In linear programming, it is when some basic variable is at one of its bound values (canonically zero). Without any given qualification, a ( basic) solution is degenerate if one or more of its basic values is zero (the canonical lower bound). A solution can be dual degenerate, where one of its nonbasic variables has zero reduced cost. In general, a solution is degenerate if it is not strictly complementary. Geometrically, this corresponds to a degenerate polyhedron. Suppose we have {x: Ax = b} (where A is m by n). This polyhedron is degenerate if there exists an extreme point that is an element of the intersection of more than n hyperplanes. The pyramid is degenerate because four planes meet at a point. ( Example - need graphic browser or postscript viewer/printer.) Algorithmically, it could cause cycling in the simplex method.
A situation in which something is so strange, compared to the usual situation, that its characteristics are completely different from the norm (see degenerate ellipse, degenerate energy state, and degenerate gas). For a degenerate gas, a number which compares the degeneracy pressure of the gas to the total gas pressure. If the degeneracy pressure is zero, the degeneracy is zero, and the gas is said to be non-degenerate. If the degeneracy pressure is equal to the total gas pressure (in other words, so high that the normal gas pressure is inconsequential), the degeneracy is 1.00, and the gas is said to be totally degenerate.
A state of matter characterized by extreme compactness or density (e.g., matter that is a million times or more as dense as water). The pressure of degenerate matter is dominated by quantum mechanical effects and is largely independent of temperature (unlike the pressure of air in a tire, the terrestrial atmosphere, or the gases in normal stars, such as the Sun).