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

## TL;DR

This paper introduces a novel deflation method for refining and verifying isolated singular zeros in polynomial systems, utilizing linear combinations and derivatives to improve accuracy and efficiency, especially for high multiplicity zeros.

## Contribution

It presents the first deflation approach based on linear combinations of polynomials, with strategies to reduce system size and enhance numerical stability.

## Key findings

- Effective for high multiplicity zeros in large systems
- Works well for non-polynomial systems with isolated singular zeros
- Provides strategies to accelerate the deflation process

## Abstract

In this paper, we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems. Starting from a polynomial system with an isolated singular zero, by computing the derivatives of the input polynomials directly or the linear combinations of the related polynomials, we construct a new system, which can be used to refine or verify the isolated singular zero of the input system. In order to preserve the accuracy in numerical computation as much as possible, new variables are introduced to represent the coefficients of the linear combinations of the related polynomials. To our knowledge, it is the first time that considering the deflation problem of polynomial systems from the perspective of the linear combination. Some acceleration strategies are proposed to reduce the scale of the 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 experiments show that our method is effective and efficient. Especially, it works well for zeros with high multiplicities of large systems. It also works for isolated singular zeros of non-polynomial systems.