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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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