Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class,[1] and the term degeneracy is the condition of being a degenerate case.[2]

The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a degenerate triangle if at least one side length or angle is zero. Equivalently, it becomes a “line segment”.[3]

Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line,[3] which makes its dimension one. This is similar to the case of a circle, whose dimension shrinks from two to zero as it degenerates into a point.[1] As another example, the solution set of a system of equations that depends on parameters generally has a fixed cardinality and dimension, but cardinality and/or dimension may be different for some exceptional values, called degenerate cases. In such a degenerate case, the solution set is said to be degenerate.

For some classes of composite objects, the degenerate cases depend on the properties that are specifically studied. In particular, the class of objects may often be defined or characterized by systems of equations. In most scenarios, a given class of objects may be defined by several different systems of equations, and these different systems of equations may lead to different degenerate cases, while characterizing the same non-degenerate cases. This may be the reason for which there is no general definition of degeneracy, despite the fact that the concept is widely used and defined (if needed) in each specific situation.

A degenerate case thus has special features which makes it non-generic or special cases. However, not all non-generic or special cases are degenerate. For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. In fact, degenerate cases often correspond to singularities, either in the object or in some configuration space. For example, a conic section is degenerate if and only if it has singular points (e.g., point, line, intersecting lines).[4]

Contents
1 In geometry
1.1 Conic section
1.2 Triangle
1.3 Rectangle
1.4 Convex polygon
1.5 Convex polyhedron
1.6 Standard torus
1.7 Sphere
1.8 Other
2 Elsewhere
3 See also