reguljära

Reguljära is a Swedish adjective meaning regular or non-singular, and it is used primarily in mathematical and scientific contexts to describe objects or points that behave in a well-defined, smooth way. The term appears in phrases such as reguljära punkter or reguljära kartor, depending on the domain, and serves to distinguish ordinary, well-behaved cases from singular or degenerate ones.

In algebraic geometry, a key usage is reguljär punkt (regular point). A point on a geometric object is reguljär if it is non-singular, meaning the local structure is smooth. For a plane curve defined by F(x, y) = 0, a point p is reguljär if the gradient ∇F(p) is not zero. By the implicit function theorem, the curve is then locally a smooth one-dimensional manifold near p. More generally, a point on a variety is reguljär when the local ring is regular, which is equivalent to having a well-behaved tangent space whose dimension matches the dimension of the variety.

In differential geometry and topology, reguljära can describe maps with good differential properties. A map between manifolds is reguljär if its differential has maximal rank at every point, making it a submersion or immersion in the appropriate context. Such regularity conditions guarantee stable behavior under perturbations and enable constructions like transversality and the formation of smooth submanifolds.

The concept of regularity is used across disciplines to avoid pathological cases and to apply standard theorems that assume smoothness or non-singularity. While the specifics of what counts as reguljär can vary by field, the core idea is the same: regular, well-behaved, and free of singularities.