Multiple zeros of nonlinear systems

As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity strucuture, proposes a depth-deflation method for accurate computation of multiple zeros, and introduces the basic algebraic theory of the multiplicity. Furthermore, this paper elaborates and proves some fundamental theorems of the multiplicity, including local finiteness, consistency, perturbation invarance, and depth-deflatability. The proposed algorithms can accurately compute the multiplicity and the multiple zeros using floating point arithmetic even if the nonlinear system is perturbed.

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