https://en.wikipedia.org/wiki/Bernstein%E2%80%93Sato_polynomial#Examples

If tij are n^2 variables, then the Bernstein–Sato polynomial of det(tij) is given by
(s+1)(s+2)⋯(s+n)
which follows from
Ω(det(tij)^s)=s(s+1)⋯(s+n−1)det(tij)^(s−1)
where Ω is Cayley's omega process, which in turn follows from the Capelli identity.

https://en.wikipedia.org/wiki/Cayley%27s_%CE%A9_process
In mathematics, Cayley's Ω process, introduced by Arthur Cayley (1846),
is a relatively invariant differential operator on the general linear group,
that is used to construct invariants of a group action.

As a partial differential operator acting on functions of n2 variables xij,
the omega operator is given by the determinant

Ω=
|∂/∂x11⋯∂/∂x1n
⋮　⋱　⋮
∂/∂xn1⋯∂/∂xnn|.
For binary forms f in x1, y1 and g in x2, y2 the Ω operator is
∂^2fg∂x1∂y2−∂2fg∂x2∂y1.

https://en.wikipedia.org/wiki/Capelli%27s_identity
In mathematics, Capelli's identity, named after Alfredo Capelli (1887),
is an analogue of the formula det(AB) = det(A) det(B),
for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra gl^n.
It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω process.