Computing zeros of analytic mappings: A logarithmic residue approach
Let \(D\) be a polydisk in \(\mathbb{C}^n\) and \(f:\overline D\to \mathbb{C}^n\) a mapping that is analytic in \(\overline D\) and has no zero on the boundary of \(D\). Then \(f\) has only a finite number of zeros in \(D\) and these zeros are all isolated. 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 is obtained by means of information about the location of these zeros. This integral formula involves the integral of a differential form, which is transformed into a sum of \(n\) Riemann integrals of dimension \(2n-1\). The authors 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.
scientific article; zbMATH DE number 1228556
scientific article