Math brain teaser

[Thomas Klemm]

(12-27-2013 05:18 AM)Han Wrote:  A computer generates a random polynomial whose coefficients are non-negative integers.

(12-29-2013 01:46 AM)Les_Koller Wrote:  Say you choose just pi. If our polynomial is p(x) = x we get pi. If our polynomial is p(x) = pi, then we also get pi, thus not identifying the polynomial completely.

That can not be: \(p(x)=\pi\) doesn't have coefficients that are non-negative integers. \(\pi\) isn't an integer.
OTOH: if \(p(\pi)=q(\pi)\) for some distinct polynomials \(p(x)\) and \(q(x)\) then \(p(\pi)-q(\pi)=0\).
But \(\pi\) can't be the root of a polynomial with integer coefficients since it's transcendent.
Thus you can distinguish these kind of polynomials based on the evaluation at \(\pi\). Or any other transcendent number.

Cheers
Thomas