Degeneracy (mathematics) - Wikipedia

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Abstraction: Limiting case of a mathematical object qualitatively simpler than the general class

Key points:

A degenerate case arises when inequalities in the definition of an object become equalities, often reducing dimension or cardinality (e.g., a triangle where all vertices are collinear)
Degeneracies often correspond to singularities in the object or its configuration space; a conic section is degenerate iff it has singular points
Examples include: point as degenerate circle (radius=0), line segment as degenerate ellipse (semi-minor axis=0), two lines as degenerate hyperbola
In probability, a random variable taking only one value has a degenerate distribution (Dirac delta); a multiple root of a polynomial is a degenerate root
In quantum mechanics, degenerate energy levels arise when eigenvalues of the Hamiltonian have multiplicity > 1, typically indicating an underlying symmetry

Connections: Degeneracy · Geometry · Singularity