On isolation of singular zeros of multivariate analytic systems

Abstract

We give a separation bound for an isolated multiple root $x$ of a square multivariate analytic system $f$ satisfying that an operator deduced by adding $Df(x)$ and a projection of $D^{2}f(x)$ in a direction of the kernel of $Df(x)$ is invertible. We prove that the deflation process applied on $f$ and this kind of roots terminates after only one iteration. When $x$ is only given approximately, we give a numerical criterion for isolating a cluster of zeros of $f$ near $x$. We also propose a lower bound of the number of roots in the cluster.

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References in corpus (4)

• Multiple zeros of nonlinear systems
• Verified Error Bounds for Isolated Singular Solutions of Polynomial Systems: Case of Breadth One
• Computing Simple Multiple Zeros of Polynomial Systems
• Numerical approximation of multiple isolated roots of analytic systems

Cited by in corpus (1)

• Punctual Hilbert Schemes and Certified Approximate Singularities