Deflation Conjecture and Local Dimensions of Brent Equations

Abstract

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

2020 Mathematics Subject Classification:

• Primary: 14Q15
• Secondary: 65H10

Keywords:

• deflation
• singularity
• Brent equations
• local dimension
• Jacobian

Acknowledgments

We are very grateful to the editors and referees for their valuable comments and suggestions. In particular, we are grateful to the anonymous referee who kindly reminds us the linear group actions we have overlooked, which improves the Remark 5.1. The discussion with Professor Zhonggang Zeng of Northeastern Illinois University inspired us a lot. We are grateful to him for so many valuable comments and suggestions. We are grateful to Yuan Feng, Xiaodong Ding, Liaoyuan Zeng, Changfeng Ma and Jinyan Fan for their kind supports and suggestions. Many thanks to Professor J. Hauenstein, J. Landsberg, Petr Tichavský and Ke Ye for many helpful discussions when we prepare this paper. Xin Li is grateful to Yaqiong Zhang for her consistent support and understanding.

Notes

1 We are grateful to the anonymous referee for improving this remark.

Additional information

Funding

Xin Li is supported by National Natural Science Foundation of China (Grant No.11801506). Liping Zhang is supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY21A010015) and Grant 11601484, the National Natural Science Foundation, People’s Republic of China.