For my bachelor thesis my goal is to understand the reasoning behind "Hilbert series" and how they connect to the idea of "dimension".


<https://en.wikipedia.org/wiki/Hilbert_series_and_Hilbert_polynomial>


Concepts like modules and graded algebras are rather unfamiliar to me. I have not encountered these before.


I am familiar with:


* Rings, groups and fields, both from linear algebra and a standard course in abstract algebra.
* Gröbner basis, monomials ideals. From a course on computational algebraic geometry.


I am looking for some books/notes that could help me understand the concepts that surround Hilbert Series, preferably an introductory book along with a recommendation for further reading. It seems that I am requesting an "introductory commutative algebra" book. 


If anybody also knows a good place for video lectures, this would be greatly appreciated.



Reference book for understanding Hilbert Series/functions
In addition to Eisenbud's book which has already been pointed out, I would like to suggest also the following references.


* Chapter 11 of **Atiyah-MacDonald's "Introduction to Commutative Algebra"** develops the theory of dimension for graded and local rings and its relation with the Hilbert function. The prerequisites you mention should be enough for reading the book, but for Chapter 11 you may need notions developed in the previous chapters such as Noetherian rings and the length of a module.
* Personally, I like a lot the approach of **Bruns-Herzog's "Cohen-Macaulay Rings"**. Chapter 4 is dedicated to the Hilbert function. Although this book is more advanced and requires in several places a certain familiarity with homological algebra, I think that Chapter 4 is more accessible. So you may try to give a look also there.
* Since you mentioned Gröbner bases, I recommend also **Kreuzer-Robbiano's "Computational Commutative Algebra 2"**. They follow a more computational approach (as the title suggests) and they dedicate a very dense Chapter (the fifth) to Hilbert functions of graded and multigraded modules. The introduction of that chapter is nice and gives some simple examples of the geometric importance of the Hilbert function.


