space. More advanced examples include the cross ratio, a projective invariant that does not change under projective transformations, and the determinant of a matrix, which remains the same under similarity transformations. In group theory and linear algebra, an invariant can be a quantity f(x) that satisfies f(g·x) = f(x) for all elements g of a group acting on x.

Topological invariants are properties preserved under continuous deformations, such as Euler characteristic and genus. In dynamical systems, conserved quantities like energy or momentum remain constant along trajectories, and invariant measures describe distribution properties preserved by the dynamics.

In computer science, invariants play a practical role in program verification and algorithm design. Loop invariants are properties that hold at each iteration of a loop, helping to establish correctness, while data invariants ensure consistency across transformations or operations.

The concept derives from Latin invariantem and appears across many disciplines. Invariants provide a unifying idea: by identifying fixed features, one can understand, compare, and reason about complex systems despite changes in appearance or state.