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function semi-algebraic over real closed sub-field?
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Martin Ziegler
science forum beginner


Joined: 12 Dec 2005
Posts: 2

PostPosted: Sat Jul 08, 2006 8:17 am    Post subject: function semi-algebraic over real closed sub-field? Reply with quote

Hi. Getting right to the point:

Let F be a real-closed subfield of the reals R.
Let f : [0,1]^n -> R denote a continuous real
function, semi-algebraic (over R), satisfying that
f(x) lies in F for every rational x (i.e. on Q).

Does this imply f to be semi-algebraic over F?

(It does under the additional hypothesis
that f be a rational function: solve the
interpolation problem for (x,f(x)), x in Q )

If it does not,
how about if f is differentiable or analytic?

Thanks for listening
Martin
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