We study applications of discrete valuations to ideals in analytically irreducible domains, in particular applications to zero divisors modulo powers of ideals. We prove a uniform version of Izumi's theorem and calculate several examples illustrating it, such as for rational singularities. The paper contains a new criterion of analytic irreducibility, a new criterion of one-beredness, and a val...

We study a result of Postnikov-Shapiro concerning the Hilbert series of a family of ideals Jφ generated by powers of linear forms in k[x1, . . . , xn]. Using the results of Emsalem-Iarrobino, we formulate this as a problem about fatpoints in P. In the three variable case this is equivalent to studying the dimension of a linear series on a blow up of P. The ideals that arise have the points in v...