Claudia Polini

This is joint work with Ulrich and Vasconcelos. Finding the integral closure of an ideal $I$ is a fundamental problem. The only theoretical approach is through the Rees algebra ${\mathcal R}$ of $I$ -- it requires to compute the normalization $\overline{\mathcal R}$ of ${\mathcal R}$. In the first part of the talk we measure the complexity of this construction by relating it to the Hilbert coefficients of the filtrations of the integral closures of the powers of $I$. We then bound these coefficients through the Brian\c{c}on-Skoda number of the ambient ring $R$. In the second part of the talk we present a strategy to compute the integral closure of $I$, or at least part of it, by a {\it direct} construction, i.e. via an algorithm whose steps take place entirely in the ambient ring. Indeed, by imposing strong residual properties on $I$, we approximate the integral closure of $I$ by a residual intersection.