Nullform

Nullform is a term used in mathematics, particularly in invariant theory and algebraic geometry, to denote a vector or homogeneous polynomial that lies in the nullcone of a group action. Given a linear action of a reductive algebraic group G on a finite-dimensional vector space V over an algebraically closed field of characteristic zero, the nullcone N is the set of vectors v in V for which every G-invariant polynomial function on V vanishes at v. A nonzero element of N is often called a nullform. Equivalently, a v is a nullform if the G-orbit of v has 0 in its Zariski closure, meaning invariant functions cannot separate v from the origin.

Nullforms are central to geometric invariant theory because they characterize unstable points under the group action and determine the structure of orbits and quotients. They arise in the study of classical invariant theory of forms, as well as modern problems in representation theory and algebraic geometry. Methods to identify nullforms include evaluating invariant polynomials, studying the action on associated graded spaces, and using the Hilbert–Mumford criterion for stability.

Historically the term appears in 19th and early 20th century invariant theory literature and was later formalized within Mumford's framework of geometric invariant theory. While the notion is specialized, it provides a concise way to describe elements indistinguishable from zero by invariants and plays a role in computational invariant theory and the analysis of moduli spaces.

See also: nullcone; invariant theory; geometric invariant theory; orbit closure; stability.