Find a subset in constant many queries

Question

Black box of $f(x)$ means I can evaluate the polynomial $f(x)$ at any point.

• Input: A black box of monic polynomial $f(x) \in\mathbb{S}[x]$ of degree $d$.

• Question : $\mathbb{S} \subseteq \mathbb{Z}$, unbounded from both sides

Here $\mathbb{S} \subseteq \mathbb{Z} $ is an interval unbounded from both sides. For example interval $(-b,\infty]$, where $b$ is any fixed value and this interval is bounded from left side. I am allowed to make only constant many queries to oracle. I want to find such an subset $\mathbb{S}$.

Unbounded interval: An interval is said to be left-bounded or right-bounded if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be bounded if it is both left- and right-bounded; and is said to be unbounded otherwise

Please give a reference to material also if possible. If anything is not clear please ask in the comments I will try clear it.

Motivation : Find a polynomial in two or three queries

Answer 1

If you don't know any bounds on the coefficients beforehand, then you cannot give them any bound without $d$ queries. Suppose you have made queries of $f(x)$ on $x=x_1,\ldots,x_{d-1}$, and claim that the coefficients of $f(x)$ are all at least $-b$. However, a different monic polynomial $f(x)-N(x-x_1)\cdots(x-x_{d-1})$ has the same evaluation on all the $d-1$ queries, and when $N$ is large enough the coefficient of $x^{d-1}$ in this polynomial must be smaller than $-b$, leading to a contradiction.