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In this paper, we develop a new deflation technique for refining or verifyingthe isolated singular zeros of polynomial systems. Starting from a polynomialsystem with an isolated singular zero, by computing the derivatives of theinput polynomials directly or the linear combinations of the relatedpolynomials, we construct a new system, which can be used to refine or verifythe isolated singular zero of the input system. In order to preserve theaccuracy in numerical computation as much as possible, new variables areintroduced to represent the coefficients of the linear combinations of therelated polynomials. To our knowledge, it is the first time that consideringthe deflation problem of polynomial systems from the perspective of the linearcombination. Some acceleration strategies are proposed to reduce the scale ofthe final system. We also give some further analysis of the tolerances we use,which can help us have a better understanding of our method.The experimentsshow that our method is effective and efficient. Especially, it works well forzeros with high multiplicities of large systems. It also works for isolatedsingular zeros of non-polynomial systems.

A New Deflation Method For Verifying the Isolated Singular Zeros of Polynomial Systems