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 linear combinations. 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.

在本文中，我们开发了一种新的放气技术，用于完善或验证多项式系统的孤立奇异零点。从具有孤立奇异零的多项式系统开始，通过直接计算输入多项式的导数或相关多项式的线性组合，我们构建了一个新系统，该系统可用于细化或验证输入的孤立奇异零系统。为了尽可能地保持数值计算的准确性，引入了新的变量来表示相关多项式的线性组合的系数。据我们所知，这是第一次从线性组合的角度考虑多项式系统的通缩问题。提出了一些加速策略以减小最终系统的规模。我们还对使用的公差进行了进一步的分析，这可以帮助我们更好地了解我们的方法。实验表明，该方法是有效的。特别是，它适用于大型系统的高度多重性的零。它也适用于非多项式系统的孤立奇异零。