Is there a general relationship between number of variables and of constraints?

Question

This comes up most notably in linear algebra and differential equations - usually unique solutions come about when the number of constraints matches the number of variables, or in the case of differential equations, the highest order of differentiation. Is there a more general result or deeper theory that I'm unfamiliar with behind this? I've thought about it and looked around, and it occurred to me that many problems can be linearized in some way. That can give a partial explanation, but it still leaves the general question open.

Answer 1

Consider $n$ equations in $m$ unknowns expressed as $F(x) = b$, where $F$ is a continuously differentiable function from an open subset of $\mathbb R^m$ to $\mathbb R^n$. If $m < n$, then the range of $F$ has Hausdorff dimension at most $m$, and therefore for almost all $b$ there is no solution. If $m > n$, then by the Implicit Function Theorem, in the neighbourhood of any solution where the derivative $D(F)$ has full rank there is a manifold of solutions. If $m=n$, then by the Inverse Function Theorem solutions where $D(F)$ has full rank are isolated.