Find a polynomial in two or three 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 $f(x) \in\mathbb{Z}^+[x]$ of degree $d$.

• Output: The $d$ coefficients of polynomial $f(x)$.

My algorithm: let

$$f(x) = x^{d} + a_{d-1} x^{d-1} + \cdots + a_1 x + a_0$$

Evaluate polynomial $\mathcal{f(x)}$ at $d$ many points using the black box and get a system of linear equations. Now I can solve the system of linear equations to get the desired coefficients.

However, in this case, I need $\mathcal{O(d)}$ many queries to the black box. I want to minimize the number of queries. Is there any way to reduce the number of queries to just two or three?