I've been doing research on the Molien function for a project, and the statements of it often starts with the Hilbert series
$$\Phi(z) = \sum_{d=0}^\infty h_m z^d$$
In different contexts I've heard of this $h_m$ as the number of algebraically independent invariants of degree m, while in others I've heard of it as the dimension of the subspace of degree m polynomials. So clearly there is a relation between algebraic and linear independence, but I cannot find any resource that has a proof or even mention of this relation. Is there a text that contains something about this?
I don’t know of any connection between linear and algebraic independence other than the trivial ones (like linear dependence implies algebraic dependence) but the two description of the Molien series you mentioned don’t coincide (or I misunderstand something).
The reason why I think they won’t coincide is because any maximal algebraically independent set has the same cardinality hence you would get that $h_m$ is bounded from above using the first description while in the second description (generally) it isn’t. For a trivial example let’s look at the trivial action on $\mathbb{C}^2$. In this case every polynomial is invariant, hence the Molien series is: $\sum_{i=0}^\infty (i+1)t^i$, while using the algebraically independent description it is $\sum_{i=1}^\infty 2t^i$.