Let $N \in \mathbb{N}$.

Discriminant the cat proposes the following game to you:

- You choose a finite sequence (of length of your choice and kept secret) of natural numbers $\left(a_{0}, \ldots, a_{n}\right) \in \mathbb{N}^{n+1}$
- At each turn of the game, Discriminant gives you an integer $\beta \in \mathbb{N}$ and you respond with the value of $\sum_{k=0}^{n} a_{k} \beta^{k}$.
- The game continues until Discriminant has guessed the sequence of integers or until fewer than $N$ turns have taken place.
- If at the end of the game Discriminant has guessed the sequence of integers, he wins; otherwise, he loses.

If Discriminant wins, you owe him a can of cat pate.

For which values of $N$ would you accept to play this game?
Discriminant can always win in two moves.

Indeed, with $M$ such that all coefficients are strictly less than $10^{M}$, the value at $10^{M}$ allows us to conclude.

To estimate $M$, it suffices to know the evaluation at any point, for example, at 1.