TITLE: What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?
QUESTION [11 upvotes]: (From MSE)
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for  

"... is the correspondence between combinatorial problems and problems of the location of the zeroes of polynomials."

Also, a refence [1] is given for this quote. Upon reading through the interview, though, I didn't discover any more about this correspondence, nor did I find a lot by searching for it on the web.
Question 1: What is this correspondence called? 
I am also interested in how these two (which seem to me) disparate problems in mathematics relate to one another, so:
Question 2: How does this correspondence work? Any references? 
Finally:
3 more questions: To what extent has this correspondence been developed any further since Rota's discovery? Are there any other connections between zeroes of polynomials and combinatorics? I know algebraic geometry is concerned with the study of zeroes of polynomials, so is there any connection between (algebraic) combinatorics and algebraic geometry? 
Reference: 
[1]  http://web.archive.org/web/20070811172343/http://www.rota.org/hotair/rotasharp.html

REPLY [3 votes]: I'm not sure why such an old question suddenly bubbled up to the (my) first page after several years, but I won't let that stop me answering it.
As explained above, Rota's critical problem is usually phrased as finding certain (integer) values where the characteristic polynomial of something (often a graph, or representable matroid) is non-zero.
But I think that describing these problems/results as "location of zeros of polynomials" is a bit disingenuous - graph colouring, like the 4-colour theorem, is essentially a combinatorial subject and the chromatic polynomial and its roots played no role in the eventual proof of the 4CT. Saying the "any planar graph is 4-colourable" is logically equivalent to "4 is not a zero of the chromatic polynomial" but I'd argue that it in this case the equivalence is superficial.
Subjects that really use the non-integer real and/or complex roots of polynomials seem to better meet the "spirit" of the question (though anyone else is free to disagree). Here the obvious candidate application is the study of the partition function of the Potts model, where statistical physicists specifically wish to compute the (limiting curves of) complex roots of these partition functions for various families of graphs. These curves provide information about phase transitions in the underlying Potts model.
The chromatic polynomial is a special limiting case of this partition function and the entire partition function is equivalent to the Tutte polynomial, so these connections somehow seem more fundamental.