I thought a lot at this problem: a polynomial $p(x) = a_{n}x^n + a_{n – 1}x^{n – 1} + … + a_1x + a_0$, where $\forall i : a_{i} > 0$ (so that means that $p(x)$ includes all non-negative integer powers of $x$ lesser than $n + 1$) is given. But you don't know any of $a_{i}$ and you don't know $n$ too. You may choose any $x_0$ and ask a question: "what is the value of this polynomial at point $x=x_0$?". And the question is what is the least count of questions you have to ask to guaranteed get to know all coefficients?
I find out that 2 questions is enough: firstly we asking a value of $p(1)$ and getting $s = \sum_{i = 0}^{n} a_i$. Then we ask $p(s + 1) = q$ and simply getting all coefficients converting $q$ into $s + 1$ base system. Coefficients will be represented at digits in this notation. (This works because $s + 1$ is greater than any coefficient for sure).
But I heard that this solution is not optimal and 1 question is actually enough. As I understood, the solution with one question is not useful and can't be used in real life because it is correct but formal. I could not think of this solution (and I don't know it as well) but I know that it somehow uses irrational numbers. I thought that maybe there is such a theorem which claims that at irrational number all $p(x)$ that satisfies our limitations have got different values or something like this, but I did not find it.
So I ask you to help me with that and either explicate the solution or tell the title of such theorem (if it exists). And a big request for you to attach proof or links to useful related materials. Thanks!
Best Answer
Rather looking at only irrational numbers, look at transcendental numbers which are not roots of a polynomial with rational coefficients. Let $P(X)$ and $Q(X)$ be two polynomials of degree $n$ having coefficients in $\mathbb{N}$. Let $P(\pi)=Q(\pi)$. If $P$ and $Q$ are different polynomials then $P-Q$ is a non-constant polynomial in $\mathbb{Q}[X]$ with $\pi$ as a root. A contradiction to the fact that $\pi$ is transcendental. So different polynomials in $\mathbb{Q}[X]$ have different values at transcendental numbers.
But the question is how can we get the coefficients from the value of $P(\pi)$.