It is possible to choose different gauges in electrodynamics. I am familiar with two of them: Coulomb gauge and Lorenz gauge. Let us stick to the Coulomb gauge. It sets $$\nabla\cdot\vec{A}=0.$$ The wisdom is that with this choice the physical electric and magnetic fields $\vec{E},\vec{B}$ do not change. But there is more to it. It is also important for me to understand why this gauge condition implies that there are superfluous degrees of freedom.
What are these superfluous and non-superfluous degrees of freedom? With which mathematical quantities should we identify them?
First of all, at each spacetime point, we have four numbers $$\phi(\vec{x},t),A_1(\vec{x},t),A_2(\vec{x},t),A_3(\vec{x},t).$$ I understand these four numbers as the four degrees of freedom. Now, Coulomb gauge means that the latter three can be related, without any loss of generality, through the differential equation $$\partial_1A_1+\partial_1A_2+\partial_3A_3=0.$$ Given this, how to understand the rest?