TITLE: For a system of PDEs, how many equations are needed generally for the system to have unique solution?
QUESTION [6 upvotes]: For an algebraic system of equations or a system of ordinary differential equations the following rule holds:(right?) 
the total number of unknown variables must be equal to the number of equations (and also the same number of boundary conditions are needed, but that's not my question)
Is it generally correct for a system of PDEs too?
I asked this specific question on physics.SE, where one of the users presented the following example in comments saing that it is not the case for a system of PDEs :
$$\partial_x f=0 \,\,\,\,\,\partial_y f=0  \,\,\,\,\text{(two equations)}$$
$$\to f(x,y)=0 \,\,\,\text{(unique solution)}$$
(I've seen this question (without answer), which asks about number of needed boundary conditions; my question is about the number of independent equations needed)

REPLY [2 votes]: First, it is not true for algebraic systems that you need the same number of unknowns and equations to have a unique solutions. $$(x,y)\in\mathbb{R}^2,\ x^2 + y^2 =0 \Rightarrow x=y=0.$$
Second, it is not obvious how to characterize the dimension of the solution space of a PDE system. See chapter 8 of Seiler's book, "Involution: The Formal Theory of Differential Equations and its Applications".