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An improved PV-reduction method for one-loop integrals with auxiliary vector$R$ has been proposed in \cite{Feng:2021enk,Hu:2021nia}. It has also been shownthat the new method is a self-completed method in \cite{Feng:2022uqp}. Analyticreduction coefficients can be easily produced by recursion relations in thismethod, where the Gram determinant appears in denominators. The singularitycaused by Gram determinant is a well-known fact and it is important to addressthese divergences in a given frame. In this paper, we propose a systematicalalgorithm to deal with this problem in our method. The key idea is that now themaster integral of the highest topology will be decomposed into combinations ofmaster integrals of lower topologies. By demanding the cancellation ofdivergence for obtained general reduction coefficients, we solve decompositioncoefficients as a Taylor series of the Gram determinant. Moreover, the sameidea can be applied to other kinds of divergences.

Reduction with Degenerate Gram matrix for One-loop Integrals