Discovery of Hilbert polynomial

Question

Presumably it was Hilbert who discovered Hilbert polynomials - where did they first appear?

The basic theorem is that for a finitely generated graded module $M = \bigoplus_k M_k$ over the ring of polynomials in finitely many variables over a field, the dimension of $M_k$ is given by a polynomial in $k$ when $k$ is sufficiently large. This statement is generally attributed to Hilbert, e.g. by Mumford (Algebraic Geometry I) and Eisenbud (Commutative Algebra with a View Toward Algebraic Geometry).

Atiyah-Macdonald attribute a more general statement, where the polynomial ring is replaced by an arbitrary Noetherian graded ring and the dimension is replaced by length, to Hilbert and Serre. Where did this general statement appear first?

Edit: The statement in Atiyah-Macdonald is more general than the above paragraph - the dimension is replaced by an arbitrary additive function $\lambda$ on the class of all finitely generated $A_0$-modules (where $A$ is the base (graded) ring) and the conclusion is the following: The generating function $\sum_{k=0}^\infty \lambda(M_k)t^k$ is a rational function in $t$ of the form $f(t)/\prod_{i=1}^s (1 - t^{k_i})$ where $s$ is the number of homogeneous generators of $A$ as an $A_0$-algebra, and $k_i$ is the degree of the $i$-th generator, $i = 1, \ldots, s$.

In particular, Atiyah-Macdonald consider all values of $k$, not only sufficiently large ones.

Answer 1

In the 1st chapter of Eisenbud's book that you had mentioned, he discusses four fundamental theorems of Hilbert that appeared in 1890 and 1893 (see p. 26): the basis theorem, the Nullstellensatz, the polynomial nature of Hilbert series, and the syzygy theorem. The history section of the Wikipedia page on the syzygy theorem here says

The syzygy theorem first appeared in Hilbert's seminal paper "Über die Theorie der algebraischen Formen" (1890). The paper is split into five parts: part I proves Hilbert's basis theorem over a field, while part II proves it over the integers. Part III contains the syzygy theorem (Theorem III), which is used in part IV to discuss the Hilbert polynomial. The last part, part V, proves finite generation of certain rings of invariants.

I provided a link to the paper in the quote. Look at part IV, which starts on p. 509.

What was Serre's contribution to Hilbert polynomials? Dieudonné writes in his Topics in Local Algebra (p. 24) that

the geometric significance of the polynomial was discovered by Serre,

the point being that the Hilbert polynomial for all $n$ (not just large $n$) has a sheaf-theoretic interpretation using Euler characteristics. See, for instance, Section 1 of Kedlaya's notes here. That is rather more advanced than what Atiyah-MacDonald write about, as they don't aim to interpret values of the Hilbert polynomial at small $n$, so I'm not sure why they attribute a theorem in their book to "Hilbert-Serre". Maybe their method of proof is based on Serre's work. If someone else can speak to that, please do so in the comments below.