Constraint counting

In mathematics, constraint counting is counting the number of constraints in order to compare it with the number of variables, parameters, etc. that are free to be determined, the idea being that in most cases the number of independent choices that can be made is the excess of the latter over the former.

For example, in linear algebra if the number of constraints (independent equations) in a system of linear equations equals the number of unknowns then precisely one solution exists; if there are fewer independent equations than unknowns, an infinite number of solutions exist; and if the number of independent equations exceeds the number of unknowns, then no solutions exist.

In the context of partial differential equations, constraint counting is a crude but often useful way of counting the number of free functions needed to specify a solution to a partial differential equation.

Consider a second order partial differential equation in three variables, such as the two-dimensional wave equation

It is often profitable to think of such an equation as a rewrite rule allowing us to rewrite arbitrary partial derivatives of the function ${\displaystyle u(t,x,y)}$ using fewer partials than would be needed for an arbitrary function. For example, if ${\displaystyle u}$ satisfies the wave equation, we can rewrite

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