We study the residue current of Bochner-Martinelli type associated with a tuple of holomorphic germs at , whose common zero set equals the origin. Our main results are a geometric description of in terms of the Rees valuations associated with the ideal generated by and a characterization of when the annihilator ideal of equals .
Let 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 ; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results.
We study the integral representation of solutions to the Cauchy problem for a differential equation with constant coefficients. The Cauchy data and the right-hand of the equation are given by entire functions on a complex hyperplane of . The Borel transformation of power series and residue theory are used as the main methods of investigation.