  Musta\c{t}\u{a} defined Bernstein-Sato polynomials in prime characteristic
for principal ideals and proved that the roots of these polynomials are related
to the $F$-jumping numbers of the ideal. This approach was later refined by
Bitoun. Here we generalize these techniques to develop analogous notions for
the case of arbitrary ideals and prove that these have similar connections to
$F$-jumping numbers.
