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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 :

  1. 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.)

  2. 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...

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Many geometry problems can be tarnslated in polynomial problems. Lines, planes, circles, elipses, and many other geometric figures can be described by equations, thus instead of studying the geometric figure you coudl study instead the polynomial(s) describing it.

The application of Galois Theory you mentioned is exactly because of this.

Here is an interesting article I found

http://www.math.ru.nl/~bosma/Students/KenMadlenerBscthesis.pdf

Check the third section.

A google search for Geometry and Grobner basis will lead to many other similar articles, I just picked one at random :D

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    Thanks, I'll have a look at it !2011-10-18