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Algebra Level 5

I am thinking of a polynomial, $f(x)$ with integer coefficients. You can ask me the value of the polynomial at any value of $x$ and I will tell you the exact mathematical value. Then you can do this over and over again. What is the minimum number of questions you need to ask to exactly determine what my polynomial is?

Inspiration

1 3 Insufficient information 4 2

1 solution

Agnishom Chattopadhyay
Jun 8, 2015

[Copied from Callvin's comment]

I would argue that the answer is 1. If we are told that $f( \pi ) = 3 \pi^2 + 2 \pi + 1$ , then we know for certain that $f(x) = 3x^2 + 2x + 1$ , because it has integer coefficients and $\pi$ is transcendental. Where's the flaw in my argument?

Unfortunately, I am not exactly convinced. I have a feeling that some trickery is involved

Agnishom Chattopadhyay - 6 years ago

There's no flaw in my argument. The flaw was in the phrasing of the previous problem.

The trickery that is involved is that a transcendental number, by definition, doesn't satisfy any polynomial equation, which allows us to pull this off.

Given any algebraic number $\alpha$ with minimial polynomial $h( \alpha) = 0$ , then the set of polynomials with integer coefficients that will satisfy $f ( \alpha) = 0$ would be of the form $g(x) \times h(x)$ , where $g(x)$ is any polynomial with integer coefficients. Hence, that will make it inappropriate for our purposes (in this problem).

Calvin Lin Staff - 6 years ago

What output do you give when asked the value of the polynomial at pi?It wouldn't be a numerical one as 1.63566357 or something, because it's transcendental.So , for instance, if the polynomial is x^3+2x+1, you would answer just $\pi^3+2\pi+1$ ?I know that this is trivial to the problem, but I was curious.

Bogdan Simeonov - 6 years ago

@Bogdan Simeonov – "I will tell you the exact mathematical value" Yes.

Pi Han Goh - 6 years ago

@Pi Han Goh – Shouldn't be in logic?

Figel Ilham - 6 years ago

@Figel Ilham – I don't think it's suitable there as you need to know the properties of a transcendental number.

Pi Han Goh - 6 years ago

@Pi Han Goh – I've got trapped also by him twice @Calvin Lin

Figel Ilham - 6 years ago

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