Dear GAP Forum,
On Mon, Jul 28, 2003 at 05:37:51PM +0200, Frank Luebeck wrote:
[...]
>
> > I have an expression, z, say, involving the n-th roots of unity E(n), which
> > evaluates to a real number (i.e. ComplexCojugate(z) = z.) For example
> > z := E(5) + E(5)^-1 + E(3) + E(3)^-1;
> >
> > Is there any way to test within GAP whether z is greater than 0, numerically?
>
> The only way I see is to use (as you say) numerical approximations.
As a remark for theoretically inclined, :)
there are certainly purely symbolic ways for checking such things,
although there is no ready-to use GAP code for this
(nor any readily available code, AFAIK :))

This is a problem of deciding whether a particular semialgebraic set in
R^n (n=2 in this case) is nonempty.
There are procedures known that would answer this in
polynomial time, for fixed n. See e.g.
@book{BPR03,
AUTHOR = {Basu, Saugata and Pollack, Richard and Roy, Marie-Fran{\c{c}}oise},
TITLE = {Algorithms in Real Algebraic Geometry },
YEAR = "2003", ISBN = {3-540-00973-6},
ADDRRESS = {Berlin--Heidelberg--New York},
PUBLISHER = {Springer-Verlag}, }

Regards,
Dmitrii
http://www.thi.informatik.uni-frankfurt.de/~dima/

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