Let $j$ be a positive integer, and $(a_i)_{1\leq i\leq j}$ be a sequence such that $$a_i = \prod_{t=1}^j |t^\alpha - i^\alpha|, \quad\, \alpha \geq 2.$$ Find $i$ such that $a_i$ is the minimum.
For $\alpha=2$, I have $a_i = \prod_{t=1}^j |t^2 - i^2|=\frac{(t+i)!(t-i)!}{2i^2}$, where $a_i$ reaches its minimum when $i \approx \sqrt{j}$. But for $\alpha \geq 2$, how to get a general result?
Related question: $\alpha$=2 case, and Prove convergence for division of two sums.