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 section

Main article: Degenerate conic

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.

Triangle

• A degenerate triangle has collinear vertices and zero area, 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).

Standard torus

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

Sphere

• When the radius of a sphere goes to zero, the resulting degenerate sphere of zero volume is a 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 multiple coinciding root of the characteristic polynomial) is one that has more than one linearly independent eigenvector.