I've been working with a collaborator (Arek Goetz) on a dynamics problem involving
piecewise isometries (a map $T$ on a domain $X$ (say a subset of the plane)
such that $X$ is divided into a finite number of polygonal regions and a
separate isometry is applied to each).


In the case where the defining isometries are all rotations through rational
angles, it is often possible to use exact computer arithmetic to define
points, lines, regions etc. in terms of integer linear combinations of roots
of unity.


The following algorithmic question then arises:



> 
> Let $(a\_i)\_{i=1}^n$ be a finite sequence of integers and $(p\_i/K)\_{i=1}^n$
> be a sequence of rationals. Is there a *non-analytic* algorithm to
> decide the sign of $\sum a\_i\sin(2\pi p\_i/K)$?
> 


More generally, given a real algebraic number $\zeta$, is there any nice way to decide the sign of an integer polynomial in $\zeta$ (I can do the quadratic case!)



Algorithm to determine sign of a polynomial
See


<http://cgi.di.uoa.gr/~et/papers/et-computations-alg-numbers2.pdf>


particularly section 3.3


