Check for common element in two arrays using FFT
This one is a bit esoteric, but:
$$
\prod_{1\leq i,j \leq n} (x_i-y_j)
$$
Could be decomposed to (as noted by @NotDijkstra)
$$
\prod_{i=1}^{n} p(x_i)
$$
where
$$
p (x) = \prod_{j=1}^{n} (x - y_j)
$$
reduces to a polynomial with roots $y_i$. Coefficients of this polynomial could be computed with FFT, see https://arxiv.org/pdf/1608.01357.pdf. It could be done even faster using divide and conquer strategy - recursively split the formula into two polynomials and multiply them with FFT (see answers).
Polynomial representation in its turn would allow fast exponentiation of x, or even multi-point evaluation
Although, I don't see any advantages over traditional approach in terms of speed or complexity except if it's implemented on specialised ASICs/FPGAs/GPUs.