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The Briançon-Skoda number of a ring $R$ is defined as the smallest integer k, such that for any ideal $I\subset R$ and $l\ge 1$, the integral closure of $I^{k+l-1}$ is contained in $I^l$. We compute the Briançon-Skoda number of the local ring of any analytic irreducible planar curve in terms of its Puiseux characteristics. It turns out that this number is closely related to the Milnor number.

We give an elementary proof of the Briançon-Skoda theorem. The theorem gives a criterionfor when a function $\phi$ belongs to an ideal $I$ of the ring of germs of analytic functions at $0\in {ℂ}^n$; more precisely, the ideal membership is obtained if a function associated with $\phi$ and $I$ is locally square integrable. If $I$ can be generated by $m$ elements,it follows in particular that $\overline{I^{min(m,n)}}\subset I$, where $\overline{J}$ denotes the integral closure of an ideal $J$.

Let $Z$ be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform Briançon-Skoda theorem for the local ring ${𝒪}_Z$; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results.