I know the question will feel rather vague, but here it goes anyway.
In some research I've seen, people often translate their problem into polynomial ones. The theory of linear feedback shift registers (LFSR) is one example. Another example, since I'm interested in music theory, is rhythmic canons (see here and here for some examples). It reminds me strongly of how one can turn a group theoretic problem into a linear algebra one by using representation theory.
Thus, my questions are :
What kind of problems can be translated into polynomial ones? (I know Galois theory can be used for telling if a figure can be constructed with only a compass and a ruler; I'm looking for other examples such as above.)
Is there a framework for doing it (an analogue of group representation?), and if so, where can I find references?
Thank you for your help...