A limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. For example, the point
is a degenerate case of the circle as the radius
approaches 0, and the circle is a degenerate form of an
ellipse as the eccentricity
approaches 0. Another example is the two identical roots
of the second-order polynomial . Since the roots of an th degree polynomial are usually
distinct, roots which coincide are said to be degenerate.
Degenerate cases often require special treatment in numerical and analytical solutions.
For example, a simple search for both roots of the above
equation would find only a single one: 1.

The word degenerate also has several very specific and technical meanings in different branches of mathematics.