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In this paper, a classical deflation process raised by Dayton, Li and Zeng isrealized for the Brent equations, which provides new bounds for localdimensions of the solution set. Originally, this deflation process focuses onisolated solutions. We generalize it to the case of irreducible components anda related conjecture is given. We analysis its realization and apply it to theBrent equations. The decrease of the nullities is easily observed. So thedeflation process can be served as a useful tool for determining the localdimensions. In addition, our result implies that along with the decrease of thetensor rank, the singular solutions will become more and more.

Deflation conjecture and local dimensions of Brent equations