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 anon-analyticalgorithm 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!)

analytic algorithm– Suvrit Jul 8 '11 at 18:53