Uniform bounds on symbolic powers

We study the question of obtaining uniform bounds for the growth of symbolic powers of ideals in Noetherian rings. First we use a generalized version of Artinʼs strong approximation theorem to obtain uniform bounds for prime ideals in the integral closure of a finite Galois field extension of the fraction field of an excellent Henselian regular local ring. As a corollary, we obtain a similar result for a complete Noetherian local domains. We also obtain linear uniform bounds for the growth of symbolic powers in certain normal subrings of equicharacteristic regular rings under mild conditions.