Well, perhaps you are not stating your question in a very precise manner to help us understand what exactly you mean, but I believe scalar field decomposition is easy. Suppose we have scalar field ##\phi## and ##\vec{A}## is any vector field, then we can write the equation
$$\phi=\nabla\cdot \vec{A}+\rho$$ where ##\rho## is another scalar field. Since in this equation both ##\phi## and ##\nabla\cdot\vec{A}## are scalar their difference ##\rho=\phi-\nabla\cdot \vec{A}## is well defined and is a scalar too. So in this way a scalar field can be decomposed in infinitely many ways, each way for each random vector field A we choose.
On the other hand if we first chose the scalar field ##\rho## randomly and then we seek for a vector field ##\vec{A}## that satisfies the equation
$$\nabla\cdot\vec{A}=\phi-\rho$$ then this vector field A is not uniquely defined since knowing only the divergence of a vector field does not uniquely determine the field (we also must know its curl and know some other conditions as well to uniquely determine it).