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What is an extremely important problem that requires one to solve large systems of polynomial equations? I've heard of a number of "general areas" where the problems crop up (robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, machine learning, control theory), but never an explicit problem.

Perhaps one that is easy to state without too much domain knowledge?

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    [This book](http://www.cs.amherst.edu/~dac/iva.html) mentions quite a number of applications. Also [this](http://books.google.com/books?id=wGryl4uDCSIC).2011-10-16
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    One of the big blocks in my current line of research boils down to quickly computing vast sets of polynomial equations (to be honest, I'm currently trying to use generating polynomials, so the polynomials are a bit strange). It has to do with calculating the parity of the number of lattice points under a curve very fast - and this relates to lots of things. If we can do it fast enough, we can factor numbers a bit faster than before, for example. (Not so far though)2011-10-16
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    @J.M. the second one even offers Fortran code! But seriously, the first is a good reference.2011-10-16
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    Well, Gröbner bases (which is what the first book pretty much talks about) are standard for either symbolic parameters and/or exact arithmetic. For inexact arithmetic, Gröbner doesn't work that well, and that's why I threw in the second book... if you can, borrow those two from the nearest library.2011-10-16
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    yes, well I am particularly looking for applications of numerical solutions. i.e. PHCpack, et. al2011-10-16
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    Multivariate polynomials model many geometric objects and curves well. So anything that could have anything to do with geometry, surfaces or paths on surfaces.2016-05-04

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