I'm a current undergrad looking to go into Theoretical Computer Science research (yes, I know about TCS Stack Exchange, but I thought the question would better fit here). In a few months, a professor and I are going to work on a research problem involving Multivariate Polynomials. I was hoping to get some of the basic background research digested and learned before we started to be able to hit the ground running with him, to possibly be more productive. In addition, I thought this would help me learn how to read and apply papers, and things that are needed for graduate school and so on.
He has said that he will send me the pre-requisite papers for the specific problem that we are studying as we get closer, but, again, some of the more basic stuff I would like to get out of the way before this project begins.
In searching around, I could find adding and subtracting polynomials practice galore, but I'm trying to look into the more in depth research about the topic. After looking through some papers, I find myself in a prerequisite loop, which I'm sure many can sympathize with. I realize that "Multivariate Polynomial" is not the best subject to search for a paper or self-study material about because it's applications are so wide and diverse. So, that's what I'm really asking : Where would I find resources giving some "background" research into the manipulation of Multivariate Polynomials?
It's hard to recommend anything without having a good sense for how much algebra you already know, but I think Dummit and Foote's Abstract Algebra is a great introductory textbook for any abstract algebra. Depending on your background, you might be able to jump right into Chapters 7-9. Then reference the earlier chapters as necessary for anything you don't understand.