nonvanishing

Nonvanishing is a term used in mathematics to describe a function, vector field, or matrix that never assumes the value zero on a given domain. If a function f is defined on a domain D and f(x) ≠ 0 for every x in D, f is said to be nonvanishing on D; when this holds for every point of the domain, the function is often described as nowhere vanishing. The phrase is widely used across analysis, algebra, and geometry.

In complex analysis, a holomorphic function that is nonvanishing on a connected domain has a holomorphic logarithm, since log f can be defined without branch cuts. Zeros of analytic functions are central to many theorems, and a nonvanishing function has no zeros and thus possesses a simple, zero-free behavior on the domain.

In linear algebra, a matrix or linear operator is often described as having a nonvanishing determinant, meaning its determinant is nonzero. This implies invertibility and a well-defined inverse, with important consequences for solving linear systems and understanding linear transformations.

In differential geometry and topology, a nonvanishing (or nowhere vanishing) vector field is a vector field with no zero vectors at any point. The existence of such fields on a manifold depends on its topology; for example, the sphere S^2 has Euler characteristic 2 and cannot support a nowhere-vanishing continuous tangent vector field.

Examples include the exponential function e^x, which never equals zero, and a matrix with det ≠ 0. The concept is often paired with notions like zeros, orders of vanishing, and zero sets to describe how and where a function or section may vanish.