# Degeneracy (mathematics)

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In mathematics, a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class. Degeneracy is the condition of being a degenerate case.

A degenerate case thus has special features, which depart from the properties that are generic in the wider class, and which would be lost under an appropriate small perturbation.

## In geometry

### Conic sections

A degenerate conic is a conic section (a second-degree plane curve, the points of which satisfy an equation that is quadratic in one or the other or both variables) that fails to be an irreducible curve.

### Sphere

• A sphere is a degenerate standard torus where the axis of revolution passes through the center of the generating circle, rather than outside it.

### Triangle

• A degenerate triangle has collinear vertices, and thus coincides with a segment covered twice.

### Rectangle

• A segment is a degenerate case of a rectangle, if this has a side of length 0.
• For any non-empty subset $S \subseteq \{1, 2, \ldots, n\}$, there is a bounded, axis-aligned degenerate rectangle
$R \triangleq \left\{\mathbf{x} \in \mathbb{R}^n: x_i = c_i \ (\text{for } i\in S) \text{ and } a_i \leq x_i \leq b_i \ (\text{for } i \notin S)\right\}$

where $\mathbf{x} \triangleq [x_1, x_2, \ldots, x_n]$ and $a_i, b_i, c_i$ are constant (with $a_i \leq b_i$ for all $i$). The number of degenerate sides of $R$ is the number of elements of the subset $S$. Thus, there may be as few as one degenerate "side" or as many as $n$ (in which case $R$ reduces to a singleton point).

## Elsewhere

• A set containing a single point is a degenerate continuum.
• Similarly, roots of a polynomial are said to be degenerate if they coincide, since generically the n roots of an nth degree polynomial are all distinct. This usage carries over to eigenproblems: a degenerate eigenvalue (i.e. a multiply coinciding root of the characteristic polynomial) is one that has more than one linearly independent eigenvector.