  We prove a uniform bound on the growth of symbolic powers of arbitrary (not
necessarily radical) ideals in arbitrary (not necessarily excellent) regular
rings of all characteristics. This gives a complete answer to a question of
Hochster and Huneke. In equal characteristic, this result was proved by Ein,
Lazarsfeld, and Smith and by Hochster and Huneke. For radical ideals in
excellent regular rings of mixed characteristic, this result was proved by Ma
and Schwede. We also prove a generalization of these results involving products
of various symbolic powers and a uniform bound for regular local rings related
to a conjecture of Eisenbud and Mazur, which are completely new in mixed
characteristic. In equal characteristic, these results are due to Johnson and
to Hochster-Huneke, Takagi-Yoshida, and Johnson, respectively.
