Given $ n$ roots, $ x_1, x_2, \dotsc, x_n$ , the corresponding monic polynomial is $ y = (x-x_1)(x-x_2)\dotsm(x-x_n) = \prod_{i}^n (x – x_i)$ . To get the coefficients (i.e. $ y = \sum_{i}^n a_i x^i$ ), a straightforward expansion requires $ O(n^2)$ steps.
Alternatively, if $ x_1, x_2, \dotsc, x_n$ are distinct with each other. The problem is equivalent to polynomial interpolation with $ n$ points: $ (x_1, 0), (x_2, 0), \dotsc, (x_n, 0)$ . The fast polynomial interpolation algorithm can be run in $ O(n\log^2(n))$ time.
I want to ask whether there is any more efficient algorithm better than $ O(n^2)$ ? Even if there are duplicated values among $ \{x_i\}$ ? If it helps, we can assume that the polynomial is over some prime finite field, i.e. $ x_i \in \mathbf{F}_q$ .
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