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P. Kravanja, R. Cools, A. Haegemans
Computing zeros of analytic mappings: a logarithmic residue approach
Let D be a polydisk in Cn and f : D → Cn an analytic mapping that has no zeros on the boundary of D. Then f has only a finite number of zeros in D and these zeros are all isolated. We consider the problem of computing these zeros. A multidimensional generalization of the classical logarithmic residue formula from the theory of functions of one complex variable will be our means of obtaining information about the location of these zeros. This integral formula involves the integral of a differential form, which we will transform into a sum of n Riemann integrals of dimension 2n-1. We will show how the zeros and their multiplicities can be computed from these integrals by solving a generalized eigenvalue problem that has Hankel structure, and n Vandermonde systems. Numerical examples are included.Appeared in BIT vol 38, number 3, pp. 583-596, September 1998.
mailto: P. Kravanja