Deflation conjecture and local dimensions of Brent equations

TL;DR

This paper extends a classical deflation process to analyze local solution set dimensions of Brent equations, providing new bounds and insights into the behavior of singular solutions as tensor rank decreases.

Contribution

It generalizes the deflation process to irreducible components and applies it to Brent equations, offering a new tool for studying local dimensions.

Findings

Decreases in nullities observed during deflation.

Deflation process helps determine local dimensions.

Singular solutions become more prevalent as tensor rank decreases.

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.