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Topic: Does polynomial P need to be an affine mapping
Replies: 7   Last Post: May 10, 2013 6:45 AM

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Timothy Murphy

Posts: 496
Registered: 12/18/07
Re: Does polynomial P need to be an affine mapping
Posted: May 9, 2013 9:40 AM
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quasi wrote:

> quasi wrote:

>>>Let P be a polynomial with real coefficients. Suppose there
>>>are nonempty intervals I and J such that P maps surjectively
>>>the rationals of I into the rationals of J. Does this imply
>>>P is an affine mapping?

>>
>>Yes.


My argument is similar to yours, but possibly simpler.

As has been established, P(x) has rational coefficients.
Choose a prime p which does not divide any of the coefficients.
Consider x = n/p^e where n is not divisible by p.
Then it is easy to see that P(x) = m/p^{re},
where p does not divide m, and r is the degree of P.
It follows that if x is rational, and r > 1,
P(x) cannot be of the form c/p
(with c not divisible by p).

--
Timothy Murphy
e-mail: gayleard /at/ eircom.net
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College Dublin




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