Say $F(X) \in \mathbb{Z}[X]$ is an even degree polynomial of degree $2n$.

One needs to evaluate $F(X)$ at $O(n)$ points to interpolate and get all the coefficients of $F(X)$.

However say I need **only the coefficient of $X^{n}$ or $X^{2n}$** (the mid coefficient or the largest), do I still have to evaluate at $O(n)$ points?

Will having coefficient of $X^{t}$ same as coefficient of $X^{2n-t}$ help in reducing the number of points from $O(n)$ to detect $X^{n}$ coefficient (mid-coefficient in the symmetric case)?

Is this a well studied problem that has some good references - that is interpolating for only one or few coefficients?

There is one way to do this - evaluating at one large prime and reduction via modulo operations. However, this gives way too much information(that is I can get all the coefficients) and when I evaluate at a large prime, the word size become the order of $O(n\log(nM))$ where M is the largest coefficient size. So in a way we are still using $O(n^{1+\epsilon})$ operations.

I am guessing there should be a way to get only information about the single coefficient I am interested in while getting the operations down to $O(n^{1-\epsilon})$ at the 'cost of not-getting' information about other coefficients.

Say you have a polynomial of odd degree (even coeffcients). Evaluating at $1$ and $-1$ and adding the results or subtracting the results, localizes the information into groups of two coefficients. My question could be can we localize further? Supposing in addition I have $F(x) = A(x)B(x)$ where I know $A(x)$ and $B(x)$, is it possible to represent $A(x)$ and $B(x)$ in a different way so that I can somehow target the mid coefficient of $F(x)$ without getting other coefficients?